# Fixed Income Concept Card: [Your elective, portfolio or bond] > A concept card for the fixed income part of an MBA investments elective: bond pricing from a yield and from a spot > curve, bootstrapping spot rates, implied forward rates, Macaulay and modified duration, convexity, DV01, > immunisation of a liability, credit spreads and floating-rate notes on the new overnight reference rates. It builds on > the time-value and bond-pricing material of the core course (mba/mba-npv-irr-card and session 2 of > mba/mba-corporate-finance-guide) and on the cost of debt in mba/mba-capm-wacc-card; it does not re-teach present > value. It is for second-year students in an investments or fixed income elective, candidates for markets, asset > management and treasury roles, and managers who run a corporate or institutional bond portfolio. Worked example A > prices a bond, measures its rate risk and bootstraps a curve; Worked example B (half solved) immunises a liability at > Tallow Creek Medical Center; Exercise C (unsolved) analyses a Northfield Outdoor Co. corporate bond. Source basis: > Macaulay (1938), Redington (1952), Fisher and Weil (1971), Merton (1974), Vasicek (1977), Nelson and Siegel (1987) and > Litterman and Scheinkman (1991), written up in original words. Reference-rate facts are dated (October 2026) with the > official page to check. Every yield, price and rate in the examples is an invented *Example* value, not market data. > Replace every [bracketed] value and every row marked *Example*. This is a learning aid, not investment advice. ## Document control | Field | Value | |---|---| | Elective, portfolio or bond | [Investments elective, session 5 / bond or portfolio name] | | Learner or analyst | [Your name] | | Pricing date | YYYY-MM-DD | | Currency and market | [USD Treasury / EUR government / corporate] | | Curve and price source with date | [Named data source, close of YYYY-MM-DD] | | Compounding and day-count convention | [Annual / semiannual; 30/360, ACT/360 or ACT/ACT] | | Reference rate for floating legs | [SOFR / SONIA / €STR / other, source and date] | | Reviewed by | [Study partner, tutor or portfolio manager] | | Version | [0.1] | | Last updated | YYYY-MM-DD | ## Contents - [When to use it](#when-to-use-it) - [The idea in plain language](#the-idea-in-plain-language) - [Formulas and notation](#formulas-and-notation) - [Assumptions and limits](#assumptions-and-limits) - [Reference rates after LIBOR (dated facts)](#reference-rates-after-libor-dated-facts) - [Worked example A (fully solved): pricing, duration, convexity and a bootstrapped curve](#worked-example-a-fully-solved-pricing-duration-convexity-and-a-bootstrapped-curve) - [Worked example B (half solved): immunising a Tallow Creek liability](#worked-example-b-half-solved-immunising-a-tallow-creek-liability) - [Exercise C (unsolved): a Northfield corporate bond](#exercise-c-unsolved-a-northfield-corporate-bond) - [Decision diagram: which rate-risk measure](#decision-diagram-which-rate-risk-measure) - [How managers use fixed income analytics](#how-managers-use-fixed-income-analytics) - [Connections to other cards](#connections-to-other-cards) - [Common mistakes and misconceptions](#common-mistakes-and-misconceptions) - [Self-test questions](#self-test-questions) - [My notes and next steps](#my-notes-and-next-steps) - [Sources and further reading](#sources-and-further-reading) - [Answer key](#answer-key) - [Revision history](#revision-history) ## When to use it **One-sentence rule:** price a bond by discounting each cash flow at the spot rate for its date, measure its sensitivity to small parallel yield moves with modified duration (and to larger moves with convexity), and treat the yield above the government curve as compensation for expected default loss, liquidity and risk, not as free return. | Use when | Use something else when | |---|---| | Pricing an option-free government or corporate bond and quoting its yield | The bond is callable, putable or mortgage-backed (use effective duration from a rate model, see mba/mba-options-real-options-card) | | Estimating the price impact of a small parallel shift in yields | The curve twists or steepens (use key-rate durations) | | Matching assets to a known liability stream (pension, insurance, a committed payment) | Liabilities are uncertain in size or timing (use scenario and cash-flow testing) | | Comparing a corporate bond's spread with its expected default loss | Assessing an issuer's credit quality in depth (use finance/bond-credit-analysis) | | Hedging rate risk with futures or swaps sized by DV01 | Pricing the futures or swaps themselves (use mba/mba-forwards-futures-swaps-card) | ## The idea in plain language **Price and yield move in opposite directions.** A bond is a fixed promise: coupons and principal on set dates. Its price is the present value of that promise. If the market's required return rises, the same fixed cash flows are worth less today, so the price falls. The **yield to maturity** is the single discount rate that equates the promised cash flows to the price; it is the bond's internal rate of return if you hold it to maturity, receive every payment and reinvest each coupon at the same yield. **The term structure.** Money lent for one year and money lent for five years need not earn the same rate. The **spot rate** for maturity $t$ is the yield on a zero-coupon bond paying only at $t$. A coupon bond is a bundle of zero-coupon bonds, so its fair price discounts each cash flow at its own spot rate. You can recover spot rates from coupon bond prices one maturity at a time (**bootstrapping**). The spot curve also implies **forward rates**: the rate for a future period that you can lock in today by borrowing short and lending long. **Duration.** Macaulay (1938) proposed the present-value-weighted average time to a bond's cash flows as a better measure of its "length" than its final maturity. Hicks (1939) and, for insurers, Redington (1952) showed that the same number measures sensitivity to interest rates: modified duration is the percentage price change for a one-unit change in yield. A zero-coupon bond's Macaulay duration equals its maturity; coupons pull it shorter, because part of the value comes back earlier. **Convexity.** The price-yield curve is curved, not straight. Duration is the slope at the current yield, so it overstates the fall when yields rise and understates the gain when they fall. Convexity is the second-order correction. For an option-free bond it is positive: the holder gains more from a fall in yields than they lose from an equal rise. **Immunisation.** An institution that owes a fixed amount at a future date can hold bonds whose present value and duration match the liability's. Redington (1952) showed that a small parallel shift then leaves the surplus roughly unchanged, and that if the assets' convexity exceeds the liability's, the surplus rises for small shifts in either direction. Fisher and Weil (1971) showed that a portfolio whose duration, measured with the term structure rather than a single yield, equals the investor's holding horizon locks in a return close to the one promised at the start, even when the curve is not flat and shifts in parallel, and tested the strategy on historical bond data. **Credit.** A corporate bond's yield exceeds the government yield of the same maturity by a **credit spread**. Merton (1974) modelled corporate debt as risk-free debt minus a put option on the firm's assets: the spread rises with leverage and asset volatility. In practice, the spread pays for expected default loss (probability of default times loss given default), for the risk that defaults cluster in bad times, and for the lower liquidity of corporate bonds. ## Formulas and notation $$ P = \sum_{k=1}^{nf} \frac{CF_k}{(1 + y/f)^{k}} \qquad P = \sum_{k=1}^{nf} \frac{CF_k}{(1 + s_{t_k}/f)^{k}} $$ $$ D_{Mac} = \frac{1}{P} \sum_{k=1}^{nf} t_k \, \frac{CF_k}{(1 + y/f)^{k}} \qquad D_{Mod} = \frac{D_{Mac}}{1 + y/f} \qquad DV01 = D_{Mod} \times P \times 0.0001 $$ $$ C = \frac{1}{P\, f^2} \sum_{k=1}^{nf} \frac{k (k+1)\, CF_k}{(1 + y/f)^{k+2}} \qquad \frac{\Delta P}{P} \approx -D_{Mod}\, \Delta y + \tfrac{1}{2}\, C\, (\Delta y)^2 $$ $$ (1 + s_2)^2 = (1 + s_1)(1 + f_{1,2}) \quad \Rightarrow \quad f_{1,2} = \frac{(1 + s_2)^2}{1 + s_1} - 1 $$ $$ \text{Immunisation: } PV_A = PV_L, \quad D_A = D_L, \quad C_A > C_L \qquad \text{Expected loss} \approx PD \times (1 - R) $$ | Symbol | Meaning | Unit | |---|---|---| | $P$ | Full price of the bond per 100 of face value | Currency per 100 face | | $CF_k$ | Cash flow at payment $k$ (coupon, plus face value at maturity) | Currency per 100 face | | $y$ | Yield to maturity, quoted per year | Rate per year | | $f$ | Payments per year (1 annual, 2 semiannual) | Number | | $t_k$ | Time of payment $k$ in years, $t_k = k/f$ | Years | | $s_t$ | Spot (zero-coupon) rate for maturity $t$ | Rate per year | | $f_{1,2}$ | One-year forward rate starting in one year | Rate per year | | $D_{Mac}$ | Macaulay duration: present-value-weighted average time to cash flows | Years | | $D_{Mod}$ | Modified duration: percentage price change per unit change in yield | Years | | $DV01$ | Price change for a one basis point change in yield | Currency per 100 face | | $C$ | Convexity | Years squared | | $PD$, $R$ | Annual probability of default and recovery rate | Rate, fraction | ## Assumptions and limits | Assumption | What it means | When it fails and what to do | |---|---|---| | Fixed, known cash flows | Coupons and principal do not depend on rates | Callable bonds, mortgage-backed securities and puttable bonds: use effective duration and convexity from a rate model | | Flat curve, parallel shifts | One yield moves and every maturity moves by the same amount | Real curves twist and change slope; Litterman and Scheinkman (1991) found that level, slope and curvature factors explain most Treasury return variation. Use key-rate durations | | Small changes | The duration approximation is a tangent line | For moves above about 50 to 100 basis points add convexity or reprice in full | | Hold to maturity, reinvest at the yield | Yield to maturity is the realised return only under these conditions | Reinvestment rates change; compute the horizon return with explicit reinvestment assumptions | | No default | The promised cash flows are paid | Corporate bonds: model expected loss and the spread, see Exercise C and finance/bond-credit-analysis | | Immunisation holds for one shift | Duration drifts as time passes and yields move | Rebalance the portfolio; Redington's condition is local, not permanent | Yield-curve models go further. Vasicek (1977) derived the term structure from a mean-reverting short rate; Nelson and Siegel (1987) fit the whole curve with level, slope and curvature terms, a parsimonious form many central banks and data providers use to publish fitted curves. An elective usually stops at using such fitted curves, not deriving them. ## Reference rates after LIBOR (dated facts) As of October 2026, check each official page before you rely on it: - **LIBOR has ended.** According to the Bank of England's transition page, the last synthetic 3-month sterling LIBOR setting continued until end-March 2024, and the remaining synthetic US dollar settings (1, 3 and 6 months) ceased at end-September 2024 (last publication 30 September 2024). Legacy contracts that referenced LIBOR rely on their fallback clauses or on legislation; read the contract and ask counsel, do not assume. Check [bankofengland.co.uk, transition from LIBOR](https://www.bankofengland.co.uk/markets/transition-to-sterling-risk-free-rates-from-libor). - **SOFR** (Secured Overnight Financing Rate): a secured overnight rate measuring the cost of borrowing cash overnight against US Treasury collateral (Treasury repo). The Federal Reserve Bank of New York administers and publishes it each business day at about 8:00 a.m. Eastern time. Check [newyorkfed.org, SOFR](https://www.newyorkfed.org/markets/reference-rates/sofr). - **SONIA** (Sterling Overnight Index Average): the average rate banks pay to borrow sterling overnight, based on actual transactions; the Bank of England is the administrator. Check [bankofengland.co.uk, SONIA](https://www.bankofengland.co.uk/markets/sonia-benchmark). - **€STR** (euro short-term rate): reflects the wholesale euro unsecured overnight borrowing costs of banks located in the euro area; the ECB first published it on 2 October 2019 for trading on 1 October 2019. Check [ecb.europa.eu, €STR](https://www.ecb.europa.eu/stats/financial_markets_and_interest_rates/euro_short-term_rate/html/index.en.html). **Why it matters for analysis.** LIBOR was a forward-looking term rate with bank credit risk built in. The new rates are overnight and (for SOFR) secured, so a floating-rate note pays a compounded overnight rate over the coupon period, known only at the end of the period, plus a fixed spread. A floating-rate note that resets to the market rate each period has a duration close to the time to the next reset, so its price is almost insensitive to rate moves; its spread duration, however, can be long, because the fixed margin does not reset. ## Worked example A (fully solved): pricing, duration, convexity and a bootstrapped curve **Setting** *(Example)*. Corvane Robotics' treasury team holds surplus cash in investment-grade bonds. One holding is a 5-year bond with a 5% annual coupon and face value 100, priced at a 6% yield to maturity. The team wants the price, the duration, the convexity, the DV01 on a USD 10m face position, and the price change if yields move by 100 and 200 basis points. It also wants the spot curve from three government bonds. **Step 1: price.** Discount each cash flow at 6%. | Year $t$ | Cash flow | PV at 6% | Weight $PV/P$ | $t \times$ weight | $t(t+1) \times PV$ | |---|---|---|---|---|---| | 1 *(Example)* | 5 | 4.7170 | 0.0492 | 0.0492 | 9.4340 | | 2 *(Example)* | 5 | 4.4500 | 0.0465 | 0.0929 | 26.6999 | | 3 *(Example)* | 5 | 4.1981 | 0.0438 | 0.1315 | 50.3772 | | 4 *(Example)* | 5 | 3.9605 | 0.0413 | 0.1654 | 79.2094 | | 5 *(Example)* | 105 | 78.4621 | 0.8191 | 4.0956 | 2,353.8632 | | **Total** | | **95.7876** | **1.0000** | **4.5347** | **2,519.5836** | Price $P = 95.7876$. Excel: `=PV(6%,5,-5,-100)` or `=PRICE(DATE(2026,1,1),DATE(2031,1,1),5%,6%,100,1)` = 95.7876. The bond trades below par because its coupon (5%) is below the market yield (6%). **Step 2: duration.** Macaulay duration is the sum of $t \times$ weight: $D_{Mac} = 4.5347$ years (`=DURATION(DATE(2026,1,1),DATE(2031,1,1),5%,6%,1)`). Modified duration $D_{Mod} = 4.5347 / 1.06 = 4.2780$ (`=MDURATION(DATE(2026,1,1),DATE(2031,1,1),5%,6%,1)`). Reading: a 1 percentage point rise in yield cuts the price by about 4.28%. **Step 3: DV01.** $DV01 = 4.2780 \times 95.7876 \times 0.0001 = 0.0410$ per 100 face. On USD 10m face (100,000 units of 100), the position loses about USD 4,098 for each basis point rise in yield. Traders size hedges with this number: a futures or swap hedge with the same DV01 and the opposite sign neutralises small parallel moves (mba/mba-forwards-futures-swaps-card). **Step 4: convexity.** $C = 2{,}519.5836 / (95.7876 \times 1.06^2) = 23.4103$. **Step 5: price changes.** Compare the full repricing with the two approximations. | Yield change | Actual new price | Actual change | Duration only | Duration plus convexity | |---|---|---|---|---| | +100 bp *(Example)* | 91.7996 | -4.1634% | -4.2780% | -4.1609% | | -100 bp *(Example)* | 100.0000 | +4.3976% | +4.2780% | +4.3950% | | +200 bp *(Example)* | 88.0219 | -8.1073% | -8.5559% | -8.0877% | Excel for the actual prices: `=PV(7%,5,-5,-100)` = 91.7996, `=PV(5%,5,-5,-100)` = 100.0000, `=PV(8%,5,-5,-100)` = 88.0219. Duration alone is symmetric and overstates the loss; convexity closes almost all of the gap at 100 bp and most of it at 200 bp. The gain from a fall exceeds the loss from a rise of the same size: that asymmetry is positive convexity. **Step 6: bootstrapping the spot curve.** Three government bonds pay annual coupons *(Example)*: | Bond | Maturity | Coupon | Price | |---|---|---|---| | G1 *(Example)* | 1 year | 4.0% | 99.52 | | G2 *(Example)* | 2 years | 4.5% | 99.30 | | G3 *(Example)* | 3 years | 5.0% | 99.60 | - 1-year spot: $99.52 = 104 / (1 + s_1)$, so $s_1 = 4.5016\%$ (`=104/99.52-1`). - 2-year spot: the first coupon is worth $4.5 / 1.045016 = 4.3062$, so $104.5 / (1 + s_2)^2 = 99.30 - 4.3062$ and $s_2 = 4.8843\%$ (`=(104.5/(99.3-4.5/(1+s1)))^(1/2)-1`). - 3-year spot: the first two coupons are worth $9.3298$ at $s_1$ and $s_2$, so $105 / (1 + s_3)^3 = 99.60 - 9.3298$ and $s_3 = 5.1675\%$. Implied forwards: $f_{1,2} = 1.048843^2 / 1.045016 - 1 = 5.2684\%$ and $f_{2,3} = 1.051675^3 / 1.048843^2 - 1 = 5.7363\%$. The curve slopes upward, so forwards lie above spots. **Step 7: pricing off the curve.** A 3-year 6% annual-coupon bond is worth $6/1.045016 + 6/1.048843^2 + 106/1.051675^3 = 5.7415 + 5.4542 + 91.1299 = 102.3257$. Its yield to maturity is 5.1437% (`=IRR({-102.3257,6,6,106})` or `=RATE(3,6,-102.3257,100)`), a blend of the three spot rates weighted towards the 3-year rate, where most of the value sits. Two bonds with the same maturity but different coupons have different yields on the same curve: the yield is a summary, the spot curve is the price. **Chart: price-yield curve and the duration tangent** (data from Step 5 and the run in the calculation script). ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Price of a 5-year 5% annual-coupon bond against yield, with the modified-duration tangent at 6% (fictional Example)", "width": "container", "height": 260, "data": { "values": [ {"yield": 2, "price": 114.14, "series": "Actual price"}, {"yield": 3, "price": 109.16, "series": "Actual price"}, {"yield": 4, "price": 104.45, "series": "Actual price"}, {"yield": 5, "price": 100.00, "series": "Actual price"}, {"yield": 6, "price": 95.79, "series": "Actual price"}, {"yield": 7, "price": 91.80, "series": "Actual price"}, {"yield": 8, "price": 88.02, "series": "Actual price"}, {"yield": 9, "price": 84.44, "series": "Actual price"}, {"yield": 10, "price": 81.05, "series": "Actual price"}, {"yield": 2, "price": 112.18, "series": "Duration tangent at 6%"}, {"yield": 3, "price": 108.08, "series": "Duration tangent at 6%"}, {"yield": 4, "price": 103.98, "series": "Duration tangent at 6%"}, {"yield": 5, "price": 99.89, "series": "Duration tangent at 6%"}, {"yield": 6, "price": 95.79, "series": "Duration tangent at 6%"}, {"yield": 7, "price": 91.69, "series": "Duration tangent at 6%"}, {"yield": 8, "price": 87.59, "series": "Duration tangent at 6%"}, {"yield": 9, "price": 83.49, "series": "Duration tangent at 6%"}, {"yield": 10, "price": 79.40, "series": "Duration tangent at 6%"} ] }, "mark": {"type": "line", "point": true}, "encoding": { "x": {"field": "yield", "type": "quantitative", "title": "Yield to maturity (%)"}, "y": {"field": "price", "type": "quantitative", "title": "Price per 100 face", "scale": {"zero": false}}, "color": {"field": "series", "type": "nominal", "title": null}, "strokeDash": {"field": "series", "type": "nominal", "title": null} } } ``` The actual curve lies above the tangent everywhere except at 6%: that gap is convexity, and it grows with the size of the move. ## Worked example B (half solved): immunising a Tallow Creek liability **Setting** *(Example)*. Tallow Creek Medical Center has committed to pay USD 25.0m to a contractor in exactly 6 years for a new wing. The finance committee wants to set aside bonds today so that the payment is covered whatever happens to rates, at least for small parallel moves. Assume a flat 5% annual yield curve. Two instruments are available: Bond X, a 2-year zero-coupon bond, and Bond Y, a 10-year bond with a 6% annual coupon. 1. Present value of the liability: $PV_L = 25.0 / 1.05^6 = 18.6554$ (USD m). Excel: `=PV(5%,6,0,-25)`. Its Macaulay duration is 6 years (a single payment) and its convexity is $6 \times 7 / 1.05^2 = 38.0952$. 2. Bond X: Macaulay duration = [ ] years; price per 100 face = $100 / 1.05^2 = 90.7029$; convexity 5.4422. 3. Bond Y: price at 5% = [ ] (`=PV(5%,10,-6,-100)`); Macaulay duration = 7.8921 years (`=DURATION(DATE(2026,1,1),DATE(2036,1,1),6%,5%,1)`); convexity 72.1737. 4. Weight in Bond X so that the portfolio duration equals 6: $w \times 2 + (1 - w) \times 7.8921 = 6$, so $w$ = [ ]. 5. Amounts to invest: Bond X USD 5.9908m and Bond Y USD [ ]m. Face values: Bond X 6.6049m, Bond Y [ ]m. 6. Redington's convexity check: asset convexity = [ ] versus liability convexity 38.0952. Is the condition met? [ ] 7. Stress test at 4% and 6% flat yields: | Flat yield | Asset value (USD m) | Liability PV (USD m) | Surplus (USD m) | |---|---|---|---| | 4% *(Example)* | 19.7705 | 19.7579 | [ ] | | 6% *(Example)* | 17.6351 | [ ] | 0.0110 | 8. In one sentence, explain why the surplus is positive in both directions, and name two reasons this hedge will not stay perfect over the next six years: [ ] ## Exercise C (unsolved): a Northfield corporate bond **Setting** *(Example)*. Northfield Outdoor Co. has a 7-year bond outstanding with a 6.5% coupon paid semiannually, quoted at 97.80. The 7-year government yield is 4.30% (semiannual bond-equivalent basis). A credit analyst estimates an annual probability of default of 1.2% and a recovery rate of 40%. Northfield is also considering a floating-rate note at compounded SOFR plus 1.80%, paid quarterly on an ACT/360 basis. 1. Compute the bond's yield to maturity on a bond-equivalent basis. Excel: `=RATE(14,3.25,-97.8,100)*2` or `=YIELD` with dates. 2. Compute the credit spread over the government yield in basis points. 3. Compute the approximate annual expected default loss and compare it with the spread. What does the remainder pay for? 4. Compute the Macaulay and modified duration (`=DURATION` and `=MDURATION` with frequency 2) and estimate the price change if the spread widens by 50 basis points. Compare with full repricing. 5. If compounded SOFR over a 91-day coupon period is 4.10% *(Example)*, what is the coupon on USD 10m face of the floating-rate note? Why is the floating-rate note's rate duration short while its spread duration is not? 6. Which would you hold if you expected government yields to rise sharply but Northfield's credit to stay stable, and why? ## Decision diagram: which rate-risk measure Start from the cash flows, not from the measure you know best: the type of instrument and the size and shape of the rate move decide which number is meaningful. ```mermaid flowchart TD A["Holding to analyse"] --> B{"Cash flows fixed and known?"} B -- "No: callable, MBS, puttable" --> C["Effective duration and convexity from a rate model"] B -- Yes --> D{"Floating coupon?"} D -- Yes --> E["Rate duration about time to next reset, plus spread duration"] D -- No --> F{"Size of the rate move"} F -- "Small, under about 50 bp" --> G["Modified duration and DV01"] F -- "Large" --> H["Duration plus convexity, or full repricing"] G --> I{"Parallel shift a fair assumption?"} H --> I I -- No --> J["Key-rate durations by maturity bucket"] I -- Yes --> K["Portfolio duration as value-weighted average"] J --> L["Hedge sized by DV01 per bucket"] K --> L C --> L E --> L ``` ## How managers use fixed income analytics - **Corporate treasurers** set a duration limit for the surplus-cash portfolio so that a rate shock cannot wipe out a quarter's interest income, and choose between fixed and floating debt with the cost of debt from mba/mba-capm-wacc-card and the hedging policy in finance/fx-interest-rate-hedging-policy. - **Pension funds and insurers** run liability-driven investment: they match the duration (often key-rate durations) of assets to liabilities, then decide how much surplus risk to take in return-seeking assets. Redington's idea is the core of that practice; the asset allocation side is in finance/strategic-asset-allocation-study. - **Banks** measure interest rate risk in the banking book with repricing gaps, duration of equity and scenario shifts of the curve (finance/irrbb-policy-and-report). - **Portfolio managers** express views as active duration (longer or shorter than the benchmark), curve positions and spread positions, and are then evaluated on them (mba/mba-performance-evaluation-card and finance/performance-attribution-report, which splits fixed income excess return into income, curve, spread and selection effects). - **Credit investors** compare the spread with the expected loss and with the spreads of similarly rated issuers, using the analysis in finance/bond-credit-analysis and finance/credit-rating-rationale. - **Risk managers** convert DV01 and convexity into value at risk and stress losses (mba/mba-value-at-risk-card). ## Connections to other cards | Card or template | Connection | |---|---| | mba/mba-npv-irr-card | Present value and IRR: a bond price is an NPV and its yield is an IRR | | mba/mba-capm-wacc-card | The cost of debt is the yield lenders require today, not the coupon | | mba/mba-portfolio-theory-card | Bonds as the low-risk asset in a mean-variance portfolio; correlation of bonds and equities | | mba/mba-factor-models-card | Level, slope and curvature as factors for bond returns, by analogy with equity factors | | mba/mba-forwards-futures-swaps-card | Interest rate swaps and bond futures for hedging DV01 | | mba/mba-value-at-risk-card | Turning duration and convexity into a loss distribution | | mba/mba-options-real-options-card | Embedded options in callable bonds; Merton's view of debt as a short put | | mba/mba-performance-evaluation-card | Judging a bond manager's return against a benchmark and the risks taken | | mba/mba-investments-guide | The elective roadmap that places this card | ## Common mistakes and misconceptions 1. **Treating the yield to maturity as the expected return.** It is the return only if every payment arrives and coupons are reinvested at the same yield; for a risky bond the expected return is lower than the promised yield. 2. **Confusing Macaulay and modified duration.** Macaulay is in years (a weighted time); modified is the price sensitivity. Use modified duration, or $D_{Mac}/(1 + y/f)$, for price changes. 3. **Using duration for large moves.** The tangent line overstates losses and understates gains; for 200 basis points in Worked example A the error was about 0.45 percentage points of price. 4. **Discounting every cash flow at one yield when the curve slopes.** Price coupon bonds off spot rates; two bonds of the same maturity can have different yields on the same curve. 5. **Mixing compounding conventions.** A semiannual bond-equivalent yield is not an annual effective rate; convert with $(1 + y/2)^2 - 1$ before comparing. 6. **Assuming a floating-rate note has no risk.** Its rate duration is short, but its spread is fixed, so credit widening still hits its price. 7. **Reading the whole credit spread as expected default loss.** In Exercise C most of the spread pays for risk and liquidity, not expected loss. 8. **Setting an immunised portfolio and forgetting it.** Duration drifts as time passes and yields change; rebalance. 9. **Still quoting LIBOR.** It ceased (final synthetic settings on 30 September 2024, as of October 2026); use the current reference rate and read legacy contract fallbacks. 10. **Adding durations instead of value-weighting them.** Portfolio duration is the market-value-weighted average of the holdings' durations (an approximation that assumes a flat curve). ## Self-test questions 1. Why does a bond's price fall when market yields rise? Answer in two sentences. 2. An 8-year zero-coupon bond yields 5% (annual). What are its Macaulay and modified durations? 3. A bond has a modified duration of 6.5 and a convexity of 60. Estimate the percentage price change for a 20 basis point rise in yield, with and without convexity. 4. A position has USD 20m face, price 101 and modified duration 4.2. What is its DV01 in dollars? 5. The 1-year spot rate is 3% and the 2-year spot rate is 4% (annual). What is the one-year forward rate one year from now? 6. A 10-year bond with a 5% annual coupon yields 5%. What is its price, and why? 7. State Redington's three conditions for immunising a liability. 8. Give three reasons a corporate bond's spread exceeds its expected default loss. 9. What changed in floating-rate notes when markets moved from LIBOR to SOFR, SONIA or €STR? 10. When should you use effective duration instead of modified duration? ## My notes and next steps | Item | My entry | |---|---| | The one idea I could not explain without notes | [ ] | | The formula I mixed up | [ ] | | A real bond I will price with =PRICE and =YIELD, with source and date | [ ] | | Questions for the next class or office hours | [ ] | | Next card | [mba/mba-forwards-futures-swaps-card or mba/mba-value-at-risk-card] | | Review dates (2 days, 1 week, 1 month) | YYYY-MM-DD, YYYY-MM-DD, YYYY-MM-DD | ## Sources and further reading - Macaulay, F. R. (1938). *Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856*. National Bureau of Economic Research. - Hicks, J. R. (1939). *Value and Capital*. Oxford: Clarendon Press. - Redington, F. M. (1952). Review of the principles of life-office valuations. *Journal of the Institute of Actuaries*, 78(3), 286-340. - Fisher, L., and Weil, R. L. (1971). Coping with the risk of interest-rate fluctuations: returns to bondholders from naive and optimal strategies. *The Journal of Business*, 44(4), 408-431. - Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. *The Journal of Finance*, 29(2), 449-470. - Vasicek, O. (1977). An equilibrium characterization of the term structure. *Journal of Financial Economics*, 5(2), 177-188. - Nelson, C. R., and Siegel, A. F. (1987). Parsimonious modeling of yield curves. *The Journal of Business*, 60(4), 473-489. - Litterman, R., and Scheinkman, J. (1991). Common factors affecting bond returns. *The Journal of Fixed Income*, 1(1), 54-61. - Official pages for reference rates (checked October 2026): Bank of England (LIBOR transition, SONIA), Federal Reserve Bank of New York (SOFR), European Central Bank (€STR); links in the dated-facts section. - Textbooks (by name only): Fabozzi, F. J., *Bond Markets, Analysis, and Strategies*; Tuckman, B., and Serrat, A., *Fixed Income Securities: Tools for Today's Markets*; Bodie, Z., Kane, A., and Marcus, A. J., *Investments*. - Open resource: OpenStax, *Principles of Finance* (CC BY-NC-SA 4.0), chapter on bonds and bond valuation. ## Answer key **Worked example B.** (2) Bond X has Macaulay duration 2 years (a zero-coupon bond's duration equals its maturity). (3) Bond Y price = 107.7217. (4) $w = (7.8921 - 6)/(7.8921 - 2) = 0.3211$ in Bond X and 0.6789 in Bond Y. (5) Bond Y USD 12.6646m (face 11.7567m, since $12.6646 / 107.7217 \times 100$). (6) Asset convexity $= 0.3211 \times 5.4422 + 0.6789 \times 72.1737 = 50.7442 > 38.0952$: the condition is met, because the barbell (cash flows at 2 and about 10 years) is more dispersed than the single 6-year payment. (7) Surplus at 4% = 0.0126; liability at 6% = 17.6240. (8) With matched value and duration and higher convexity, the assets gain more than the liability when yields fall and lose less when they rise. The hedge decays because durations change at different speeds as time passes (rebalance), and because real curve moves are not parallel (a steepening hurts a barbell against a bullet liability). **Exercise C.** (1) Semiannual rate 3.4508%, so YTM = 6.9016% bond-equivalent (`=RATE(14,3.25,-97.8,100)*2`). (2) Spread = 6.9016% - 4.30% = 260 basis points (260.2). (3) Expected loss $\approx 1.2\% \times (1 - 0.40) = 0.72\%$ a year, or 72 basis points; the other 188 basis points or so pay for the risk that defaults cluster in recessions, for lower liquidity and for uncertainty about the default estimate. (4) Macaulay duration 5.7167 years, modified 5.5260 (`=MDURATION(...,6.5%,6.9016%,2)`); a 50 basis point widening gives about $-5.5260 \times 0.005 = -2.763\%$, or -2.70 on a price of 97.80; full repricing gives -2.66, so convexity reduces the loss slightly. (5) Coupon rate $4.10\% + 1.80\% = 5.90\%$; payment $10{,}000{,}000 \times 5.90\% \times 91/360 =$ USD 149,139. The coupon resets to the market rate each period, so the rate duration is about the time to the next reset; the 1.80% margin is fixed for the life of the note, so if Northfield's credit spread widens the note's price falls as if it were a fixed-rate bond of similar life. (6) The floating-rate note: it avoids most of the loss from rising government yields while keeping the credit exposure you are comfortable with. **Self-test.** 1. The bond's cash flows are fixed. A higher required return discounts the same cash flows more heavily, so their present value (the price) falls. 2. Macaulay duration 8 years; modified $8 / 1.05 = 7.6190$. 3. Duration only: $-6.5 \times 0.002 = -1.30\%$. With convexity: $-1.30\% + 0.5 \times 60 \times 0.002^2 = -1.288\%$. 4. $4.2 \times 1.01 \times 20{,}000{,}000 \times 0.0001 =$ USD 8,484 per basis point. 5. $1.04^2 / 1.03 - 1 = 5.0097\%$. 6. 100 (par): when the coupon rate equals the yield, the bond prices at par (`=PV(5%,10,-5,-100)` = 100). 7. Present value of assets equals present value of liabilities; duration of assets equals duration of liabilities; convexity (dispersion) of assets exceeds that of liabilities. 8. Default risk is systematic (defaults cluster in recessions, so investors require a premium); corporate bonds are less liquid than government bonds; there is uncertainty about default and recovery estimates (also taxes in some markets). 9. The reference rate became an overnight risk-free rate (SOFR secured, SONIA and €STR unsecured) compounded over the period and known at its end, instead of a forward-looking term rate that contained bank credit risk; any credit premium now sits in the fixed spread. 10. When cash flows depend on rates (callable, putable or mortgage-backed bonds): effective duration reprices the bond with a rate model after up and down shifts, so it captures the embedded option. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First pass through the card | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Example yields with a dated curve for a real portfolio | ## Standard and licence note Based on bond pricing, duration, immunisation and credit-spread theory in the primary works listed above (Macaulay 1938, Hicks 1939, Redington 1952, Fisher and Weil 1971, Merton 1974, Vasicek 1977, Nelson and Siegel 1987, Litterman and Scheinkman 1991), and on the official reference-rate pages of the Bank of England, the Federal Reserve Bank of New York and the European Central Bank as checked in October 2026. The text, examples, numbers and questions are original; no textbook text is copied or paraphrased, and Corvane Robotics, Northfield Outdoor Co. and Tallow Creek Medical Center are fictional. All yields, prices and rates are *Example* values, not market data. The template is independent and not affiliated with or endorsed by any business school, publisher, central bank or data provider. Check current market data and the current official pages before you rely on a figure. Template licence: MIT.