3. Financial Markets and Instruments

Treasury does not create value in isolation: it borrows, lends, invests and hedges through financial markets, using a fairly small set of instruments again and again. This chapter explains how capital markets are organized, the time-value-of-money mathematics behind every price, and the money market, fixed income, FX and derivative instruments a bank treasury uses daily. For engineers, these instruments are the objects your systems capture, value, settle and report, so understanding their cash flows and conventions is the foundation for everything in later chapters.

<a id="31-capital-markets"></a>

3.1 Capital Markets

This subchapter describes where securities are created and traded, who takes part, and how instruments and entities are identified in systems.

Primary vs Secondary Markets

Why the distinction exists. A company or government that needs money for many years has a problem: most investors do not want to lock up their money for that long. Capital markets solve this in two stages. In the first stage, the issuer sells new securities to investors and receives cash. In the second stage, investors can sell those securities to other investors whenever they want, so they are never locked in. Without the second stage, far fewer investors would buy in the first, and issuers would pay much more to borrow.

What they are.

• Primary market: where new securities are created and sold by the issuer to investors. The issuer receives the proceeds. Examples: an initial public offering (IPO), a government bond auction, a bank issuing a new senior bond.
• Secondary market: where existing securities are traded between investors. The issuer receives nothing; ownership and cash simply change hands between buyer and seller.
• Liquidity link: a deep secondary market lowers the yield investors demand in the primary market, because they know they can exit. Issuers therefore care about secondary liquidity even though they do not trade there.
• Price discovery link: secondary market prices tell issuers what yield or valuation they can expect for their next primary issue.

Analogy. A car manufacturer selling a new car to a dealer is the primary market: the manufacturer gets paid. The used-car market is the secondary market: cars change owners many times, and the manufacturer earns nothing from those resales, but a strong resale value makes new cars easier to sell.

Example: one bond, two markets.

  PRIMARY (once, at issuance)                SECONDARY (any time until maturity)
  ---------------------------                -----------------------------------
  Issuer: Bank X                             Investor A ---- sells EUR 5m ----> Investor C
  New 5y bond, EUR 500m, 3.25% coupon                   <--- pays cash ------
        |                                    Investor B ---- sells EUR 2m ----> Investor D
        | sells bonds via syndicate                     <--- pays cash ------
        v
  Investors A, B, ... pay EUR 500m           Bank X: no cash in or out,
  Bank X receives EUR 500m (less fees)       only pays coupons to whoever holds the bond

Aspect	Primary market	Secondary market
Seller	Issuer	Existing investor
Cash goes to	Issuer	Selling investor
Frequency per security	Once (or a few taps)	Continuously, until maturity
Price set by	Auction or bookbuilding	Supply and demand on venues or OTC
Main purpose	Raising capital	Liquidity and price discovery

Key takeaways

• The primary market raises new money for issuers; the secondary market transfers existing securities between investors.
• Issuers receive cash only in the primary market.
• Secondary market liquidity and prices directly affect the cost and success of primary issuance.
• A bank treasury participates in both: as an issuer of its own debt and as an investor and trader of other issuers' securities.

🧪 Practice

1. Beginner: Classify each event as primary or secondary: a government sells new 10-year bonds at auction; a pension fund sells shares it has held for five years; a company sells new shares in a rights issue; a bank buys a covered bond from another bank.
2. Intermediate: A bank's existing 5-year bonds trade at a yield of 4.10% in the secondary market. Explain why it is unlikely to issue a new 5-year bond at 3.80%, and what "new issue premium" means in this context.
3. Interview: "Our treasury system records a bond purchase. What data tells you whether it was bought in the primary or secondary market, and why would downstream systems care?" Hint: think about counterparty (issuer or syndicate vs dealer), settlement date and price conventions at issuance, fees, accounting classification, and regulatory reporting.

Equity and Debt Capital Markets (ECM, DCM)

Why they exist. Organizations raise long-term money in two fundamentally different ways: by selling ownership (equity) or by borrowing (debt). Each has different investors, legal terms, risks and processes, so investment banks organize their origination teams into Equity Capital Markets (ECM) and Debt Capital Markets (DCM). A bank treasury interacts mainly with DCM, because banks fund themselves largely through debt issuance; equity raising is rarer and is usually driven by capital planning (Chapter 7).

What they are.

• Equity Capital Markets (ECM): issuance of shares and equity-linked instruments.
  • IPO: first sale of shares to the public and listing on an exchange.
  • Follow-on offering: a listed company sells additional shares.
  • Rights issue: new shares offered first to existing shareholders, in proportion to their holdings.
  • Convertible bonds: debt that can be converted into shares; sits between ECM and DCM.
• Debt Capital Markets (DCM): issuance of bonds and notes by governments, supranationals, financial institutions and corporates.
  • Syndicated issuance: a group of banks (the syndicate) markets the bond, collects orders and allocates bonds to investors.
  • Auctions: most government debt is sold by auction to primary dealers.
  • Private placements: bonds sold directly to a small number of investors without public marketing.

Aspect	ECM	DCM
Instrument	Shares, equity-linked	Bonds, notes, commercial paper
Investor's claim	Ownership, residual profits, votes	Fixed contractual payments
Issuer's obligation	None to pay dividends	Must pay coupons and principal
Investor risk	Highest; last in insolvency	Lower; ranks ahead of equity
Pricing quoted as	Share price, discount to last close	Spread over a benchmark (e.g. mid-swaps + 85bp)
Typical frequency for a bank	Rare	Regular, often under an issuance programme

Example: a syndicated bank bond in DCM.

  Day -5    Mandate: Bank X hires 5 lead managers; announces investor calls
  Day  0    08:30  Initial price thoughts (IPTs):   mid-swaps + 110bp area
            10:00  Order book EUR 2.1bn
            11:00  Guidance revised:                mid-swaps + 95bp area
            12:30  Book EUR 3.0bn, size set at EUR 750m
            14:00  Pricing: mid-swaps + 90bp; 5y mid-swaps 2.35% -> re-offer yield 3.25%
            15:30  Allocations sent to investors
  Day +5    Settlement (T+5): investors pay, bonds delivered via ICSDs

The book was four times covered, which let the issuer tighten the spread by 20bp from IPTs, saving EUR 1.5m per year on EUR 750m.

Key takeaways

• ECM raises ownership capital; DCM raises borrowed capital with contractual repayment.
• Debt ranks ahead of equity in insolvency and is cheaper for the issuer, but must be repaid.
• DCM deals are usually priced as a spread over a benchmark such as mid-swaps or a government bond.
• Strong demand (book coverage) allows the issuer to tighten pricing during bookbuilding.

🧪 Practice

1. Beginner: For each instrument, state whether it belongs to ECM or DCM: IPO, covered bond, rights issue, senior preferred bond, commercial paper.
2. Intermediate: In the example, compute the annual coupon saving if the issuer had priced at mid-swaps + 110bp instead of + 90bp on EUR 750m, and explain why an issuer might still leave a small new issue premium on the table.
3. Interview: "Why do banks issue far more debt than equity, even though equity absorbs losses better?" Hint: consider cost of capital, tax deductibility of interest, dilution of existing shareholders, regulatory capital requirements, and loss-absorbing debt instruments such as AT1 and Tier 2.

Market Participants (Issuers, Investors, Broker-Dealers, Infrastructure)

Why it matters. Every trade involves several institutions with different roles, incentives and systems. When a trade fails to settle or a price looks wrong, an engineer needs to know who each party is and what they are responsible for. Participant roles also determine regulatory obligations, such as who reports a trade and who holds the securities.

What they are.

• Issuers: entities that create securities to raise money: governments, supranationals and agencies (SSAs), banks, corporates, special purpose vehicles.
• Investors (buy-side): entities that invest money: pension funds, insurers, asset managers, hedge funds, central banks, bank treasuries, retail investors.
• Broker-dealers (sell-side): firms that intermediate trading.
  • Broker (agent): executes orders on behalf of clients, without taking the position; earns commission.
  • Dealer (principal): buys and sells from its own inventory, quoting bid and ask prices; earns the spread and bears market risk.
  • Market maker / primary dealer: a dealer with an obligation to quote continuously, or to bid in government auctions.
• Market infrastructure: neutral utilities that let trades happen safely.
  • Trading venues: exchanges, MTFs, OTFs.
  • CCPs: novate and guarantee trades (Chapter 2.3).
  • CSDs and ICSDs: hold securities and settle transfers.
  • Custodians: hold securities for investors (Chapter 2.4).
  • Trade repositories and data vendors: store reports and distribute prices.
• Regulators and central banks: supervise participants and markets, and in the case of central banks, conduct monetary policy through the markets.

Analogy. A property market has sellers building new houses (issuers), buyers (investors), estate agents who match them for a fee (brokers), developers who buy and resell from stock (dealers), and the land registry and notaries that make the transfer legally safe (infrastructure).

Example: who touches a single bond trade.

  Asset manager (buy-side)            Dealer bank (sell-side)
  "buy EUR 10m Bund 2034"   ---RFQ--> quotes 98.42, trades on MTF (venue)
          |                                   |
          | allocation to fund                | trade report (MiFIR)
          v                                   v
  Custodian of the fund  <-- settlement --> Dealer's settlement agent
          \                                  /
           \--- both instruct the CSD ------/
                (Clearstream Banking Frankfurt, DvP in T2S)

Participant	Main objective	Bears market risk?	Typical systems
Issuer	Cheap, reliable funding	On its liabilities	Debt issuance, treasury management
Buy-side investor	Return for clients	Yes	Order management, portfolio management
Broker	Commission	No	Order routing, execution
Dealer / market maker	Bid-ask spread, flow	Yes, on inventory	Pricing engines, risk, trading
CCP / CSD / custodian	Safe, efficient processing	No (CCP only on default)	Clearing, settlement, asset servicing

Key takeaways

• Issuers supply securities, investors supply money, broker-dealers intermediate, infrastructure makes trades safe and final.
• Brokers act as agents and carry no position; dealers trade as principal and carry market risk.
• A bank treasury plays several roles at once: issuer of its own debt, investor in its liquidity buffer, and client of dealers and infrastructure.
• Knowing each party's role is essential to diagnose breaks, fails and reporting obligations.

🧪 Practice

1. Beginner: Classify each as issuer, buy-side, sell-side or infrastructure: Euroclear, an insurance company, the Italian Treasury, a primary dealer, LCH.
2. Intermediate: Explain why a dealer quoting bid and ask prices needs real-time risk systems, while a pure agency broker does not.
3. Interview: "Model the counterparty entities for a trade capture system that must support a bank acting both as principal and as agent." Hint: separate legal entity, role on the trade (principal, agent, broker, custodian, clearing member), settlement parties and identifiers such as LEI, BIC and internal IDs.

Exchanges, MTFs and OTC Markets

Why it matters. The same bond or share can be traded in very different ways: on a public order book, by requesting quotes from a few dealers, or by phone directly with one bank. The trading mechanism determines transparency, cost, counterparty risk and regulatory reporting. Treasury trades most of its instruments OTC (money market deposits, FX, swaps), but uses venues for futures and increasingly for bonds and FX.

What they are.

• Exchange (regulated market): a multilateral system operated under strict rules, with listing standards and usually a central limit order book (CLOB) where buy and sell orders from many participants are matched by price-time priority. Examples: Deutsche Börse Xetra, NYSE, CME, Eurex.
• Multilateral Trading Facility (MTF): an EU (MiFID II) category of multilateral venue run by an investment firm or market operator, with non-discretionary rules but lighter listing requirements. Many bond and FX platforms are MTFs, often using request for quote (RFQ) rather than an order book. In the US, the equivalent concept is an alternative trading system (ATS).
• Organised Trading Facility (OTF): an EU venue for non-equity instruments where the operator has discretion over how orders are matched.
• Systematic Internaliser (SI): not a venue but an investment firm that trades frequently against its own book with clients, with transparency obligations.
• Over-the-counter (OTC): trades negotiated bilaterally between two parties outside a venue, by phone, chat or single-dealer platforms. Terms can be customized; counterparty risk is bilateral unless the trade is centrally cleared.

Feature	Exchange	MTF / OTF	OTC
Counterparties	Many-to-many	Many-to-many	One-to-one
Matching	CLOB, non-discretionary	CLOB or RFQ (OTF discretionary)	Bilateral negotiation
Contracts	Standardized	Mostly standardized	Fully customizable
Pre-trade transparency	High	Medium	Low
Counterparty risk	CCP	CCP or bilateral	Bilateral or cleared
Typical treasury use	Futures	Bonds, FX, swaps on platforms	Deposits, FX swaps, bespoke swaps

Example: a central limit order book.

         BID (buyers)                    ASK (sellers)
   Qty     Price                    Price     Qty
   500    99.48  <- best bid        99.52     300   <- best ask
   1200   99.47                     99.53     800
   2000   99.45                     99.55    1500

   Best bid 99.48 / best ask 99.52  -> spread 0.04
   New market order: BUY 600
     fills 300 @ 99.52, then 300 @ 99.53  (average 99.525)

A market order consumes liquidity at progressively worse prices; a limit order adds liquidity to the book and waits.

Key takeaways

• Exchanges and MTFs are multilateral and rule-based; OTC trades are bilateral and negotiated.
• Venues offer standardization and transparency; OTC offers flexibility at the cost of transparency and bilateral risk.
• RFQ protocols dominate bond and FX platforms, while CLOBs dominate equities and futures.
• Venue type drives clearing, reporting (MiFIR, EMIR) and best execution obligations.

🧪 Practice

1. Beginner: In the order book example, what is the average price for a market order to SELL 1,000?
2. Intermediate: Explain why most interest rate swaps traded by a bank treasury are OTC but still centrally cleared, and what changes for the treasury when they are.
3. Interview: "Design the core data structures for a matching engine with price-time priority." Hint: think about one sorted structure per side keyed by price, FIFO queues per price level, constant-time access to best bid and ask, and handling of partial fills and cancellations.

Securities Identifiers (ISIN, CUSIP, LEI)

Why they exist. Systems across banks, custodians, CSDs and regulators must refer to the same bond or the same legal entity without ambiguity. Names are not unique ("Bank X 3.25% 2031" could match several issues), and every institution's internal codes differ. Standard identifiers make it possible to match trades, reconcile positions and report to regulators automatically.

What they are.

Identifier	Identifies	Standard / issuer	Format	Example
ISIN	A security (issue)	ISO 6166, national numbering agencies	12 chars: 2-letter country + 9-char national ID + check digit	US0378331005
CUSIP	A security (US, Canada)	CUSIP Global Services	9 chars: 6 issuer + 2 issue + check digit	037833100
SEDOL	A security (UK and others)	London Stock Exchange	7 chars
FIGI	A security at listing level	Open standard (OMG)	12 chars
LEI	A legal entity	ISO 17442, GLEIF network	20 chars: 4 LOU prefix + 14 entity + 2 check digits
BIC	A financial institution for messaging	ISO 9362, SWIFT	8 or 11 chars
MIC	A trading venue	ISO 10383	4 chars	XETR

• The US ISIN embeds the CUSIP: US + 037833100 + check digit 5.
• An ISIN identifies an issue, not a venue: the same share listed on several exchanges keeps one ISIN, so systems combine ISIN with MIC to identify a specific listing.
• The LEI identifies who an entity is (legal name, address, parent); it is mandatory for counterparties in EU regulatory reporting under EMIR, MiFIR and SFTR.

Example: validating an ISIN check digit (Luhn algorithm).

  ISIN body: U S 0 3 7 8 3 3 1 0 0       check digit to verify: 5
  1. Letters to numbers (A=10 ... Z=35):  U=30, S=28
     digit string: 3 0 2 8 0 3 7 8 3 3 1 0 0
  2. From the right, double every second digit, starting with the rightmost:
     digits:  3  0  2  8  0  3  7  8  3  3  1  0  0
     doubled: 6     4     0     14    6     2     0
     (14 -> 1 + 4 = 5)
  3. Sum: doubled 6+4+0+5+6+2+0 = 23, others 0+8+3+8+3+0 = 22, total 45
  4. Check digit = (10 - 45 mod 10) mod 10 = 5   -> valid

Key takeaways

• ISIN is the global security identifier; CUSIP is the US national code embedded in US ISINs.
• An ISIN identifies the issue, not the trading venue; combine it with MIC for a listing.
• LEI identifies legal entities and is required for regulatory reporting; BIC identifies institutions in payment and settlement messaging.
• Always validate check digits at the system boundary to catch keying and mapping errors early.

🧪 Practice

1. Beginner: Name the identifier you would use for each: the counterparty in an EMIR report, a bond in a settlement instruction, the venue on which a trade was executed.
2. Intermediate: Write pseudocode for an ISIN validator, including format checks (length, country prefix, allowed characters) and the Luhn check digit.
3. Interview: "Our security master has two records for the same bond, one keyed by ISIN and one by an internal code. How would you design security master identity?" Hint: consider a stable internal surrogate key, a cross-reference table of external identifiers with validity dates, identifier changes after corporate actions, and source precedence rules between vendors.

<a id="32-time-value-of-money"></a>

3.2 Time Value of Money

This subchapter builds the mathematics used to price every instrument in the rest of the chapter: how money grows with interest, and how future cash flows are converted into today's value.

Simple vs Compound Interest

Why it matters. A euro today is worth more than a euro next year, because today's euro can be lent out and earn interest. To compare cash flows at different dates, we need precise rules for how interest accrues. Markets use two: simple interest for short-term instruments and compound interest for longer ones. Using the wrong convention, even by a few days, causes real P&L and reconciliation breaks.

What they are.

• Simple interest: interest is computed only on the original principal, in proportion to time.

  Interest = P × r × t, so FV = P × (1 + r × t)

• Compound interest: interest earned in each period is added to the principal and itself earns interest in later periods.

  FV = P × (1 + r)^n for n annual periods.

• Year fraction (t) and day count conventions: in practice, t is not "months / 12" but a day count fraction defined by convention:

Convention	Year fraction	Typical use
ACT/360	Actual days / 360	EUR and USD money markets, SOFR, €STR, EURIBOR, floating swap legs
ACT/365 (Fixed)	Actual days / 365	GBP money markets, SONIA
30/360	Days with each month counted as 30 / 360	EUR fixed swap legs, many corporate bonds
ACT/ACT (ICMA)	Actual days / actual days in coupon period	Government bonds (Bunds, OATs, BTPs, US Treasuries)

Analogy. Simple interest is a savings jar where the bank adds the same amount every year based on your original deposit. Compound interest is a snowball rolling downhill: the bigger it gets, the more snow it picks up.

Example: EUR 10,000 at 5% for 3 years.

  Year   Simple: interest   Balance         Compound: interest   Balance
  1           500.00       10,500.00              500.00        10,500.00
  2           500.00       11,000.00              525.00        11,025.00
  3           500.00       11,500.00              551.25        11,576.25
                                           difference after 3 years: 76.25

Example: same rate, different day count. A EUR 1,000,000 deposit at 4% for 92 days earns 1,000,000 × 0.04 × 92/360 = 10,222.22 under ACT/360, but 1,000,000 × 0.04 × 92/365 = 10,082.19 under ACT/365. The 140.03 difference is a common source of booking breaks between systems.

Key takeaways

• Simple interest grows linearly; compound interest grows exponentially because interest earns interest.
• Money market instruments (under one year) almost always use simple interest; bonds and longer-term instruments use compounding.
• The year fraction is defined by a day count convention, which is part of the instrument's terms and must be stored with it.
• ACT/360 pays more interest than ACT/365 for the same quoted rate.

🧪 Practice

1. Beginner: Compute the interest on a USD 5,000,000 deposit at 5.10% for 30 days under ACT/360.
2. Intermediate: An investor can choose 4.00% simple interest for 2 years or 3.95% compounded annually for 2 years on EUR 1,000,000. Which pays more, and by how much?
3. Interview: "Two systems compute interest on the same deposit and disagree by a few euros. What do you check?" Hint: day count convention, inclusion or exclusion of start and end dates, business day adjustment of the end date, rounding rules (per period or at the end), and rate precision.

Present and Future Value

Why it matters. Almost every pricing question in finance reduces to "what is a payment at a future date worth today?" A bond is a set of future payments; a loan is a set of future repayments; a swap is two sets of future payments. Present value (PV) is the common language that makes them comparable and lets treasury decide which is cheaper or more valuable.

What they are.

• Future value (FV): what an amount invested today grows to at a future date.

  FV = PV × (1 + r)^n

• Present value (PV): what a future amount is worth today, obtained by reversing the growth. This is called discounting.

  PV = FV / (1 + r)^n

• Net present value (NPV): the sum of the present values of all cash flows of an investment, inflows positive and outflows negative. An NPV greater than zero means the investment earns more than the discount rate.

• Discount rate: the return available on an alternative investment of similar risk; it represents the opportunity cost of money.

Analogy. PV is like converting currencies across time: "1,000,000 euros in two years" and "924,556 euros today" are the same amount expressed in different "time currencies", with the interest rate as the exchange rate.

Example: EUR 1,000,000 received in 2 years, discount rate 4% (annual compounding).

   today                          year 1                     year 2
     |------------------------------|--------------------------|
   PV = ?                                                   1,000,000
     <--------- divide by 1.04 ------<------ divide by 1.04 ----

   PV = 1,000,000 / 1.04^2 = 1,000,000 / 1.0816 = 924,556.21

   Check (forward):  924,556.21 x 1.04 = 961,538.46   (year 1)
                     961,538.46 x 1.04 = 1,000,000.00 (year 2)

Example: NPV decision. A project costs 900,000 today and pays 1,000,000 in two years. At a 4% discount rate, NPV = -900,000 + 924,556.21 = +24,556.21, so it is worth doing. At 6%, PV = 1,000,000 / 1.1236 = 889,996.44 and NPV = -10,003.56, so it is not.

Key takeaways

• FV compounds an amount forward; PV discounts it back to today.
• PV falls as the discount rate rises or the payment date moves further away.
• NPV adds up the present values of all cash flows; a positive NPV beats the discount rate.
• Pricing almost any instrument means projecting its cash flows and summing their present values.

🧪 Practice

1. Beginner: What is the PV of USD 500,000 received in 3 years at 5% annual compounding?
2. Intermediate: An investment pays 300,000 at the end of each of the next three years and costs 820,000 today. Compute its NPV at 4% and at 6%.
3. Interview: "Why does a rise in interest rates reduce the market value of a bank's fixed-rate loan book?" Hint: the contractual cash flows do not change, but each is discounted at a higher rate; connect this to the economic value of equity in Chapter 6.

Discount Factors

Why they matter. Using one interest rate for all maturities is a simplification. In reality, money borrowed for one year costs a different rate from money borrowed for ten years. A discount factor captures, for one specific date, exactly what one unit of currency paid on that date is worth today. Pricing systems store curves of discount factors because they make valuation a simple multiply-and-sum.

What they are.

• Discount factor DF(t): the present value today of 1 unit paid at time t. It is always between 0 and 1 for positive rates, and decreases as t increases.

• Zero rate (spot rate) z(t): the single rate that applies from today to t with no intermediate payments. With annual compounding:

  DF(t) = 1 / (1 + z(t))^t

• Discount curve: the set of discount factors for all dates, built (bootstrapped) from market instruments such as deposits, futures and swaps. Values between known points are interpolated.

• Valuation with discount factors:

  PV = Σ CF_i × DF(t_i)

Analogy. A discount curve is a price list for future money: "1 euro in one year costs 0.9709 today, 1 euro in three years costs 0.8890 today". To value any set of payments, you buy each one at its listed price and add up the bill.

Example: pricing a 3-year 5% annual bond (nominal 1,000,000) on a zero curve.

  t   Zero rate   DF = 1/(1+z)^t     Cash flow      PV = CF x DF
  1     3.00%       0.970874          50,000          48,543.69
  2     3.50%       0.933511          50,000          46,675.54
  3     4.00%       0.888996       1,050,000         933,446.18
                                                  ------------
                                       Price       1,028,665.40   (102.87% of par)

Each cash flow is discounted with its own rate, rather than one rate for all.

Key takeaways

• A discount factor is the price today of one unit paid at a given future date.
• Discount factors and zero rates are two views of the same curve; the conversion depends on compounding convention.
• Any fixed set of cash flows is valued as the sum of cash flow times discount factor.
• Curves are bootstrapped from market quotes and interpolated between maturities; interpolation choices affect valuations.

🧪 Practice

1. Beginner: Convert a 2-year zero rate of 3.20% (annual compounding) into a discount factor.
2. Intermediate: Using the curve in the example, price a 3-year bond paying a 2% annual coupon on 1,000,000 nominal. Is it above or below par, and why?
3. Interview: "Design the data model and API for a curve service used by several pricing engines." Hint: consider curve identity (currency, index, purpose), as-of date and snapshot versioning, pillar instruments and their quotes, interpolation method, and returning discount factors for arbitrary dates.

Annuities and Amortization

Why they matter. Mortgages, consumer loans and leases are usually repaid in equal instalments that mix interest and principal. A large part of a retail bank's balance sheet consists of such loans, and their changing principal profile drives interest income, liquidity forecasts and interest rate risk. Understanding the annuity formula lets you generate repayment schedules and reason about how balances run off.

What they are.

• Annuity: a series of equal payments at regular intervals. Its present value is:

  PV = PMT × [1 - (1 + r)^-n] / r

• Annuity payment (instalment): solving for the payment that repays a loan P over n periods at rate r per period:

  PMT = P × r / [1 - (1 + r)^-n]

• Amortization: the gradual repayment of principal. In each period:

  • interest = outstanding balance × r
  • principal repaid = PMT − interest
  • new balance = old balance − principal repaid

• Repayment profiles:

Profile	Instalment	Principal repaid	Typical product
Annuity (French)	Constant	Increases over time	Mortgages, consumer loans
Linear (constant principal)	Decreases over time	Constant	Some corporate and mortgage loans
Bullet	Interest only, principal at end	All at maturity	Bonds, interbank loans

Example: EUR 200,000 loan, 5% annual, 5 annual instalments.

  PMT = 200,000 x 0.05 / (1 - 1.05^-5) = 46,194.96

  Year   Opening bal.   Instalment    Interest    Principal    Closing bal.
  1       200,000.00    46,194.96    10,000.00    36,194.96    163,805.04
  2       163,805.04    46,194.96     8,190.25    38,004.71    125,800.33
  3       125,800.33    46,194.96     6,290.02    39,904.94     85,895.39
  4        85,895.39    46,194.96     4,294.77    41,900.19     43,995.20
  5        43,995.20    46,194.96     2,199.76    43,995.20          0.00
                       ----------    ---------   ----------
                       230,974.80    30,974.80   200,000.00

The interest share falls and the principal share rises every year, while the instalment stays constant.

Key takeaways

• An annuity is a series of equal payments; its PV formula gives the instalment that repays a loan.
• In an annuity loan, early instalments are mostly interest, later ones mostly principal.
• Linear and bullet profiles repay principal differently and produce different liquidity and interest rate risk profiles.
• Rounding of instalments must be handled explicitly, usually by adjusting the final instalment.

🧪 Practice

1. Beginner: Compute the monthly instalment of a EUR 150,000 mortgage at 3.6% nominal annual rate (0.3% per month) over 20 years.
2. Intermediate: Build the first three rows of a linear repayment schedule for the example loan (EUR 40,000 principal per year) and compare total interest paid with the annuity version.
3. Interview: "Implement a loan schedule generator. What edge cases must it handle?" Hint: rounding and the final instalment, day count and irregular first periods, business day adjustments, rate resets on variable loans, partial prepayments that change either the instalment or the term, and payment holidays.

Effective vs Nominal Rates

Why it matters. "6% per year" means different things depending on how often interest is compounded. A deposit paying 6% compounded monthly earns more than one paying 6% compounded annually. Regulators require banks to disclose an effective rate so consumers can compare products, and treasury systems must convert between quoting conventions before comparing or combining rates.

What they are.

• Nominal (stated) rate r_nom: the annual rate quoted together with a compounding frequency m (times per year). The rate per period is r_nom / m.

• Effective annual rate (EAR): the rate that, compounded once a year, gives the same result.

  EAR = (1 + r_nom / m)^m - 1

• Converting nominal rates between frequencies: go through the effective rate.

  r_m2 = m2 × [(1 + EAR)^(1/m2) - 1]

• Consumer terminology: in the EU, consumer credit must disclose the annual percentage rate of charge (APRC), which includes fees; savings products often quote an annual equivalent rate (AER). In the US, APR is a nominal rate and APY is the effective rate.

Example: 6% nominal at different compounding frequencies.

  Compounding       m       Rate per period      EAR
  Annual            1        6.0000%             6.0000%
  Semi-annual       2        3.0000%             6.0900%
  Quarterly         4        1.5000%             6.1364%
  Monthly          12        0.5000%             6.1678%
  Daily           365        0.0164%             6.1831%
  Continuous        -          -                 6.1837%

On EUR 10,000,000 for one year, monthly compounding pays EUR 16,778 more than annual compounding at the same nominal rate.

Key takeaways

• A nominal rate is meaningless without its compounding frequency.
• The effective annual rate is the true annual growth and is the right basis for comparison.
• More frequent compounding increases the effective rate, with diminishing increments.
• Convert between conventions via the effective rate, never by simple scaling.

🧪 Practice

1. Beginner: Compute the EAR of 4.8% compounded quarterly.
2. Intermediate: A swap pays 3.00% annually. What semi-annual nominal rate is equivalent?
3. Interview: "Product teams want to display 'interest rate' on a savings app. What would you ask before implementing it?" Hint: nominal vs effective, compounding and payment frequency, day count, fees, regulatory disclosure requirements, and consistency with how interest is actually accrued in the ledger.

Continuous Compounding

Why it matters. If compounding becomes more and more frequent, the effective rate converges to a limit. This limit, continuous compounding, has convenient mathematical properties: growth over consecutive periods simply adds up in the exponent, and it makes derivative pricing formulas clean. Many pricing libraries store curves as continuously compounded zero rates, so engineers working with them must convert correctly.

What it is.

• Continuous compounding: the limit of (1 + r/m)^m as m tends to infinity, which equals e^r.

  FV = PV × e^(r × t), DF(t) = e^(-r × t)

• Conversion to and from annual compounding:

  r_cont = ln(1 + r_annual), r_annual = e^(r_cont) - 1

• Additivity: rates over consecutive periods add: e^(r1 × t1) × e^(r2 × t2) = e^(r1 × t1 + r2 × t2). This makes forward rates and averaging easy to compute.

Property	Discrete compounding	Continuous compounding
Growth factor	(1 + r/m)^(m × t)	e^(r × t)
Combining periods	Multiply factors	Add exponents
Used for	Quoting market instruments	Internal curve storage, option models

Example.

  5% continuously compounded:
    EAR = e^0.05 - 1 = 5.1271%
    EUR 1,000,000 for 3 years: 1,000,000 x e^(0.15) = 1,161,834.24

  5% annual compounding as a continuous rate:
    r_cont = ln(1.05) = 4.8790%
    DF(3) = e^(-0.048790 x 3) = 1 / 1.05^3 = 0.863838

Key takeaways

• Continuous compounding is the limit of ever more frequent compounding; growth is e^(r × t).
• Continuous rates convert to annual rates with e^r - 1 and back with ln(1 + r).
• Exponents add over consecutive periods, which simplifies curve and forward calculations.
• Libraries often store continuous zero rates internally; always confirm the convention before using a rate from another system.

🧪 Practice

1. Beginner: Convert a 3.5% annually compounded rate to a continuously compounded rate.
2. Intermediate: The 1-year continuous zero rate is 3.0% and the 2-year rate is 3.4%. Compute the continuously compounded forward rate from year 1 to year 2.
3. Interview: "A risk system receives a curve from the front office library and valuations are off by a few basis points. What do you suspect?" Hint: compounding convention of the zero rates, day count basis of the time axis, interpolation method, and whether the curve values are discount factors, zero rates or forward rates.

<a id="33-money-market-instruments"></a>

3.3 Money Market Instruments

This subchapter covers instruments with maturities of up to one year, which treasury uses every day to manage short-term liquidity, and the overnight benchmark rates that underpin them.

Interbank Deposits

Why they exist. At the end of every day, some banks have more central bank reserves than they need and others have less, because customer payments flow unevenly between them. Rather than leaving surplus cash idle or borrowing expensively from the central bank, banks lend to each other for short periods. The interbank deposit market is the oldest and simplest tool for redistributing liquidity across the banking system.

What they are.

• Interbank deposit (loan): an unsecured loan of central bank money from one bank to another, at a fixed rate, for a fixed term. The lender takes full credit risk on the borrower.
• Tenors:

Tenor	Starts	Ends
Overnight (O/N)	Today	Next business day
Tom-next (T/N)	Tomorrow	Business day after tomorrow
Spot-next (S/N)	Spot date (T+2 for EUR)	Next business day after spot
1W, 1M, 3M, 6M, 12M	Spot date	Spot date + term, business day adjusted

• Interest: simple interest, paid at maturity, with ACT/360 for EUR and USD and ACT/365 for GBP.
• Quotes: dealers quote a two-way price as bid / offer (the rate at which they borrow / lend). In Europe the convention is often the reverse order (offer / bid), so systems must know which convention a source uses.
• Post-crisis reality: since 2008, unsecured interbank volumes have fallen sharply, because of credit concerns, abundant central bank reserves and regulatory liquidity rules (LCR, NSFR) that penalize short-term interbank funding. Much activity moved to secured repo.

Analogy. Neighbours lending each other a cup of sugar until tomorrow: it relies purely on trust that the neighbour will pay back, with no collateral.

Example: Bank A lends EUR 50,000,000 to Bank B for 3 months at 3.20%.

  Trade date   Mon 13 Oct      Value date (spot)  Wed 15 Oct
  Maturity     Thu 15 Jan      Days               92
  Interest     50,000,000 x 3.20% x 92/360 = 408,888.89

  15 Oct:  Bank A  --- EUR 50,000,000 ------------>  Bank B   (via T2)
  15 Jan:  Bank A  <-- EUR 50,408,888.89 ----------  Bank B   (principal + interest)

Key takeaways

• Interbank deposits are unsecured, fixed-rate, simple-interest loans between banks.
• Tenors are defined relative to today or spot, with business day adjustments.
• The lender bears the full credit risk of the borrower, which is why volumes shrank after 2008.
• Always know the quote order convention (bid/offer vs offer/bid) and the day count of each currency.

🧪 Practice

1. Beginner: Compute the interest on a GBP 20,000,000 overnight deposit at 4.50% over a weekend (Friday to Monday) using ACT/365.
2. Intermediate: A dealer quotes 1M EUR deposits with a bid of 3.05% and an offer of 3.15%. Which rate does a bank receive if it lends to the dealer, and which does it pay if it borrows? Explain the dealer's profit.
3. Interview: "Implement a function that computes the maturity date for a money market tenor." Hint: spot lag per currency, holiday calendars of the relevant currencies, business day conventions (modified following), and the end-of-month rule.

Repo and Reverse Repo

Why it exists. Lending unsecured is risky, and many institutions holding bonds need cash only temporarily. A repurchase agreement (repo) solves both problems: the borrower hands over securities as collateral, so if it defaults the lender keeps the securities. Repo is now the largest segment of the money market, the main transmission channel for central bank policy, and the way treasury turns its bond portfolio into cash.

What it is.

• Repo: a sale of securities with an agreement to buy them back at a fixed price on a later date. Economically, it is a collateralized loan: the cash borrower (repo) delivers securities, the cash lender (reverse repo) delivers cash.
• Reverse repo: the same trade seen from the cash lender's side.
• Repo rate: the interest rate implicit in the difference between the sale price and the repurchase price (ACT/360 for EUR and USD).
• Haircut: a discount applied to the collateral's market value to protect the cash lender against a fall in collateral price. Cash lent = market value × (1 − haircut).
• Title transfer: legal ownership of the securities passes to the cash lender, who may reuse them; coupons paid during the term are passed back to the original owner (manufactured payments).
• Types:

Type	Description
General collateral (GC)	Any security from a basket is acceptable; rate close to market overnight rates
Special	A specific security is in demand; the cash lender accepts a lower rate to obtain it
Bilateral	Parties manage collateral and settlement themselves
Tri-party	An agent (e.g. Euroclear, Clearstream, BNY) selects, values and substitutes collateral
Cleared	Novated to a CCP such as LCH, Eurex or FICC

Analogy. Pawning a watch: you hand over the watch, receive cash worth somewhat less than the watch, and buy it back later for a bit more than you received.

Example: Bank A borrows cash against a government bond for 7 days at 3.00%.

  Collateral market value (dirty)   10,200,000.00
  Haircut 2%: cash lent             10,200,000 x 0.98 = 9,996,000.00
  Repo interest                     9,996,000 x 3.00% x 7/360 = 5,831.00
  Repurchase price                  10,001,831.00

  Start leg (DvP):  Bank A --- bond (MV 10.2m) --->  Bank B
                    Bank A <--- cash 9,996,000 ----  Bank B
  End leg (DvP):    Bank A <--- bond --------------  Bank B
                    Bank A --- cash 10,001,831 --->  Bank B

Key takeaways

• A repo is legally a sale and repurchase but economically a secured loan.
• The haircut protects the cash lender; the repo rate is the interest on the cash.
• GC repo rates track overnight benchmarks; specials trade at lower rates because the security itself is in demand.
• Both legs settle DvP, so repo depends on the securities settlement infrastructure of Chapter 2.3.

🧪 Practice

1. Beginner: Compute the cash lent and the repurchase price for collateral worth EUR 25,000,000, a 5% haircut, a 3.10% repo rate and a 30-day term.
2. Intermediate: During the repo in the example, the bond pays a coupon of EUR 150,000. Who receives it from the issuer, and what must happen next?
3. Interview: "Why do central banks set or target overnight rates through repo rather than unsecured lending, and what does SOFR measure?" Hint: consider credit risk, volume and robustness of the underlying transactions, and the link between Treasury repo and the Fed's policy implementation.

Commercial Paper and Certificates of Deposit

Why they exist. Large corporates and banks need short-term funding beyond what interbank or bank loans offer. Instead of borrowing from a single lender, they can issue short-term debt securities directly to money market funds, corporates and other investors. Because the securities are tradable, investors can sell them before maturity, which makes them more attractive than deposits.

What they are.

• Commercial paper (CP): unsecured short-term notes issued by corporates, banks and public entities.

  • US CP (USCP): maturities up to 270 days (to avoid SEC registration), usually issued at a discount to face value.
  • Euro CP (ECP): issued in the international market, maturities typically up to 364 days, in several currencies.
  • Programmes: issuers set up a CP programme once (e.g. USD 5bn) and draw on it continuously through dealers.

• Certificate of deposit (CD): a time deposit with a bank, documented as a negotiable security. Usually pays interest at maturity on a face amount (interest-bearing), though some are issued at a discount.

• Discount instruments: the investor pays less than face value and receives face value at maturity; the difference is the return.

  Price = Face × (1 − d × days/360) where d is the discount rate.

  Money market yield = (Face / Price − 1) × 360/days

Feature	Commercial paper	Certificate of deposit
Issuer	Corporates, banks, public entities	Banks
Typical form	Discount	Interest-bearing
Security	Unsecured (except ABCP)	Unsecured bank liability
Main investors	Money market funds, corporates	Money market funds, corporates, other banks
Treasury use	Funding (issuer) or investment	Funding (bank issuer) or investment

Example: USCP, face USD 10,000,000, 90 days, discount rate 5.00%.

  Price  = 10,000,000 x (1 - 0.05 x 90/360) = 9,875,000.00
  Return = 10,000,000 - 9,875,000            =   125,000.00
  Money market yield = (10,000,000 / 9,875,000 - 1) x 360/90 = 5.0633%

The discount rate understates the investor's true return, because the discount is computed on face value while the investor only paid 9,875,000.

Key takeaways

• CP and CDs are tradable short-term debt securities; CP comes from corporates and banks, CDs from banks.
• Discount instruments are bought below face and repaid at face.
• Discount rate and money market yield differ; always compare yields on the same basis.
• A bank treasury issues CDs and ECP for funding and invests in other issuers' CP and CDs.

🧪 Practice

1. Beginner: Compute the price of a 60-day CP with face EUR 5,000,000 at a 3.40% discount rate.
2. Intermediate: An investor can buy a 90-day CD paying 5.05% interest at maturity, or the CP in the example. Which offers the higher return?
3. Interview: "During the 2008 crisis and in March 2020, CP markets froze. Why does that matter to a bank treasury, even if it does not issue CP?" Hint: consider backstop liquidity lines that banks provide to CP issuers, money market fund outflows, and the resulting draws on bank liquidity.

Treasury Bills

Why they exist. Governments receive tax revenue unevenly and need to smooth their cash flows. Treasury bills (T-bills) are short-term government debt that fills this gap. Because they are backed by the government and are highly liquid, T-bills are one of the safest assets in the market and a core component of bank liquidity buffers.

What they are.

• Treasury bill: a short-term government security, usually issued at a discount, with no coupon. Typical maturities are 4 to 52 weeks.
• Issued by auction: governments auction bills regularly (weekly in the US), with primary dealers bidding.
• Examples: US Treasury bills, UK Treasury bills, German Bubills, French BTFs, Italian BOTs, Spanish Letras.
• Quoting (US):
  • Discount rate: d = (Face − Price) / Face × 360/days.
  • Investment rate (bond-equivalent yield, BEY): (Face − Price) / Price × 365/days, comparable with coupon bond yields.
• Regulatory role: T-bills of high-rated sovereigns are Level 1 HQLA under the LCR (Chapter 5.4) and are eligible collateral in central bank operations.

Example: US 26-week bill (182 days), face USD 1,000, discount rate 4.50%.

  Price           = 1,000 x (1 - 0.045 x 182/360)        = 977.25
  Discount        = 1,000 - 977.25                        =  22.75
  Investment rate = 22.75 / 977.25 x 365/182              = 4.6687%

  A bank investing USD 100,000,000 face:
    pays 97,725,000.00 at settlement, receives 100,000,000.00 at maturity

Measure	Formula base	Value
Discount rate	Face value, 360 days	4.5000%
Money market yield	Price, 360 days	4.6048%
Investment rate (BEY)	Price, 365 days	4.6687%

Key takeaways

• T-bills are short-term, zero-coupon government securities issued at a discount by auction.
• They are among the safest and most liquid assets and qualify as Level 1 HQLA.
• Discount rate, money market yield and bond-equivalent yield describe the same price differently.
• Treasury systems must store the quote basis together with the rate.

🧪 Practice

1. Beginner: A 91-day bill is quoted at a 4.20% discount rate. Compute its price per USD 100 face.
2. Intermediate: Verify the money market yield of 4.6048% in the table, and explain why it is higher than the discount rate.
3. Interview: "A treasury holds USD 2bn of T-bills in its liquidity buffer. Describe the ways it could turn them into cash on a stress day." Hint: outright sale, repo with market counterparties, central bank facilities, settlement timing and haircuts in each case.

Overnight Index Rates (SOFR, €STR, SONIA)

Why they exist. Financial contracts worth hundreds of trillions reference a benchmark interest rate. After the LIBOR scandal, regulators required benchmarks based on large volumes of real transactions rather than bank estimates. Overnight rates satisfy that requirement, because overnight markets are by far the most active. These risk-free rates (RFRs) now underpin loans, bonds, swaps and the discounting of collateralized derivatives.

What they are.

Rate	Currency	Administrator	Underlying market	Secured?	Day count
SOFR	USD	Federal Reserve Bank of New York	Overnight Treasury repo	Secured	ACT/360
€STR	EUR	European Central Bank	Overnight unsecured deposits from banks' wholesale counterparties	Unsecured	ACT/360
SONIA	GBP	Bank of England	Overnight unsecured sterling deposits	Unsecured	ACT/365
SARON	CHF	SIX	Overnight CHF repo	Secured	ACT/360
TONA	JPY	Bank of Japan	Overnight uncollateralized call market	Unsecured	ACT/365

• Each rate is a volume-weighted median (or trimmed mean) of actual transactions, published on the next business day.

• RFRs contain almost no bank credit risk or term premium, unlike IBORs.

• Using overnight rates for a term period: a loan or swap for 3 months needs one rate for the whole period. The standard method is compounding in arrears: the daily rates over the period are compounded, so the rate is only known at the end.

  Compounded rate = [Π (1 + r_i × d_i / 360) − 1] × 360 / D

  where r_i is the rate for business day i, d_i the calendar days it applies (3 over a weekend), and D the total days.

• Conventions to know the payment amount in advance: lookback (shift the observation window a few days earlier), observation shift, lockout, and payment delay.

Example: one week of SOFR on USD 100,000,000.

  Day     SOFR     Days applied   Factor (1 + r x d/360)
  Mon     4.30%         1         1.000119444
  Tue     4.32%         1         1.000120000
  Wed     4.31%         1         1.000119722
  Thu     4.33%         1         1.000120278
  Fri     4.35%         3         1.000362500   (covers Fri, Sat, Sun)
                       ---        -----------
                        7         1.000842204   (product)

  Compounded rate = (1.000842204 - 1) x 360/7 = 4.3313%
  Simple average (day-weighted)               = 4.3300%
  Interest = 100,000,000 x 0.000842204        = 84,220.45

Key takeaways

• SOFR, €STR and SONIA are overnight rates computed from large volumes of real transactions.
• SOFR is secured (repo); €STR and SONIA are unsecured.
• Term interest on RFRs is usually compounded in arrears, so it is known only near the end of the period.
• Weekends and holidays apply the previous business day's rate for multiple calendar days.

🧪 Practice

1. Beginner: Using the example, explain why the Friday rate has a weight of 3 days.
2. Intermediate: Recompute the example with a 2-day lookback: which daily rates would you use, and why might a borrower prefer it?
3. Interview: "Design a component that computes compounded RFR interest for thousands of loans daily." Hint: store the published rate series once with publication dates and corrections, precompute a cumulative compounded index, derive any period's rate as a ratio of index values, and handle holiday calendars and late rate republication.

IBOR Transition

Why it happened. For decades, most floating-rate contracts referenced LIBOR or EURIBOR: rates submitted by panel banks estimating what they would pay to borrow unsecured from other banks. After 2008, actual unsecured interbank borrowing dried up, so submissions became largely expert judgement, and in 2012 investigations revealed that some banks had manipulated them. A benchmark used for trillions of contracts rested on very few real transactions, so regulators decided to replace LIBOR with transaction-based RFRs.

What it involved.

• Cessation: most LIBOR settings (GBP, EUR, CHF, JPY, and some USD tenors) stopped at the end of 2021; the main USD LIBOR tenors stopped on 30 June 2023. Temporary "synthetic" GBP and USD LIBOR settings for legacy contracts were wound down by the end of September 2024.
• EONIA was replaced by €STR (EONIA was redefined as €STR + 8.5bp from October 2019 and ceased in January 2022).
• EURIBOR survived after a reform of its methodology (a hybrid using transactions where available), so euro markets now use both EURIBOR and €STR.
• Fallbacks: legacy contracts needed rules for what happens when LIBOR stops. ISDA's fallback protocol replaced LIBOR with the compounded RFR plus a fixed credit adjustment spread (CAS), equal to the 5-year historical median difference between LIBOR and the RFR (for example 26.161bp for 3M USD LIBOR).
• Term rates: forward-looking term rates derived from RFR derivatives (e.g. CME Term SOFR) exist for some loan markets, but derivatives mainly use compounded overnight rates.

Aspect	IBOR (e.g. LIBOR)	RFR (e.g. SOFR, €STR)
Basis	Panel bank submissions / estimates	Actual overnight transactions
Tenor	Term (1M, 3M, 6M...)	Overnight
Known when	At the start of the period	At the end (compounded in arrears)
Bank credit risk	Included	Essentially none
Behaviour in stress	Rises with bank credit spreads	Often falls (flight to quality)

Example: fallback on a legacy loan referencing 3M USD LIBOR + 2.00%.

  Before cessation:   coupon = 3M USD LIBOR (set in advance)          + 2.00%
  After fallback:     coupon = compounded SOFR (in arrears)
                               + 0.26161% (credit adjustment spread)   + 2.00%

  System changes needed:
    - new index and daily rate feed
    - compounding engine with lookback / observation shift
    - interest amount known only a few days before payment
    - new spread field (CAS) separate from the commercial margin

Key takeaways

• LIBOR was replaced because it relied on judgement rather than transactions and was manipulated.
• USD, GBP, CHF and JPY markets moved to RFRs; euro markets use both reformed EURIBOR and €STR.
• Fallbacks add a fixed credit adjustment spread to compounded RFRs to keep economics broadly unchanged.
• The transition was a major technology change: from rates known in advance to rates known in arrears.

🧪 Practice

1. Beginner: List three differences between 3M LIBOR and compounded SOFR as a reference rate.
2. Intermediate: Explain why a bank that lent at LIBOR + margin and funded itself at a rate close to its own credit spread was exposed when contracts moved to SOFR.
3. Interview: "You must migrate 50,000 floating-rate contracts from LIBOR to SOFR fallbacks. How do you plan it?" Hint: contract inventory and classification by fallback language, data model changes, parallel run and reconciliation of amounts, client communications, and handling contracts that reset mid-period.

<a id="34-fixed-income-securities"></a>

3.4 Fixed Income Securities

This subchapter explains how bonds are structured, priced and quoted, how their interest rate risk is measured, and the bond types most relevant to bank treasury.

Bond Structure and Cash Flows

Why it matters. A bond is a contract that turns a loan into a tradable security. Because it is standardized and documented in a prospectus, thousands of investors can hold pieces of the same loan and trade them. Every system that holds bonds, from pricing to settlement to accounting, needs a precise model of the bond's terms, because those terms fully determine the cash flows.

What it is.

• Bond: a debt security in which the issuer promises to pay periodic interest (coupons) and to repay the principal at maturity.
• Key terms (static data):

Term	Meaning	Example
Issuer	Who owes the money	Bank X
Face / nominal value	Principal repaid at maturity; coupons computed on it	EUR 1,000 per bond
Coupon rate	Annual interest rate on face value	4.00%
Coupon frequency	Payments per year	Annual (EUR), semi-annual (USD, GBP)
Day count	How accrued interest is computed	ACT/ACT ICMA, 30/360
Issue date, maturity date	Life of the bond	15 Mar 2026 - 15 Mar 2031
Issue price	Price paid at issuance, % of face	99.62
Business day convention	How payment dates are adjusted	Following, unadjusted accrual
Seniority	Rank in insolvency	Senior preferred, Tier 2, covered
Embedded options	Issuer call, investor put, sinking fund	Callable at par from 2030

• Coupon types: fixed, floating (Chapter 3.4 Floating Rate Notes), zero-coupon (no coupons, issued at a discount), step-up, inflation-linked.
• Redemption types: bullet (all principal at maturity, the norm), amortizing, callable, perpetual.
• Prices are quoted as a percentage of face value: a price of 98.50 means 985 per 1,000 nominal.

Analogy. A bond is like a pre-printed sheet of dated cheques from the issuer: one for each coupon, plus a large one for the principal. Buying the bond means buying the whole sheet of remaining cheques.

Example: 5-year 4% annual bond, EUR 1,000,000 nominal.

  Investor pays ~1,000,000 at issue
       |
       v
  2026 ------ 2027 ------ 2028 ------ 2029 ------ 2030 ------ 2031
              40,000      40,000      40,000      40,000      40,000
                                                            +1,000,000
                                                            ----------
                                                             1,040,000

Key takeaways

• A bond's static data (coupon, frequency, day count, dates, conventions, options) fully defines its cash flows.
• Prices are quoted in percent of face value.
• Most bonds are bullet bonds with fixed coupons, but floating, zero-coupon, amortizing and callable structures are common.
• Errors in static data cause wrong valuations, wrong accruals and failed settlements, so security master quality is critical.

🧪 Practice

1. Beginner: List the cash flows (dates and amounts) of a 3-year bond with a 2.5% semi-annual coupon and EUR 2,000,000 nominal, issued on 1 June 2026.
2. Intermediate: What is a zero-coupon bond's cash flow profile, and why is its price always below par when rates are positive?
3. Interview: "Design a cash flow generator for bonds. What inputs does it need and what can go wrong?" Hint: schedule generation (forward or backward from maturity), short or long first coupons, end-of-month rules, holiday calendars, adjusted vs unadjusted accrual dates, and embedded options.

Yield to Maturity

Why it matters. Bonds have different coupons, maturities and prices, so comparing them by price is meaningless: is a 2% bond at 95 cheaper than a 5% bond at 104? Investors need one number that summarizes the return of buying the bond at today's price and holding it to maturity. That number is the yield to maturity, and it is how bonds are actually quoted and compared in the market.

What it is.

• Yield to maturity (YTM): the single discount rate that makes the present value of all remaining cash flows equal to the bond's (dirty) price.

  Price = Σ C / (1 + y)^t + Face / (1 + y)^n

• It is the bond's internal rate of return, assuming the investor holds to maturity, all payments are made, and coupons are reinvested at the same yield.

• Inverse relationship: when yields rise, prices fall, and vice versa.

  • Coupon > yield: price above par (premium).
  • Coupon = yield: price at par.
  • Coupon < yield: price below par (discount).

• No closed-form solution: YTM is found numerically, using Newton-Raphson or bisection.

• Related quotes: current yield (coupon / price, ignores the pull to par), yield to call, spread to government bond, spread to mid-swaps (asset swap spread or I-spread).

Analogy. YTM is the "average speed" of a journey with several legs at different speeds: one figure that captures the whole trip, even though no single leg ran at exactly that speed.

Example: 3-year 5% annual bond.

  Yield    Price (% of face)
  3.00%      105.66        premium
  4.00%      102.78        premium
  5.00%      100.00        par
  6.00%       97.33        discount

  Price
   106 |*
   104 |   *
   102 |      *
   100 |         *
    98 |            *
       +---+---+---+---+---> Yield
         3%  4%  5%  6%

If the bond trades at 102.78, its YTM is 4.00%: an investor paying 102.78 earns 5% coupons but loses 2.78 of price as the bond pulls to par, for a net return of about 4% a year.

Key takeaways

• YTM is the single discount rate that equates the PV of cash flows to the market price.
• Price and yield move in opposite directions.
• A bond trades above par when its coupon exceeds market yield, and below par when it is lower.
• YTM is solved numerically; implementations need robust iteration and convergence checks.

🧪 Practice

1. Beginner: Without calculation, state whether a 2% coupon bond trades above or below par when similar bonds yield 3.5%.
2. Intermediate: Compute the price of a 2-year 3% annual bond at a 4% yield, then explain why YTM differs from the bond's current yield.
3. Interview: "Implement a YTM solver. How do you make it robust?" Hint: Newton-Raphson using the analytical derivative (related to duration), a bracketing fallback such as bisection, sensible initial guesses, tolerance and iteration limits, and handling of negative yields.

Clean vs Dirty Price and Accrued Interest

Why it matters. A bond's coupon is paid in full to whoever holds the bond on the record date, even if they bought it the day before. To be fair, a buyer must compensate the seller for the interest earned while the seller held the bond since the last coupon. If quoted prices included this accrued interest, they would jump down on every coupon date for no economic reason. Markets therefore quote the clean price and settle on the dirty price.

What they are.

• Accrued interest (AI): the coupon earned since the last coupon date up to the settlement date.

  AI = Coupon × (days since last coupon / days in coupon period) (ACT/ACT ICMA)

• Clean price: the quoted price, excluding accrued interest.

• Dirty (full, invoice) price: what the buyer actually pays: Dirty = Clean + AI.

• Settlement amount: Nominal × Dirty price / 100.

• Ex-dividend periods: some markets (e.g. UK gilts) have a period before the coupon date in which a buyer does not receive the next coupon; accrued interest becomes negative.

• Accrued interest is computed to the settlement date, not the trade date.

Analogy. Buying a flat in the middle of a month when the seller has already paid the whole month's service charge: the price is negotiated without it (clean), and the buyer then reimburses the seller for the remaining days (accrued interest).

Example: buy EUR 10,000,000 of a 3% annual government bond at a clean price of 98.50.

  Last coupon 15 Mar 2026         Next coupon 15 Mar 2027    (period 365 days)
  Settlement  10 Aug 2026         Days accrued = 148

  15 Mar 2026                 10 Aug 2026                          15 Mar 2027
      |--------- 148 days ---------|----------- 217 days ---------------|
      seller earned coupon         buyer earns coupon

  AI per 100     = 3.00 x 148/365          = 1.216438
  Dirty price    = 98.50 + 1.216438        = 99.716438
  Settlement amt = 10,000,000 x 99.716438% = 9,971,643.84
     of which principal 9,850,000.00 and accrued interest 121,643.84

Key takeaways

• Markets quote clean prices and settle at dirty prices: dirty = clean + accrued interest.
• Accrued interest runs from the last coupon date to the settlement date, using the bond's day count.
• Without the split, quoted prices would drop mechanically on each coupon date.
• Accounting and P&L systems record accrued interest separately from the clean price movement.

🧪 Practice

1. Beginner: Compute the accrued interest per 100 for the example bond if settlement is on 15 September 2026.
2. Intermediate: A corporate bond pays 4% annually on a 30/360 basis, last coupon 31 January 2026. Compute accrued interest per 100 for settlement on 15 April 2026.
3. Interview: "Our settlement amount differs from the counterparty's by a few hundred euros on a bond trade. What do you check?" Hint: settlement date used for accrual, day count convention, coupon dates in the security master, ex-coupon rules, rounding precision of accrued interest, and whether the price was clean or dirty.

Duration and Convexity

Why they matter. A bank holding bonds or fixed-rate loans loses value when rates rise. Treasury needs to answer, quickly and for the whole portfolio: by how much? Revaluing every position for every scenario is expensive, and comparing risk across bonds requires a common measure. Duration gives a first-order answer; convexity refines it. Together they are the basic tools of interest rate risk management (Chapters 6 and 8).

What they are.

• Macaulay duration: the PV-weighted average time to receive the bond's cash flows, in years.

  D_mac = Σ t × PV(CF_t) / Price

• Modified duration: the percentage change in price for a 1 percentage point change in yield (first-order approximation).

  D_mod = D_mac / (1 + y) and ΔP / P ≈ −D_mod × Δy

• DV01 / PV01: the money change in value for a 1 basis point (0.01%) change in yield. DV01 ≈ D_mod × Price × 0.0001.

• Convexity: the curvature of the price-yield relationship. Including it improves the estimate for larger moves:

  ΔP / P ≈ −D_mod × Δy + ½ × Convexity × (Δy)²

• Drivers: duration is higher for longer maturity, lower coupon and lower yield. A zero-coupon bond's Macaulay duration equals its maturity.

Analogy. Duration is the slope of a hill at your current position: it tells you how much you go down if you step forward. Convexity is how the slope changes: on a curved hill, a straight-line estimate overstates the drop downhill and understates the rise uphill.

Example: 5-year 4% annual bond priced at par (yield 4%).

  t    CF      DF @4%      PV(CF)      t x PV
  1     4    0.961538     3.8462      3.8462
  2     4    0.924556     3.6982      7.3964
  3     4    0.888996     3.5560     10.6680
  4     4    0.854804     3.4192     13.6769
  5   104    0.821927    85.4804    427.4021
              ------------------   ---------
              Price      100.0000    462.9895

  Macaulay duration = 462.9895 / 100   = 4.6299 years
  Modified duration = 4.6299 / 1.04    = 4.4518
  Convexity                            = 25.01
  DV01 per EUR 1,000,000 nominal       = 445.18 EUR

  Yield shock    Actual change    Duration only    Duration + convexity
  +100bp           -4.3295%         -4.4518%           -4.3268%
  -100bp           +4.5797%         +4.4518%           +4.5769%

Duration alone overstates the loss and understates the gain; adding convexity brings both estimates close to the actual change.

Key takeaways

• Macaulay duration is the weighted average time to cash flows; modified duration is price sensitivity to yield.
• DV01 expresses sensitivity in money per basis point and can be summed across positions.
• Convexity corrects duration for larger rate moves; positive convexity benefits the holder.
• Longer maturity and lower coupons mean higher duration and more interest rate risk.

🧪 Practice

1. Beginner: A portfolio has a modified duration of 6 and a value of EUR 200 million. Estimate the loss for a 50bp parallel rise in yields.
2. Intermediate: Compute the Macaulay duration of a 3-year zero-coupon bond and of a 3-year 8% annual bond at a 5% yield. Explain the difference.
3. Interview: "Why do risk systems compute DV01 by bumping the curve and revaluing, rather than using the modified duration formula?" Hint: non-flat curves, instruments with optionality or floating legs, key-rate (bucketed) sensitivities, and consistency with the valuation model.

Floating Rate Notes

Why they exist. A fixed-rate bond exposes its holder to interest rate risk: if rates rise, its value falls. Many investors, especially banks funding themselves at floating rates, prefer assets whose income moves with market rates. Floating rate notes (FRNs) pay a coupon that resets periodically to a reference rate, so their price stays close to par and their interest rate risk is small.

What they are.

• FRN coupon: Reference rate + quoted margin, reset each period.
  • Term reference rate fixed in advance: e.g. 3M EURIBOR fixed two business days before the period starts.
  • Overnight reference rate compounded in arrears: e.g. compounded SOFR or €STR, known just before payment (Chapter 3.3).
• Quoted margin: a fixed spread set at issuance, reflecting the issuer's credit risk.
• Floors and caps: some FRNs specify a minimum (often 0%) or maximum for the reference rate or the coupon.
• Price behaviour: on each reset date, an FRN with no change in the issuer's credit quality is worth close to par, because its next coupon matches the market rate. Its price moves mainly with credit spreads.
• Discount margin: the spread over the reference rate that, used for discounting, gives the market price. It is the FRN's equivalent of a yield.
• Interest rate duration: approximately the time to the next reset, so very low; credit spread duration remains close to the bond's maturity.

Analogy. A fixed-rate bond is like a long-term fixed-price contract, which loses value if market prices go up. An FRN is like a contract repriced every three months to the market: its value hardly changes with market prices, only with how much you trust the other party.

Example: Bank X 3-year FRN, 3M EURIBOR + 0.50%, EUR 20,000,000 held.

  Period            Fixing date    3M EURIBOR   Coupon rate   Days   Coupon (ACT/360)
  15 Jan - 15 Apr     13 Jan          2.05%        2.55%        90      127,500.00
  15 Apr - 15 Jul     13 Apr          2.20%        2.70%        91      136,500.00
  15 Jul - 15 Oct     13 Jul          2.30%        2.80%        92      143,111.11

Each coupon is fixed two business days before its period starts, so the holder always knows the next payment amount in advance.

Key takeaways

• An FRN pays a reference rate plus a fixed margin, reset periodically.
• Its interest rate sensitivity is low (roughly to the next reset), but its credit spread sensitivity is not.
• Coupons are fixed in advance for term rates and determined in arrears for compounded overnight rates.
• The discount margin plays the role of yield for FRNs.

🧪 Practice

1. Beginner: Compute the coupon on EUR 5,000,000 of an FRN paying 3M EURIBOR + 0.75% for a 91-day period when EURIBOR fixed at 2.10%.
2. Intermediate: The issuer's credit spread widens from 50bp to 90bp. Explain why the FRN's price falls even though its coupon is floating.
3. Interview: "A bank funds a fixed-rate mortgage book with FRNs. What risk does it have, and how could it hedge?" Hint: mismatch between fixed income on assets and floating cost on liabilities, NII sensitivity, and paying fixed on an interest rate swap (Chapter 3.6).

Covered Bonds and Securitizations

Why they exist. Banks hold large portfolios of loans, such as mortgages, that are illiquid and funded mostly by deposits. Two techniques let banks use these loans to raise cheaper long-term funding from capital markets: pledge them as security for the bank's own bonds (covered bonds), or sell them to a separate vehicle that issues securities backed by their cash flows (securitization). Both lower funding costs because investors receive extra protection from the loans themselves.

What they are.

• Covered bond: a bond issued by a bank and secured by a ring-fenced cover pool of high-quality assets (usually residential mortgages or public sector loans) that stay on the bank's balance sheet.
  • Dual recourse: investors have a claim on the issuing bank and, if it fails, a preferential claim on the cover pool.
  • Overcollateralization: the cover pool exceeds the bonds outstanding, by a minimum set in law.
  • Governed by specific legislation (in the EU, the Covered Bond Directive), with a cover pool monitor.
• Securitization: loans are sold (true sale) to a special purpose vehicle (SPV), which issues notes in tranches paid from the loans' cash flows.
  • Types: RMBS (residential mortgages), ABS (auto loans, credit cards, consumer loans), CMBS (commercial property), CLO (corporate loans).
  • Waterfall: cash collected is paid to the most senior tranche first; losses hit the most junior tranche first.
  • Risk retention: EU originators must retain at least 5% of the economic risk.
  • Can transfer credit risk off the bank's balance sheet (reducing capital needs) or be retained and used as central bank collateral.

Feature	Covered bond	Securitization
Assets stay on bank balance sheet	Yes	Usually no (true sale to SPV)
Investor recourse	Issuing bank + cover pool	Only the SPV's assets
Asset substitution	Yes, pool is dynamic	Usually static or limited
Tranching	No, single class	Yes, senior to equity
Credit risk transfer	No	Possible
Typical rating	AAA	Senior AAA, juniors lower
LCR treatment (EU)	High-quality covered bonds can be Level 1 or 2A	Limited (some senior RMBS in Level 2B)

Example: securitization waterfall.

  Mortgage pool EUR 1,000m  --true sale-->  SPV  --issues-->
                                                 Class A (senior)   EUR 850m  AAA
                                                 Class B (mezz)     EUR 100m  A
                                                 Class C (junior)   EUR  50m  unrated (retained)

  Cash from borrowers each quarter flows top-down:
     1. Senior fees and expenses
     2. Class A interest
     3. Class B interest
     4. Class A principal  -> then Class B -> then Class C
     5. Residual to Class C

  Losses flow bottom-up:
     pool loses EUR 30m  -> Class C absorbs 30m (A and B untouched)
     pool loses EUR 80m  -> Class C wiped out (50m), Class B loses 30m

Key takeaways

• Covered bonds keep the assets on the balance sheet and give investors dual recourse; they are among the cheapest long-term bank funding.
• Securitization moves assets into an SPV and issues tranched notes, enabling risk transfer and funding.
• In a waterfall, cash goes to senior tranches first and losses hit junior tranches first.
• Both encumber assets, which reduces what is available for other funding and for unsecured creditors (Chapter 5.4).

🧪 Practice

1. Beginner: In the waterfall example, how much does each class lose if the pool loses EUR 160m?
2. Intermediate: Explain why a covered bond investor is better protected than a senior unsecured investor in the same bank, and what that means for unsecured creditors.
3. Interview: "Design the data feed a covered bond issuer needs to produce its monthly investor report." Hint: loan-level data for the cover pool, eligibility criteria and exclusions, overcollateralization calculation, maturity profiles of assets and bonds, and reconciliation with the loan systems and general ledger.

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3.5 Foreign Exchange

This subchapter covers how currencies are quoted and exchanged, how forward prices are derived from interest rates, and the FX instruments a treasury uses to fund and hedge across currencies.

Spot FX and Quote Conventions

Why it matters. Banks have assets, liabilities and customers in many currencies, and the FX market is where they exchange one for another. It is the largest financial market in the world, trading several trillion US dollars per day, mostly OTC. Its quoting conventions are compact and easy to misread: a system that inverts a rate or confuses the base currency can produce errors of the full trade amount.

What it is.

• Currency pair BASE/QUOTE: the price of 1 unit of the base currency expressed in the quote (terms) currency. EUR/USD = 1.0850 means 1 EUR costs 1.0850 USD.
• Market convention for base currency: the priority order is roughly EUR > GBP > AUD > NZD > USD > CAD > CHF > JPY. So the market quotes EUR/USD, GBP/USD, USD/JPY and USD/CHF, never USD/EUR.
• Bid / ask (offer): a dealer quotes two prices for the base currency: the bid at which it buys and the ask at which it sells. EUR/USD 1.0850 / 1.0852.
• Pip: the standard smallest quoted increment, usually the 4th decimal (0.0001), or the 2nd for JPY pairs (0.01).
• Spot date: FX spot trades settle on T+2 business days for most pairs, and T+1 for some (e.g. USD/CAD). The spot date must be a business day in both currencies.
• Settlement risk: each side pays a different currency in a different payment system, so one party can pay and not receive. CLS (Chapter 2.2) mitigates this with payment-versus-payment settlement.

Analogy. An airport exchange booth displays "We buy / We sell" for each currency. The booth buys your euros cheaply and sells you euros expensively; the gap is its profit. Interbank FX dealers do the same, with much tighter gaps.

Example: a client trades with a dealer quoting EUR/USD 1.0850 / 1.0852.

  Client BUYS EUR 10,000,000 (dealer sells EUR at the ask 1.0852)
     Client pays    USD 10,852,000.00
     Client receives EUR 10,000,000.00     both on spot date (T+2)

  Client SELLS EUR 10,000,000 (dealer buys EUR at the bid 1.0850)
     Client pays    EUR 10,000,000.00
     Client receives USD 10,850,000.00

  Dealer round-trip profit: USD 2,000 (2 pips on EUR 10m)

Term	Meaning
Base currency	The currency being priced (left)
Quote / terms currency	The currency in which the price is expressed (right)
Dealt currency	The currency in which the notional amount is agreed (can be either)
Big figure	The leading digits that traders omit (1.08 in 1.0850)
Value date	Settlement date of both currency legs

Key takeaways

• A rate BASE/QUOTE is the price of 1 unit of base currency in quote currency; pair order follows market convention.
• Clients buy base currency at the dealer's ask and sell at the bid.
• Spot settles T+2 for most pairs, on a date that must be a business day in both currencies.
• Store the pair, direction, dealt currency and both amounts explicitly; never infer them from the rate alone.

🧪 Practice

1. Beginner: USD/JPY is quoted 152.38 / 152.42. How many yen does a client receive for selling USD 5,000,000?
2. Intermediate: A trade is done on Thursday in EUR/USD. Friday is a US holiday. What is the spot date, and why?
3. Interview: "Design the trade representation for an FX spot deal." Hint: currency pair in market convention, buy/sell from whose perspective, dealt currency and amount, contra amount, rate with precision, trade and value dates, and settlement method (CLS, gross, net).

FX Forwards and Forward Points

Why they exist. A company expecting USD revenue in three months, or a bank that will need to repay a USD bond, faces the risk that the exchange rate moves before then. An FX forward fixes today the rate for an exchange on a future date. Its price is not a forecast of the future spot rate: it is determined by the interest rates of the two currencies, because anyone can replicate a forward by borrowing in one currency and depositing in the other.

What it is.

• FX outright forward: an agreement to exchange two currency amounts on a date later than spot, at a rate fixed today.

• Forward rate (from covered interest parity, see below):

  F = S × (1 + r_quote × days/360) / (1 + r_base × days/360)

• Forward points: the difference F − S, quoted in pips. Dealers quote points rather than outright rates, because points depend only on interest rates and change slowly, while spot moves constantly.

• Premium or discount: the currency with the lower interest rate trades at a forward premium (it buys more of the other currency forward); the one with the higher rate trades at a discount.

• Common misconception: a forward premium does not mean the market expects that currency to appreciate. It only reflects the interest rate differential.

Analogy. Buying a plane ticket for travel in three months at today's agreed price: the price differs from today's walk-up fare not because of a forecast, but because of the cost of holding the seat until then.

Example: 3-month EUR/USD forward (92 days).

  Spot EUR/USD           1.0850
  USD rate (quote)       4.30%  ACT/360
  EUR rate (base)        2.00%  ACT/360

  F = 1.0850 x (1 + 0.0430 x 92/360) / (1 + 0.0200 x 92/360)
    = 1.0850 x 1.0109889 / 1.0051111
    = 1.09134

  Forward points = (1.09134 - 1.0850) x 10,000 = +63.4 pips
  EUR (lower rate) trades at a forward premium; USD at a discount.

  Quote:  spot 1.0850/1.0852, 3M points 63.0/63.8  -> outright 1.09130/1.09158

Key takeaways

• An FX forward fixes today the rate for an exchange on a future date.
• The forward rate follows from spot and the two interest rates, not from expectations.
• Forward points are added to spot; the lower-yielding currency trades at a forward premium.
• Systems store spot rate, forward points and the resulting all-in rate separately, because they are priced and risk-managed separately.

🧪 Practice

1. Beginner: Spot USD/JPY is 152.40, the USD rate is 4.30% and the JPY rate is 0.50% (both ACT/360 for simplicity). Is the 6-month (182 days) forward above or below spot?
2. Intermediate: Compute the 6-month USD/JPY forward rate and the forward points in the previous exercise.
3. Interview: "An FX forward's fair value is not zero the day after trading, even though spot did not move. Why?" Hint: changes in either interest rate, the passage of time (one day of carry), and discounting the future cash flows with each currency's curve.

FX Swaps

Why they exist. A bank often has surplus cash in one currency and a shortage in another, temporarily. For example, a eurozone bank funded in euros may need US dollars for three months to fund USD loans. Borrowing unsecured USD may be expensive or unavailable. Instead, it can exchange its euros for dollars now and reverse the exchange later. The FX swap is the world's most traded FX instrument and the main tool for short-term cross-currency funding.

What it is.

• FX swap: two opposite FX transactions with the same counterparty, traded together:
  • Near leg: exchange currencies on a near date (usually spot).
  • Far leg: reverse the exchange on a later date, at a rate fixed today (spot + swap points).
• The notional in one currency is usually the same on both legs; the amount in the other currency differs by the swap points.
• Economics: equivalent to a collateralized loan: lending one currency while borrowing the other. The swap points represent the interest rate differential.
• Implied interest rate: from the swap amounts and the known rate in one currency, the rate paid on the other can be calculated. When it differs from the direct money market rate, the gap is the cross-currency basis.
• No FX risk on the notional: both legs' rates are fixed today, so the swap moves liquidity across currencies without creating an open FX position.
• Uses: funding in a foreign currency, investing surplus liquidity, rolling forward an FX position (tom-next swaps), and hedging.

Analogy. Two companies in different countries lend each other their currencies for three months, each holding the other's money as collateral, and agree today on the exact amounts to give back.

Example: Bank A needs USD for 3 months and has EUR. Spot 1.0850, forward 1.09134.

  Near leg (spot, 15 Oct):   Bank A sells EUR 10,000,000  buys USD 10,850,000.00   @ 1.0850
  Far leg (15 Jan, 92 days): Bank A buys  EUR 10,000,000  sells USD 10,913,400.00  @ 1.09134

  Bank A                                  Counterparty
    |--- EUR 10,000,000 ------------------->|   near leg
    |<-- USD 10,850,000 --------------------|
                    ... 92 days ...
    |<-- EUR 10,000,000 --------------------|   far leg
    |--- USD 10,913,400 ------------------->|

  Bank A borrowed USD 10.85m and repays 10.9134m: USD interest 63,400
  It gave up EUR interest on 10m at 2.00%:  10,000,000 x 0.02 x 92/360 = 51,111.11 EUR
  Implied USD cost ~ 4.30%, consistent with the money market rate (no basis here)

Key takeaways

• An FX swap is a spot exchange combined with an opposite forward exchange at a pre-agreed rate.
• It is economically a secured loan in one currency against a deposit in another.
• Swap points reflect the interest differential; deviations from money market rates are the cross-currency basis.
• Treasury uses FX swaps daily to move liquidity between currencies without taking FX risk on the notional.

🧪 Practice

1. Beginner: In the example, what are Bank A's net cash flows in each currency on each date?
2. Intermediate: If the far leg were priced at 1.0930 instead of 1.09134, what USD interest rate would Bank A implicitly be paying? Is the cross-currency basis positive or negative from the USD borrower's view?
3. Interview: "How should an FX swap be represented in a treasury system and in liquidity reports?" Hint: one deal with two legs or two linked deals, each leg's cash flows by currency and value date, and how the legs appear in maturity ladders and LCR outflows and inflows.

Covered Interest Rate Parity

Why it matters. If an investor could earn more by converting money into another currency, investing it, and locking in the conversion back with a forward, everyone would do it until the opportunity disappeared. This no-arbitrage argument, covered interest rate parity (CIRP), links spot rates, forward rates and interest rates in two currencies. It explains how forwards are priced and gives a way to measure stress in dollar funding markets.

What it is.

• CIRP: two ways of investing for a period must give the same result when the FX risk is covered with a forward:

  1. Invest in the quote currency at r_quote.
  2. Convert to the base currency at spot, invest at r_base, and convert back at the forward rate F.

  (1 + r_quote × τ) = (1 / S) × (1 + r_base × τ) × F, hence F = S × (1 + r_quote × τ) / (1 + r_base × τ)

• "Covered" means the FX risk is eliminated with a forward; uncovered interest parity (no forward) is a theory about expectations and does not hold reliably.

• Cross-currency basis: since 2008, CIRP does not hold exactly when using unsecured interbank rates. The deviation is the cross-currency basis, usually negative for EUR/USD and JPY/USD, meaning that borrowing USD via FX swaps costs more than the direct USD rate. Causes include balance sheet constraints on arbitraging banks, demand for USD funding and differences in credit risk.

Example: arbitrage when the market forward deviates.

  S = 1.0850, USD 4.30%, EUR 2.00%, 92 days, CIRP forward = 1.09134
  Market forward quoted at 1.0930 (too high)

  Path 1: keep USD 1,000,000 at 4.30%  ->  1,010,988.89 USD

  Path 2: USD 1,000,000 -> EUR at spot    =   921,658.99 EUR
          invest at 2.00% for 92 days     =   926,369.69 EUR
          sell EUR forward at 1.0930      = 1,012,522.07 USD

  Arbitrage: borrow USD at 4.30%, follow Path 2, repay loan
             profit = 1,012,522.07 - 1,010,988.89 = 1,533.18 USD per USD 1m

Key takeaways

• CIRP states that a fully hedged foreign investment earns the same as a domestic one.
• It determines the theoretical forward rate from spot and the two interest rates.
• Persistent deviations, measured as the cross-currency basis, reflect funding and balance sheet constraints rather than free money.
• Treasury compares FX-swap-implied rates with direct funding rates to choose the cheapest source of foreign currency.

🧪 Practice

1. Beginner: If USD and EUR interest rates were equal, what would the forward rate be relative to spot?
2. Intermediate: Using the example data, describe the arbitrage if the market forward were 1.0890 instead of 1.0930.
3. Interview: "Why did the cross-currency basis widen sharply in March 2020, and what did central banks do?" Hint: global demand for USD funding, constrained dealer balance sheets at quarter-end, and the central bank USD swap lines.

Non-Deliverable Forwards

Why they exist. Some currencies, such as the Indian rupee, Korean won, Brazilian real and Taiwan dollar, are subject to capital controls: foreigners cannot freely hold or move them offshore. International investors and companies still need to hedge exposure to these currencies. A non-deliverable forward (NDF) gives the same economic hedge as a forward but is settled only by a cash payment in a freely convertible currency, usually USD.

What it is.

• NDF: a forward contract on a restricted currency pair where, at maturity, no exchange of notionals takes place. Instead, the difference between the contracted rate and an official fixing rate is paid in USD.

• Fixing date: usually two business days before the settlement date; the fixing source is defined in the contract (e.g. the RBI reference rate for INR).

• Settlement amount (USD/XXX pair, notional in USD):

  Settlement (USD) = Notional_USD × (Fixing − Contract rate) / Fixing

  A positive result is paid to the buyer of USD.

• Risks: fixing risk (disruption or manipulation of the fixing), basis between onshore and offshore markets, and convertibility events.

Example: Bank A buys USD 10,000,000 against INR in a 3-month NDF at 84.50.

  Fixing date: USD/INR fixing = 85.20  (USD strengthened)

  Contracted:  USD 10,000,000 at 84.50 = INR 845,000,000
  At fixing:   USD 10,000,000 at 85.20 = INR 852,000,000
  Difference in INR = 7,000,000 in Bank A's favour

  Settlement in USD = 10,000,000 x (85.20 - 84.50) / 85.20 = USD 82,159.62
  Paid by the USD seller to Bank A on settlement date. No INR changes hands.

Feature	Deliverable forward	NDF
Notionals exchanged	Yes, both currencies	No
Settlement	Gross, in both currencies	Net, in USD (or other convertible currency)
Requires fixing	No	Yes
Used for	Freely convertible currencies	Restricted currencies
Settlement risk	Full notional (unless CLS)	Only the net difference

Key takeaways

• NDFs hedge restricted currencies by cash-settling the difference in USD instead of exchanging notionals.
• The settlement amount depends on an official fixing rate observed shortly before settlement.
• Settlement risk is much lower than for deliverable forwards because only the net amount moves.
• Contract terms must specify the fixing source and fallbacks if the fixing is unavailable.

🧪 Practice

1. Beginner: In the example, who pays whom if the fixing is 83.90 instead of 85.20?
2. Intermediate: Compute the USD settlement amount for the case in the previous exercise.
3. Interview: "What data and events must a system handle for NDFs that it does not need for deliverable forwards?" Hint: fixing source and date, fixing capture and validation, the net settlement calculation and currency, disruption fallbacks, and reporting of NDFs under EMIR clearing and margin rules.

Cross Rates

Why they matter. Most currencies trade most actively against the US dollar. A rate between two non-USD currencies, such as EUR/JPY or GBP/CHF, is often derived from each currency's rate against USD. Treasury and payment systems constantly convert between currencies that are not directly quoted, and cross rate calculation, including bid/ask spreads, must be done correctly to avoid systematic losses.

What they are.

• Cross rate: an exchange rate between two currencies derived through a third (usually USD).
• Calculation depends on the position of USD in each pair:
  • USD is quote in one pair and base in the other (e.g. EUR/USD and USD/JPY): multiply. EUR/JPY = EUR/USD × USD/JPY.
  • USD is quote in both (e.g. EUR/USD and GBP/USD): divide. EUR/GBP = EUR/USD / GBP/USD.
  • USD is base in both (e.g. USD/CHF and USD/JPY): divide. CHF/JPY = USD/JPY / USD/CHF.
• Bid/ask of a cross: combine the prices that are worst for the client. A client buying the cross base currency must go through both legs at the dealer's less favourable side, so the cross spread is wider than either leg.

Analogy. If you know a train from Milan to Zurich and from Zurich to Paris, you can build a Milan to Paris journey. The total time is the sum of both legs, and any delay on either leg affects you.

Example: deriving EUR/JPY and EUR/GBP.

  EUR/USD 1.0850 / 1.0852       USD/JPY 152.38 / 152.42      GBP/USD 1.2950 / 1.2952

  EUR/JPY (multiply):
    bid = 1.0850 x 152.38 = 165.33   (sell EUR->USD at bid, sell USD->JPY at bid)
    ask = 1.0852 x 152.42 = 165.41

  EUR/GBP (divide):
    bid = 1.0850 / 1.2952 = 0.83771  (sell EUR->USD at EUR/USD bid, buy GBP with USD at GBP/USD ask)
    ask = 1.0852 / 1.2950 = 0.83799

Key takeaways

• Cross rates are derived through a common currency, usually USD.
• Multiply when the common currency is quote in one pair and base in the other; divide when it is on the same side in both.
• The cross bid uses the worse side of each leg for the client, so cross spreads are wider than the legs.
• Systems should derive crosses from a consistent snapshot of rates, at the same timestamp.

🧪 Practice

1. Beginner: With USD/CHF at 0.8800 and USD/JPY at 152.40 (mid), compute the mid CHF/JPY rate.
2. Intermediate: Using the example quotes, a client wants to buy GBP 5,000,000 against EUR. Which rate applies and how many EUR does the client pay?
3. Interview: "Our reporting converts all balances to EUR using cross rates computed from USD rates captured at different times. What problems can this cause?" Hint: inconsistent snapshots leading to triangular arbitrage in the data, reconciliation differences with official reference rates (e.g. ECB fixings), and the need for a single, timestamped rate set per reporting run.

<a id="36-derivatives-used-in-treasury"></a>

3.6 Derivatives Used in Treasury

This subchapter introduces the derivatives a bank treasury uses to hedge interest rate and currency risk, and the collateral agreements that control the counterparty risk they create.

Interest Rate Swaps

Why they exist. A bank grants 10-year fixed-rate mortgages but funds them with deposits and bonds whose cost moves with market rates. If rates rise, its funding costs go up while mortgage income stays fixed, and its margin shrinks. Refinancing the whole balance sheet to match is impossible. An interest rate swap lets the bank change the interest profile of its assets or liabilities synthetically, without touching the underlying loans or deposits. Swaps are the most important hedging instrument in bank treasury.

What it is.

• Interest rate swap (IRS): an agreement to exchange, on a notional amount, fixed-rate interest payments for floating-rate interest payments over a period.

  • Payer swap: pay fixed, receive floating. Gains when rates rise.
  • Receiver swap: receive fixed, pay floating. Gains when rates fall.

• Notional is not exchanged: it only serves to compute the interest amounts.

• Legs: each leg has its own frequency, day count and dates. A standard EUR swap: fixed leg annual 30/360; floating leg semi-annual ACT/360 on 6M EURIBOR. A standard USD swap now references compounded SOFR (making it an OIS, see next topic).

• Netting: when payment dates coincide, only the net amount is paid.

• Valuation: PV of floating leg minus PV of fixed leg (for a payer). Floating cash flows are projected from forward rates on the projection curve, and both legs are discounted on the discount curve (OIS curve for collateralized trades).

• Par swap rate: the fixed rate that gives the swap zero value at inception:

  Par rate = (1 − DF(T_n)) / Σ τ_i × DF(T_i) (single-curve approximation)

Analogy. Two homeowners: one has a fixed-rate mortgage but wants a variable rate, the other the reverse. Rather than refinancing, they agree to pay each other's interest. Each keeps their own mortgage with the bank, but their net payments now behave as they want.

Example: Bank A pays fixed 3.00% annual, receives 6M EURIBOR, notional EUR 100,000,000.

               Bank A  --- fixed 3.00%, annual, 30/360 ------->  Dealer
               Bank A  <-- 6M EURIBOR, semi-annual, ACT/360 ----  Dealer

  Year 1 cash flows:
    Month 6:   EURIBOR 2.80%, 181 days   Bank A receives  1,407,777.78
    Month 12:  EURIBOR 3.10%, 184 days   Bank A receives  1,584,444.44
               fixed 3.00% x 1 year      Bank A pays      3,000,000.00
                                         net at month 12: pays 1,415,555.56

  Hedge view: mortgages (receive fixed)  +  payer swap (pay fixed, receive floating)
              = synthetic floating-rate asset, matching floating-rate funding

Example: par rate and value. With discount factors 0.9709, 0.9335 and 0.8890 for years 1 to 3, the 3-year par rate is (1 − 0.8890) / (0.9709 + 0.9335 + 0.8890) = 3.97%. An existing swap in which the bank pays only 3.00% fixed is worth about 100m × (3.97% − 3.00%) × 2.7934 = 2.72m to the bank.

Key takeaways

• An IRS exchanges fixed for floating interest on a notional that is never exchanged.
• Payers of fixed gain when rates rise; receivers gain when rates fall.
• Swaps let treasury reshape the interest rate profile of the balance sheet without changing customer products.
• Valuation projects floating cash flows from forward rates and discounts both legs, usually on the OIS curve.

🧪 Practice

1. Beginner: A bank has issued a 5-year fixed-rate bond at 3.5% and prefers floating-rate funding. Should it enter a payer or a receiver swap?
2. Intermediate: In the example, compute the net payment at month 12 if the second EURIBOR fixing were 3.40% instead of 3.10%.
3. Interview: "Explain why modern swap valuation uses two curves, one for projection and one for discounting." Hint: after 2008, IBOR rates embedded bank credit and liquidity premiums, while collateralized cash flows should be discounted at the rate paid on collateral, typically an overnight rate.

Overnight Index Swaps

Why they exist. An overnight index swap (OIS) is an interest rate swap whose floating leg is an overnight rate (€STR, SOFR, SONIA) compounded over each period. Because the floating leg contains almost no credit risk, OIS rates reveal the market's expectation of central bank policy rates. They are also the benchmark for discounting collateralized derivatives and, after the IBOR transition, the standard swap in USD and GBP.

What it is.

• OIS: exchange of a fixed rate for the compounded overnight rate over each period, on a notional.
• Short OIS (up to 1 year): a single payment at maturity of the net difference between fixed and compounded floating interest.
• Longer OIS: annual payments on both legs.
• Payment delay: because the floating amount is known only at the end of the period, payment typically occurs one or two business days after the period ends.
• Uses in treasury:
  • Hedging or expressing views on central bank rates.
  • Hedging floating-rate assets or liabilities linked to overnight rates.
  • Building the discount curve for collateralized trades (OIS discounting).
  • Measuring term funding premiums, e.g. the spread between 3M EURIBOR and 3M €STR OIS.

Analogy. An OIS is like agreeing a fixed monthly price for electricity while your actual bill is calculated daily at spot prices: at the end of the month you settle the difference between the fixed price and the accumulated daily costs.

Example: 3-month €STR OIS, Bank A pays fixed 2.10%, notional EUR 50,000,000, 91 days.

  Fixed leg:     50,000,000 x 2.10% x 91/360            = 265,416.67
  Floating leg:  compounded €STR over the period = 2.05%
                 50,000,000 x 2.05% x 91/360            = 259,097.22
  Net payment (2 days after maturity): Bank A pays       =   6,319.44

  Interpretation: at trade date, the market expected €STR to average ~2.10%;
  the ECB cut rates earlier than expected, so realized €STR was lower and the
  fixed payer lost 5bp on the period.

Feature	IBOR swap (e.g. 6M EURIBOR)	OIS (e.g. €STR)
Floating rate	Term rate, set in advance	Overnight rate, compounded in arrears
Credit / term premium in floating rate	Yes	Almost none
Payment frequency (short tenor)	Periodic	Single payment at maturity
Main uses	Hedging EURIBOR-linked assets	Policy expectations, discounting, RFR hedging

Key takeaways

• An OIS swaps a fixed rate for a compounded overnight rate.
• OIS rates reflect expected central bank policy rates with minimal credit risk.
• The OIS curve is the standard discount curve for collateralized derivatives.
• The floating amount is known only at period end, so payments are delayed by a day or two.

🧪 Practice

1. Beginner: If the realized compounded €STR in the example were 2.20%, who would pay, and how much?
2. Intermediate: 3M EURIBOR is 2.35% and the 3M €STR OIS rate is 2.10%. What does the 25bp spread measure, and why would it widen in a banking crisis?
3. Interview: "Our valuation of a collateralized swap differs from the counterparty's by a material amount, with identical trade terms. What could explain it?" Hint: discounting curve (OIS vs other), the CSA's collateral currency and rate, curve construction and interpolation, valuation timestamp, and fixings already observed.

Cross-Currency Swaps

Why they exist. Banks often raise funds in one currency but need them in another. A eurozone bank may issue a 5-year bond in USD because USD investors offer a deep market, but it needs euros to fund euro loans. Converting with an FX spot trade would leave it exposed to the exchange rate when the bond matures. A cross-currency swap (CCS) converts both the principal and the interest of the debt into another currency for its full life.

What it is.

• Cross-currency swap: an agreement to exchange principal and interest in two different currencies:
  1. Initial exchange: notionals are exchanged at the spot rate at inception.
  2. Periodic interest: each party pays interest in the currency it received (fixed or floating in either currency).
  3. Final exchange: notionals are re-exchanged at the same rate as the initial exchange.
• Variants:
  • Fixed-fixed, fixed-floating, or floating-floating (basis swap, e.g. €STR + spread vs SOFR).
  • Mark-to-market (resettable) CCS: the notional of one leg resets periodically to the current FX rate, with a cash payment, to reduce counterparty exposure. This is the market standard for interbank floating-floating swaps.
• Cross-currency basis spread: the spread added to one floating leg, reflecting the relative demand for funding in each currency (see Covered Interest Rate Parity).
• Difference from FX swap: an FX swap is short-term and has no periodic interest payments; a CCS is typically long-term and exchanges interest periodically.

Analogy. Two companies, one in Europe and one in the US, each borrow cheaply at home and then swap the loans: each pays the interest and principal of the other's loan, so each ends up with money in the currency it needs without FX risk.

Example: Bank A issues a USD 500m 5-year bond at 4.50% fixed and swaps it to EUR.

  Spot at inception EUR/USD 1.0850  ->  USD 500,000,000 = EUR 460,829,493.09

  Start:       Bond investors --USD 500m--> Bank A --USD 500m--> Swap dealer
                                            Bank A <--EUR 460.83m-- Swap dealer

  Each year:   Bank A <--USD 22.5m (4.50% fixed)-- Swap dealer
               Bank A --USD 22.5m--> Bond investors           (coupon passed through)
               Bank A --EUR 3M EURIBOR + spread on 460.83m--> Swap dealer

  Maturity:    Bank A <--USD 500m-- Swap dealer  (same rate 1.0850)
               Bank A --EUR 460.83m--> Swap dealer
               Bank A --USD 500m--> Bond investors

  Result: Bank A has synthetic EUR floating-rate funding, with no FX risk on principal or coupons.

Key takeaways

• A CCS exchanges principal at the start and end, at the same rate, plus periodic interest in two currencies.
• It converts foreign-currency debt or assets into the home currency for their whole life.
• Large FX moves create large mark-to-market exposures, which resettable notionals and collateral reduce.
• The cross-currency basis affects the all-in cost of foreign funding and is a key input to issuance decisions.

🧪 Practice

1. Beginner: In the example, which cash flows of the CCS mirror the bond's cash flows, and which are new?
2. Intermediate: Five years later, EUR/USD is 1.2500. Explain why Bank A is still protected, and which party had the larger exposure to the other during the life of the swap.
3. Interview: "How should a liquidity reporting system treat the final exchange of a CCS?" Hint: gross inflows and outflows in different currencies on the same date, currency-specific liquidity ladders, the FX rate used for aggregation, and the risk that one leg is paid and the other not.

Futures and Forward Rate Agreements

Why they exist. Treasury often needs to fix today the interest rate for a period starting in the future, for example the rate on a deposit it will roll over in three months. A forward rate agreement (FRA) does this bilaterally and with custom dates; an interest rate future does it on an exchange with standardized terms and daily margining. Both are building blocks of short-end interest rate curves.

What they are.

• Forward rate agreement (FRA): an OTC contract to exchange, on a notional, a fixed rate for a reference rate over a future period. Named by months: a 3x6 FRA covers the period from 3 months to 6 months from now.

  • The buyer pays fixed and gains if the reference rate is higher.
  • Settled in cash at the start of the period, so the payment is discounted:

  Settlement = N × (L − K) × τ / (1 + L × τ)

  where L is the fixed reference rate, K the FRA rate and τ the period's year fraction.

• Interest rate future: an exchange-traded contract on a short-term rate, e.g. 3M SOFR futures (CME) or 3M EURIBOR futures (ICE).

  • Price = 100 − rate: a futures price of 96.50 implies a 3.50% rate. Rising rates mean falling prices.
  • Standardized size (USD 1m for 3M SOFR, EUR 1m for 3M EURIBOR); 1bp is worth USD 25 or EUR 25 per contract.
  • Daily variation margin through the CCP; no bilateral credit risk.

• Convexity adjustment: because futures are margined daily and FRAs are not, futures-implied rates are slightly higher than equivalent forward rates for longer expiries.

Feature	FRA	Interest rate future
Venue	OTC	Exchange
Terms	Customizable dates and notional	Standardized contracts and dates
Credit risk	Bilateral (or cleared)	CCP, daily margining
Settlement	Single discounted payment at period start	Daily margin, final cash settlement
Quote	Rate	100 − rate

Example 1: Bank A buys a 3x6 FRA at 3.00%, notional EUR 10,000,000, period 92 days.

  In 3 months, 3M EURIBOR fixes at 3.40%
  Undiscounted difference = 10,000,000 x (3.40% - 3.00%) x 92/360 = 10,222.22
  Settlement at period start = 10,222.22 / (1 + 0.034 x 92/360)     = 10,134.17
  Bank A receives 10,134.17, which offsets its higher borrowing cost for the period.

Example 2: hedging a USD 100m 3-month borrowing with SOFR futures.

  Sell 100 contracts at 95.70 (implied rate 4.30%)
  Rates rise 20bp: futures fall to 95.50
  Futures gain:  100 contracts x 20bp x USD 25   = +50,000
  Extra borrowing cost: 100,000,000 x 0.20% x 0.25 = -50,000

Key takeaways

• FRAs and short-term interest rate futures both lock in a rate for a future period.
• FRAs are OTC, customizable and settled once, discounted, at the period start.
• Futures are standardized, exchange-traded, quoted as 100 minus the rate, and margined daily.
• A borrower hedges by buying FRAs or selling futures.

🧪 Practice

1. Beginner: A 3M EURIBOR future trades at 97.85. What rate does it imply?
2. Intermediate: Recompute FRA Example 1 if EURIBOR fixes at 2.70%. Who pays whom?
3. Interview: "A treasury hedges with futures instead of FRAs. What operational processes change?" Hint: daily margin calls and funding of variation margin, clearing broker relationship, rolling contracts at expiry, mismatch between standardized dates and the hedged exposure, and hedge accounting documentation.

Caps, Floors and Swaptions

Why they exist. Swaps and FRAs lock in a rate completely: they remove losses but also give up gains. Sometimes treasury needs protection against only one direction of movement, or protection that applies only if something happens. For example, a bank offering floating-rate mortgages with a maximum rate needs protection if rates rise above that maximum, but does not want to give up income if rates fall. Interest rate options provide this asymmetric protection in exchange for an upfront premium.

What they are.

• Cap: a series of options (caplets), one per period, each paying when the reference rate exceeds a strike K.

  Caplet payoff = N × max(L − K, 0) × τ

• Floor: a series of floorlets, each paying when the reference rate falls below the strike.

  Floorlet payoff = N × max(K − L, 0) × τ

• Collar: buying a cap and selling a floor (or vice versa) to reduce or eliminate the net premium.

• Swaption: an option to enter an interest rate swap at a fixed rate on a future date.

  • Payer swaption: right to pay fixed; valuable when rates rise.
  • Receiver swaption: right to receive fixed; valuable when rates fall.
  • Notation: a "1y5y" swaption is a 1-year option on a 5-year swap.

• Premium: paid upfront by the buyer; depends on strike, maturity and implied volatility.

• Banking book uses: hedging rate caps or floors embedded in customer products (capped mortgages, deposits floored at 0%), and hedging mortgage prepayment risk with receiver swaptions (Chapter 6).

Instrument	Holder protected against	Payoff when	Typical treasury use
Cap	Rising rates	Rate above strike	Floating-rate liabilities, capped loans sold
Floor	Falling rates	Rate below strike	Floating-rate assets, deposit floors
Payer swaption	Rising rates (future swap)	Swap rate above strike	Pipeline of fixed-rate loan offers
Receiver swaption	Falling rates (future swap)	Swap rate below strike	Prepayment risk on fixed-rate loans

Analogy. A cap is insurance on your borrowing rate: you pay a premium, and if rates climb above the agreed level, the insurer pays the excess. A swaption is like a reservation for a fixed-rate loan that you may or may not take up.

Example: Bank A buys a 3-year cap on 3M EURIBOR, strike 3.00%, notional EUR 20,000,000.

  Premium paid upfront: EUR 180,000

  Quarter   EURIBOR fixing   Days   Caplet payoff
  Q1           2.40%          90    0           (below strike)
  Q2           2.90%          91    0
  Q3           3.60%          91    20,000,000 x 0.60% x 91/360 = 30,333.33
  Q4           3.25%          92    20,000,000 x 0.25% x 92/360 = 12,777.78

  Payoff by rate (per period):
      payoff
        |                 /
        |               /
        |_____________/
        +-------------+---------> EURIBOR
                     3.00%

Key takeaways

• Caps pay when rates rise above a strike; floors pay when rates fall below it; each is a strip of per-period options.
• Swaptions give the right to enter a swap in the future.
• Options give one-sided protection in exchange for an upfront premium, priced with implied volatility.
• Treasury uses options mainly to hedge optionality embedded in customer products.

🧪 Practice

1. Beginner: Compute the floorlet payoff for a floor with strike 1.00% on EUR 50,000,000 when 3M EURIBOR fixes at 0.60% for a 91-day period.
2. Intermediate: A bank pays floating rates on deposits but customers' deposit rates cannot go below 0%. What embedded option has the bank sold, and what instrument could hedge it?
3. Interview: "Why is it harder to hedge a fixed-rate mortgage book than a fixed-rate bond portfolio, and how do swaptions help?" Hint: borrowers prepay when rates fall, shortening the asset's life exactly when the bank's payer swap hedges lose value; receiver swaptions gain in that scenario.

Collateral and CSA Agreements

Why they exist. An OTC derivative can run for years, and its value can swing heavily in one party's favour. If the losing party defaults, the winning party loses that value. Before 2008, much of this exposure was uncollateralized; the default of Lehman Brothers showed how dangerous that was. Today, nearly all derivatives between financial institutions are collateralized under legal agreements that require the party with a negative mark-to-market to post collateral, usually daily.

What they are.

• ISDA Master Agreement: the standard legal framework for OTC derivatives between two parties, including events of default and close-out netting (all trades are netted into one amount on default).
• Credit Support Annex (CSA): the annex to the ISDA that defines the collateral terms:

Term	Meaning
Variation margin (VM)	Collateral covering the current mark-to-market of the portfolio
Initial margin (IM)	Extra collateral covering potential future exposure; required for uncleared trades between large firms under the Uncleared Margin Rules
Threshold	Exposure allowed before collateral is required (now usually zero for VM)
Minimum transfer amount (MTA)	Smallest amount that can be called, to avoid tiny transfers
Rounding	Calls are rounded (e.g. to the nearest 10,000 or 100,000)
Eligible collateral	Cash in specific currencies, government bonds, with haircuts
Collateral interest rate	Rate paid on cash collateral, typically the overnight rate (€STR, SOFR)
Valuation and call frequency	Usually daily
Dispute resolution	Process when the parties' valuations differ

• Margin call process: value portfolio, compute exposure net of collateral held, apply threshold, MTA and rounding, issue call, counterparty agrees or disputes, collateral settles (usually same or next day).
• Cleared derivatives: trades cleared through a CCP follow the CCP's margin rules instead of a bilateral CSA (Chapter 2.3).
• Treasury impact: margin calls are a major source of intraday and daily liquidity demand; the collateral currency and rate also affect derivative valuation (Overnight Index Swaps).

Analogy. A tenant pays a deposit against damage. A CSA is a deposit that is recalculated every day: if the potential loss grows, the tenant tops it up; if it shrinks, the landlord returns some.

Example: daily VM call for Bank A against Dealer D.

  CSA terms: threshold 0, MTA EUR 500,000, rounding up to EUR 100,000, cash EUR only

  Day 1:  Portfolio MTM (Bank A's view)        +12,400,000  (D owes A)
          Collateral already held by A          10,000,000
          Exposure after collateral              2,400,000
          > MTA, rounded                         2,400,000  -> A calls 2.4m from D

  Day 2:  MTM falls to                          +12,150,000
          Collateral held                       12,400,000
          Excess held by A                         250,000  -> below MTA, no return call

  Day 3:  MTM falls to                          +11,700,000
          Excess held by A                         700,000  -> D calls return of 700,000

Key takeaways

• The ISDA Master Agreement provides close-out netting; the CSA sets the collateral rules.
• Variation margin covers current exposure; initial margin covers potential future exposure.
• Threshold, MTA and rounding determine when and how much collateral moves.
• Margin calls create real liquidity needs, so collateral management is a core treasury function (Chapter 4.2 and 5.4).

🧪 Practice

1. Beginner: With a threshold of EUR 1,000,000 and an MTA of EUR 250,000, how much collateral can be called on an uncollateralized exposure of EUR 1,180,000?
2. Intermediate: Explain why a bank with a large payer swap portfolio might face large collateral inflows when rates rise, and outflows when rates fall. How should the liquidity buffer account for this?
3. Interview: "Design a daily margin call engine." Hint: consistent valuation snapshots, netting sets per CSA, collateral inventory with haircuts, threshold, MTA and rounding logic, workflow for disputes, messaging with counterparties, and feeding expected collateral flows to the cash forecast.

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