3.3Lesson 3.3 · Module 3

Same cash flows, same price.

Bond pricing is not just a formula exercise. If two portfolios deliver the same dollars on the same dates, they should have the same price. If they do not, arbitrage logic appears. Then duration and convexity measure how those bond prices move when yields move.

Market prices → Law of one price → Arbitrage → Duration → Convexity
Decode why identical cash flows must have identical prices
Learn how short selling and frictions constrain arbitrage
Measure interest-rate risk with duration and convexity

Start Arbitrage, Duration, and Convexity View module map

Learning objectives

By the end of this lesson, you should be able to:

1Explain why market prices are informative but not automatically correct.
2Explain the law of one price and why it does not require equilibrium.
3Identify the arbitrage direction when a coupon bond and STRIPS portfolio have different prices.
4Explain why short selling and transaction costs matter for arbitrage.
5Explain how multiple coupon bonds create an overdetermined pricing system.
6Define Macaulay duration as a present-value-weighted average payment time.
7Define modified duration as approximate price sensitivity to yield changes.
8Calculate duration from cash flows.
9Explain how coupon rate, YTM, and maturity affect duration.
10Define convexity as curvature in the price-yield relationship.
11Use duration plus convexity to approximate bond price changes.
12Explain why portfolio duration and convexity are value-weighted averages.

M3Module map

Fixed-Income Securities

Bonds move the financial system. Two lessons build the valuation machinery now; duration, convexity, credit risk, and securitization arrive in upcoming lessons.

Node 01

Bond Markets & Discount Bonds

The market system, cash flows, and zero-coupon valuation

Ready

Node 02

Spot Rates & Yield Curves

Forwards, YTM, replication, and arbitrage

Ready

Node 03

Arbitrage, Duration & Convexity

Law of one price, mispricing signals, and interest-rate risk

Ready

Node 04

Credit Risk & Securitization

Corporate bonds, default risk, spreads, and structured credit

Ready

01Chapter 1

Market prices, same cash flows, same price

Before we can talk about arbitrage or risk, we need to understand what a bond price is really telling us. A Treasury yield is not a moral truth. It is a market price translated into a rate. Market prices are a window into current sentiment, fear, liquidity, and expectations — useful, but not a crystal ball.

1.1Section 1

Market prices are current sentiment, not correctness

The lecture begins by comparing the yield curve from one week to the next during the 2008 crisis. The short end of the Treasury curve had been extremely low because investors were rushing into short-term Treasury bills. A week later, the three-month Treasury yield had moved higher, suggesting that the panic was less severe than before.

When investors rush into Treasury bills, the price of those bills rises and the yield falls. A very low short-term Treasury yield can mean extreme demand for safety and liquidity. It does not mean "the correct interest rate has been discovered forever." It means that market participants, at that moment, are willing to accept a very low yield to hold a very safe, liquid instrument.

Professor's note

Do not ask whether the price is correct. Ask what information and sentiment the price reflects, and whether you want to use that price in your own calculation.

Definition · The market is a thermometer, not a judge

A Treasury price does not announce whether it is fair. It reports what investors collectively demand right now for time, safety, and inflation risk.

Try itMarket regime bridge

What information does each price reflect?

Switch the regime. Watch which end of the curve bends. Each move is a signal about what investors want — not a verdict on whether the bond is correctly priced.

Short end (1–2y)

Treasury priceRises

Short Treasury prices rise, yields fall — flight to liquidity.

Long end (10–20y)

Treasury priceSteady

Long end barely moves; the demand is for safe, liquid cash now.

Regime reading

Flight to liquidity

Professor's note

Do not ask whether the price is correct. Ask what information the price reflects.

1.2Section 2

Recall: coupon bonds as packages of discount bonds

A 3-year 5% coupon bond with $1,000 face value has cash flows of $50 in year 1, $50 in year 2, and $1,050 in year 3. The same cash flows can be created by holding 50 one-year STRIPS, 50 two-year STRIPS, and 1050 three-year STRIPS. The coupon bond and the STRIPS portfolio produce identical future cash flows. This sets up the law of one price.

Definition · Cash-flow replication

If two investments produce identical future dollars on identical future dates, then — absent frictions — they must sell for the same price today.

Try itCash-flow match scanner

Are these two streams the same future dollars?

The coupon bond pays $50.00 in years 1 and 2, then $1,050.00 at maturity. A portfolio of zero-coupon STRIPS can be assembled to deliver exactly those dollars. Scan to confirm each date matches.

Coupon bond

$50.00

Year 1

$50.00

Year 2

$1,050.00

Year 3

STRIPS portfolio

$50.00

Year 1

$50.00

Year 2

$1,050.00

Year 3

Year 1

—

Year 2

—

Year 3

—

1.3Section 3

The law of one price

Definition · Law of one price

Two identical cash flows must have the same market price.

This does not require a full equilibrium model. It does not require a perfect theory of supply and demand. It requires only that at least one market participant prefers more money to less money.

If two identical cash-flow streams sell for different prices: buy the cheaper one, sell the more expensive one, receive money today, and use the future cash flows from the asset you bought to offset the future obligations from the asset you sold.

Definition

Arbitrage is a free lunch in the idealized model: no money down, no future net obligation, and positive money today.

Professor's note

If this law appears to fail, do not complain that finance theory broke. First ask whether you can actually trade it.

Definition · Law of One Price

If two investments produce the same future cash flows, they must trade for the same price today. Any difference is an arbitrage signal — before frictions.

Try itLaw of One Price engine

Same future dollars, same price?

Edit the cash flows and prices for stream A and stream B. The engine checks identity, then flags an arbitrage direction if the prices disagree.

Stream A

Year 1Year 2Year 3

Price today

Stream B

Year 1Year 2Year 3

Price today

Price comparison

$1,020.00

$1,000.00

Arbitrage gap$20.00

\Rightarrow

risk-free profit today

Verdict

A is expensive · sell A, buy B

Same future dollars, but A costs more. Sell the expensive one ($1,020.00), buy the cheap one ($1,000.00), lock $20.00 today with zero net future obligation.

Friction disclaimer: this is the frictionless, textbook version. Real execution pays bid-ask spreads, transaction costs, and may be blocked by short-sale constraints — see the friction switches that follow.

1.4Section 4

Arbitrage direction: coupon bond versus STRIPS

Coupon bond as STRIPS portfolio

P = C P_{0, 1} + C P_{0, 2} + \dots + (C + F) P_{0, T}

A coupon bond's price should equal the cost of buying each cash flow separately as a zero-coupon bond.

$P$: coupon bond price
$C$: coupon payment
$F$: face value
$P_{0, t}$: price today of a pure discount bond paying $1 at time t
$T$: maturity

If the coupon bond costs more than the STRIPS portfolio, short the bond and buy the STRIPS. If it costs less, do the reverse.

Definition · STRIPS replication

Strip a coupon bond into its individual cash flows. Each cash flow is a zero-coupon "STRIP." Reassembling those STRIPS must reproduce the bond's price — or the desk has a trade.

Try itSTRIPS replicator desk

Replicate the bond with zero-coupon STRIPS

Each coupon and the final principal is bought as a separate STRIPS piece. Add up the replicating cost and compare it to the bond's market price.

Replication formula

P = C P_{0, 1} + C P_{0, 2} + \dots + (C + F) P_{0, T}

The bond's price equals the sum of each cash flow times the price of a zero-coupon STRIPS paying $1 at that date.

$C$: Coupon per period
$F$: Face value (principal)
$P_{0, t}$: Today's price of a zero paying $1 at time t

Face (F)Coupon %Maturity T (yr)Bond price

P_{0, 1}

0.95

P_{0, 2}

0.90

P_{0, 3}

0.85

1.5Section 5

Why short selling matters

The arbitrage trade may require selling something you do not already own. That is short selling. To short a security, you must borrow it, sell it, and later return it. This may be costly or impossible.

If short selling is banned or expensive, the law-of-one-price force becomes weaker. Prices can remain misaligned because the trade that would normally close the gap cannot be executed easily.

Professor's note

When short sales are restricted, finance theory is not destroyed, but one of the mechanisms that enforces pricing relationships is on vacation.

Definition · Short sale

Selling a security you do not own, by borrowing it first. Short sales are essential to enforcing the Law of One Price downward: they let you sell the expensive side of a mispricing.

Try itShort-sale friction switch

What survives when you actually try to trade?

A $20.00 mispricing shows on the screen. Toggle the friction and see how much of that gap you can actually capture.

Theoretical gap vs executable opportunity

Theoretical gap$20.00

What the screen shows

Executable opportunity$20.00

Capturable after frictions

Frictionless

No costs, no constraints. The full $20.00 gap is executable. Arbitrageurs sell the expensive side, buy the cheap side, and the gap closes.

Professor's note

When short sales are restricted, one of the mechanisms that enforces pricing relationships is on vacation.

02Chapter 2

Arbitrage, short selling, and overdetermined bond systems

2.1Section 6

Multiple coupon bonds and overdetermined systems

With one coupon bond, we can compare it with a matching STRIPS portfolio. With many coupon bonds, the same idea becomes a system of equations. Each bond price should equal the value of its cash flows discounted by the appropriate pure discount bond prices.

Usually there are more bonds than maturity points — more equations than unknown discount factors. The system is overdetermined. If no solution exists, at least one price is inconsistent with the others. In finance, "no solution" can mean a mispricing signal.

Professor's note

In ordinary algebra, no solution can feel like failure. In fixed-income arbitrage, no solution may be the interesting part.

Definition · Pricing by linear system

If two bonds let you solve for the discount factors

P_{0, 1}

and

P_{0, 2}

, then every other bond paying in those periods has a single no-arbitrage price. A market price that disagrees is a mispricing.

Try itMatrix mispricing lab

Two bonds pin down the discount factors

Bond A and Bond B reveal $P_{0, 1} = 0.95$ and $P_{0, 2} = 0.90$ . Bond C pays 50 in each period, so its no-arbitrage price is $50 (0.95) + 50 (0.90) = 92.50$ . Set Bond C's market price and read the signal.

Bond A

CF [100, 0]

Price$95.00

Solves

P_{0, 1}

= 0.95

Bond B

CF [0, 100]

Price$90.00

Solves

P_{0, 2}

= 0.90

Bond C · market

CF [50, 50]

Set Bond C market price

No-arb price = $92.50

Balance scale — market vs no-arbitrage

Signal

Consistent

Bond C's price matches the no-arbitrage price. The three-bond system has a single consistent solution.

Professor's note

No solution can mean mispricing. If the system of equations has no consistent answer, the market itself is internally inconsistent — and that is information.

2.2Section 7

Fixed-income arbitrage: linear algebra plus trading reality

Fixed-income arbitrage can be seen as looking across many bonds and many implied discount factors. In a real market, there may be hundreds of bonds and far fewer maturity points. Traders search for inconsistencies. But this is not something retail investors should casually attempt — transaction costs, financing costs, short-selling constraints, liquidity, callable features, and speed all matter.

Professor's note

Do not try this at home. Institutional fixed-income arbitrage requires models, data, financing, execution, and risk controls. In the 1970s, MIT-trained quants at Salomon Brothers used simultaneous linear equations to search for bond mispricings — high-school algebra scaled up with market data and execution. One trader reportedly earned a very large bonus doing it.

Definition · Arbitrage in practice

Theoretical mispricing is not profit. Each real-world friction — transaction costs, short-sale constraints, liquidity — takes a bite out of the gap before it reaches your pocket.

Try itArbitrage desk reality panel

From theoretical gap to executable opportunity

A $30.00 mispricing appears. Add frictions one at a time and watch how much of the gap survives all the way to a real trade.

Gap breakdownExecutable: $30.00

Executable Tx cost Short-sale Liquidity

Executable opportunity?

$30.00 of the gap survives

No frictions. The full theoretical gap is executable and arbitrageurs will close it.

From the desk

In the 1970s, MIT-trained quants at Salomon Brothers used simultaneous linear equations to find bond mispricings — pricing every Treasury against every other Treasury and pouncing on the inconsistent ones.

Professor's note

Do not try this at home. The easy mispricings were traded away long ago; today's version requires speed, capital, and a tolerance for the frictions above.

03Chapter 3

Duration and price sensitivity

Once we understand how bonds are priced and why prices must be internally consistent, the next question is risk: how much does a bond or bond portfolio move when yields move? Duration and convexity provide fast measures of that sensitivity.

3.1Section 8

Bond prices and interest-rate risk

Bond prices are functions of yield. If yield changes, price changes. The relationship is inverse: yield up → price down, yield down → price up. The slope tells us how sensitive the bond price is to yield changes. A steeper slope means more price risk around that yield.

Definition · Price–yield relationship

For a coupon bond the curve is downward-sloping and convex: higher yield means lower price, and the price falls more slowly as yields rise. The slope at a point is the bond's duration.

Try itPrice vs yield · coupon bond

y = 6.00% · P = $103.51

The curve and its tangent

A 4-year, 7% semiannual coupon bond. Drag the yield. The amber dot is the price; the dashed line is the duration tangent. A steeper slope means more price risk — small yield moves produce large price moves.

Yield (y)6.00%

2%10%

Price

$103.51

Mod. duration (annual)

3.46

Slope |dP/dy|

358.39

price drop per +1.0 yield

Steeper slope = more price risk. Notice the curve is flattest at high yields (low prices) and steepest at low yields (high prices) — the same bond is riskier when it trades at a premium.

3.2Section 9

Duration intuition before the formula

A longer maturity bond is usually more sensitive to yield changes because its cash flows are farther in the future. A small change in the discount rate is compounded over more periods. If a zero-coupon bond pays only at year 30, its duration is 30 years. If a bond pays coupons along the way, some value arrives earlier, so duration is less than final maturity.

Core intuition: Duration is the average time at which the bond's value is received, weighted by present value.

Definition · Macaulay duration

The weighted-average time to receive a bond's cash flows, weighted by each flow's present value. It is the balance point of the cash-flow timeline.

Try itDuration balance scale

Where does the timeline balance?

Each cash flow is a weight hung on the timeline. The fulcrum that balances them is the duration. Move the levers and watch the balance point shift.

Coupon rate7.0%

Yield (ytm)6.0%

Maturity (years)4 yr

Bar height = present value of that cash flow · fulcrum = Macaulay duration

Macaulay duration

3.63 years

Why this value?

Coupon spreads cash flows across time, pulling the balance point below maturity.

Higher coupon

Duration falls

Higher yield

Duration falls

Longer maturity

Duration rises

3.3Section 10

Macaulay duration formula

Definition · Macaulay duration

The present-value-weighted average term to maturity.

Macaulay duration

D_{m} = k = 1 \sum T k ω_{k}

Each cash-flow date receives a weight equal to that cash flow's share of total present value. Duration averages the payment dates using those weights.

$D_{m}$: Macaulay duration (in periods)
$k$: payment period index
$ω_{k}$: present-value weight of cash flow at time k

Present-value weights

ω_{k} = \frac{P V ( C _{k} )}{P} = \frac{C _{k} / ( 1 + y ) ^{k}}{P}

The weight is each cash flow's present value divided by the bond's total price. The weights sum to one.

$C_{k}$: cash flow at time k
$y$: yield per period
$P$: bond price = sum of all PV( $C_{k}$ )

Because weights sum to one, duration is a proper weighted average — not an arbitrary sum.

3.4Section 11

MIT example: 4-year Treasury note duration

Consider a 4-year Treasury note with a face value of $100, a coupon rate of 7%, a market price of $103.50, and a yield of 6%. Coupons are paid semiannually, so there are 8 periods, the coupon per period is $3.50, and the semiannual yield is 3%.

Definition · Computing Macaulay duration

Build a table: each row is a period's cash flow, its present value, and the time-weighted present value. Sum the last two columns and divide. The result is duration in periods — divide by the frequency to annualize.

Try itDuration table builder

MIT 4yr Treasury · 7% coupon · y=6% · semiannual

Build the duration, row by row

8 half-year periods, coupon $3.50 each, per-period yield 3%. Reveal each column in turn to see how the weighted average assembles.

t (half-yr)	CF_t	PV(CF_t)	t × PV(CF_t)
1	$3.50	···	···
2	$3.50	···	···
3	$3.50	···	···
4	$3.50	···	···
5	$3.50	···	···
6	$3.50	···	···
7	$3.50	···	···
8	$103.50	···	···
Σ		···	···

Hover a row to highlight that period on the timeline below.

Half-year timeline

Duration in half-years

Reveal step 6…

Why is the last row so large?

Macaulay duration definition

D_{m} = \frac{\sum _{t = 1}^{T} t \cdot P V ( C F _{t} )}{\sum _{t = 1}^{T} P V ( C F _{t} )}

D_{m} = \frac{738.28}{103.51} = 7.13 half-years

Annual = 7.13 / 2 = 3.57 yr (lib: 3.57 yr)

Weighted-average time to receive the bond's cash flows, in periods. Divide by q to express in years.

3.5Section 12

Modified duration and price sensitivity

Macaulay duration gives weighted timing. Modified duration converts duration into approximate price sensitivity. If yield moves up by 0.1%, the bond price decreases by approximately 0.6860%.

Modified duration

\frac{1}{P} \frac{\partial P}{\partial y} = - \frac{D _{m}}{1 + y} = - D_{m}^{*}

Modified duration is Macaulay duration divided by one plus the per-period yield. The negative sign means price and yield move inversely.

$D_{m}$: Macaulay duration
$D_{m}^{*}$: modified duration
$y$: yield per period

Approximate percentage price change

\frac{Δ P}{P} \approx - D_{m}^{*} Δ y

For a small yield change, multiply modified duration by the yield change (with a negative sign) to get the approximate percentage price change.

\frac{Δ P}{P} \approx - 6.86 \times 0.001

≈ −0.686%

A 10 basis point yield increase reduces the bond price by about 0.686%.

Definition · Duration as price risk

Modified duration is the bond's interest-rate sensitivity. For a small yield shock, the percentage price change is approximately minus the modified duration times the yield change.

Try itDuration shock simulator

What happens to price when yields move?

Set the modified duration and shock the yield. A +10 bps rise on a bond with modified duration 6.86 cuts the price by roughly 0.686%.

Modified duration D*_mStarting price

Yield shock Δy+10 bps

−200 bps+200 bps

Starting price

$103.50

Estimated new price

$102.79

−$0.71 (−0.686%)

%ΔP estimate

−0.686%

−6.86 × 0.10%

Duration price-change rule

\frac{Δ P}{P} \approx - D_{m}^{*} Δ y

The percentage price change is approximately minus modified duration times the yield change. This is a first-order (linear) approximation.

\frac{Δ P}{P} \approx - 6.86 \times 0.10% = - 0.686%

$P_{n} e w$ ≈ $103.50 × (1 + -0.686%) = $102.79

Duration is symmetric and linear — it overestimates gains and underestimates losses for large shocks. Convexity corrects this (see the next interactives).

First-order only. The rule ignores convexity, so for shocks of 100+ bps the real price will differ — especially the downside.

3.6Section 13

What changes duration?

$∙$ Duration decreases with coupon rate (more value arrives earlier).
$∙$ Duration decreases with YTM (distant cash flows shrink).
$∙$ Duration usually increases with maturity.
$∙$ For bonds at par or premium, duration always increases with maturity.
$∙$ For deep discount bonds, duration can decrease with maturity — but empirically, duration usually increases.

Definition · What moves duration?

Three levers: coupon rate, yield, and maturity. Higher coupons and higher yields both shorten duration; longer maturity usually lengthens it.

Try itDuration levers

Pull the three levers of duration

Watch how each lever shifts the balance point of the cash flows and the resulting Macaulay and modified duration.

Coupon rate7.0%

YTM6.0%

Maturity (years)10 yr

Macaulay (years)

7.61

Modified (years)

7.18

Balance point

76% of maturity

010 yr

Maturity timeline · fulcrum = Macaulay duration

7.61 yr

10 yr

Coupon ↑

Duration falls

Yield ↑

Duration falls

Maturity ↑

Duration rises

Mini-check

If the coupon rate rises, duration rises or falls?

3.7Section 14

Intra-year coupons

Annual Macaulay duration (intra-year coupons)

D_{m}^{ann u a l} = k = 1 \sum T \frac{k ω _{k}}{q}

Divide the period-based duration by the number of payments per year to annualize it.

$q$: number of coupon payments per year (e.g. q=2 for semiannual)
$T$: number of payment periods

Annual modified duration

D_{m}^{*, ann u a l} = \frac{D _{m}^{ann u a l}}{1 + y / q}

The annualized modified duration adjusts for the compounding frequency.

Definition · Frequency and duration units

Duration computed in periods depends on how many periods a year (q) the bond pays. To compare bonds, convert to annual: divide the period-level Macaulay by q, then adjust modified duration by (1 + y/q).

Try itPayment frequency switch

Same bond, different compounding rhythm

A 2-year bond, coupon and yield matched. Switch the frequency: the period-level duration moves, but once annualized the durations sit close together.

Coupon rate8.0%

Yield (ytm)8.0%

4 periods · q = 2 per year

Periods (q·T)

Macaulay (periods)

3.78

Macaulay (annual)

1.89 yr

Annualizing across frequency

D_{m}^{ann u a l} = k = 1 \sum T \frac{k ω _{k}}{q}

Period-level weights k are divided by q to convert from periods to years.

$q$: Coupons per year (1, 2, or 4)
$k$: Period index
$ω_{k}$: PV-weight of period k

Annual modified duration

D_{m}^{*, ann u a l} = \frac{D _{m}^{ann u a l}}{1 + y / q}

D_{m}^{*, ann u a l} = \frac{1.89}{1 + 8.00%/2} = 1.81

Modified (annual) = 1.81 years

More frequent coupons mean a finer compounding grid; once annualized, duration is comparable across bonds regardless of frequency.

04Chapter 4

Convexity and portfolio interest-rate risk

4.1Section 15

Convexity intuition

Duration is slope. Convexity is bend. Bond price as a function of yield is not a straight line. The duration line is a local linear approximation. Convexity measures how the slope itself changes as yield changes.

For small yield moves, duration may be enough. For larger yield moves, curvature matters. Convexity improves the approximation.

Professor's note

Today Excel can reprice the bond directly. But duration and convexity still give quick intuition about risk, especially for large portfolios.

Definition · Why the curve bends

Duration is a linear, first-order estimate — it ignores the curve's curvature. Convexity is the second-order term that captures the bend. For small shocks duration is enough; for large shocks, convexity matters.

Try itBending price curve

base y = 6.00% · P₀ = $104.27

Tangent line vs the real curve

The cyan curve is the true price. The amber dashed line is the duration-only tangent. The purple mark is the convexity-adjusted estimate. Shock the yield and watch the duration line pull away while convexity stays close.

Yield shock Δy+100 bps → y = 7.00%

−300 bps+300 bps

True price

$100.00

Duration-only

$99.89

error vs true: $0.11

Duration + convexity

$100.00

error vs true: $0.00

Duration-only moves in a straight line; the real curve bends. The gap widens with larger shocks — and convexity closes most of that gap.

Professor's note

Today Excel can reprice directly, but duration and convexity give quick risk intuition — how much pain a yield move causes before you recompute anything.

4.2Section 16

Convexity formula

Convexity

V_{m} = \frac{1}{P} \frac{\partial ^{2} P}{\partial y ^{2}}

Convexity is the second derivative of bond price with respect to yield, divided by price. It captures curvature.

$V_{m}$: convexity
$P$: bond price
$y$: yield

Second derivative for a simple bond

\frac{\partial ^{2} P}{\partial y ^{2}} = \frac{1}{( 1 + y ) ^{2}} k = 1 \sum T \frac{k ( k + 1 ) C _{k}}{( 1 + y ) ^{k}}

The second derivative sums k(k+1) times each discounted cash flow. Longer-dated cash flows contribute more curvature.

Convexity captures how duration changes as yield changes.

4.3Section 17

Duration-convexity approximation

P (y_{0}) \approx P (y) [1 - D_{m}^{*} (y_{0} - y) + \frac{1}{2} V_{m} (y_{0} - y)^{2}]

Approximate the new price after a yield change using duration (linear) plus convexity (quadratic) correction.

$P (y)$: current price at yield y
$y_{0}$: new yield
$D_{m}^{*}$: modified duration
$V_{m}$: convexity

P (0.08) \approx 100 [1 - 3.509846 \times 0.02 + \frac{1}{2} (14.805972) (0.02)^{2}]

P(0.08) ≈ 93.276 vs exact 93.267

The approximation differs by about a penny. A penny is small for one bond, but meaningful for very large portfolios.

Definition · Duration + convexity approximation

The full second-order approximation combines a linear duration term with a quadratic convexity term. It tracks the true price far better than duration alone — for one bond the difference is a penny, but across a large portfolio those pennies add up.

Try itApproximation console

Compute the approximated price

Start from a price $100.00 at y = 6%. Estimate the price at y₀ = 8% (Δy = 2.00%). Step through each term of the approximation.

P(y)y (%)y₀ (%)D*_mV_m (convexity)Exact P(y₀)

Duration term

-D*_m × Δy = -3.509846 × 2.00%

-0.0701969

Convexity term

½ × V_m × Δy² = 0.5 × 14.805972 × 0.0004

0.0029612

Factor 1 + dur + conv

1 + (-0.0701969) + (0.0029612)

0.9327643

Duration-only

$92.98

error: $0.29

Duration + convexity

$93.28

error: $0.01

Exact reference

$93.27

Second-order price approximation

P (y_{0}) \approx P (y) [1 - D_{m}^{*} (y_{0} - y) + \frac{1}{2} V_{m} (y_{0} - y)^{2}]

Duration gives the linear slope; convexity adds the curvature correction. The bracket is the price factor applied to P(y).

P (8%) \approx 100.00 [1 - 3.509846 \times 2.00% + \frac{1}{2} \times 14.805972 \times 0.0004]

P(8%) ≈ $93.28 (exact ≈ $93.27)

A $0.01 error is small for one bond, but meaningful for large portfolios — especially when duration-only alone is off by $0.29.

A penny is small for one bond, but meaningful for large portfolios. Convexity shaves the duration-only error from $0.29 down to $0.01.

4.4Section 18

Portfolio duration and convexity

Portfolio modified duration

D_{m}^{*} (P) = j \sum \frac{P _{j}}{P} D_{m, j}^{*}

Portfolio modified duration is the market-value-weighted average of each bond's modified duration.

$P_{j}$: market value of bond j
$P$: total portfolio value
$D_{m, j}^{*}$: modified duration of bond j

Portfolio convexity

V_{m} (P) = j \sum \frac{P _{j}}{P} V_{m, j}

Portfolio convexity is the market-value-weighted average of each bond's convexity.

Definition · Portfolio duration and convexity

A portfolio's modified duration and convexity are value-weighted averages of the holdings. Add a long-duration bond and the whole portfolio stretches — even if it is a small slice by count.

Try itBond portfolio risk mixer

Mix the bonds, watch the portfolio risk stretch

Change each bond's market value, then shock the whole portfolio. The long-duration bond drags the portfolio duration up disproportionately.

Bond A · Short bond50.0%

modDur 1.8

conv 4.0

Market value

$50,000.00

Bond B · Medium bond30.0%

modDur 5.2

conv 25.0

Market value

$30,000.00

Bond C · Long bond20.0%

modDur 9.0

conv 90.0

Market value

$20,000.00

Portfolio composition (by market value)

A: 50.0%B: 30.0%C: 20.0%

Parallel yield shock+100 bps

−300 bps+300 bps

Portfolio MV

$100,000.00

Weighted mod. duration

4.26 yr

Weighted convexity

27.5

Est. % price impact

−4.12%

Estimated dollar impact

−$4,122.50

This portfolio behaves like a bond with approximately 4.26 modified duration and convexity 27.5. The long bond (C) is only 20% of value but contributes most of the duration.

05Mastery

Summary and mastery check

Try itLesson 3.3 mastery check

Pass with 8 of 11 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
What does the law of one price say?
02
Does the law of one price require supply-demand equilibrium?
03
A coupon bond is more expensive than its matching STRIPS portfolio. What is the idealized arbitrage direction?
04
Why can short-sale restrictions weaken arbitrage?
05
What does it mean if an overdetermined fixed-income pricing system has no solution?
06
If yield rises, what usually happens to a normal bond price?
07
What is Macaulay duration?
08
What happens to duration when coupon rate rises, all else equal?
09
What does modified duration approximate?
10
Why is convexity useful?
11
A portfolio has more weight in long-duration bonds. What happens to portfolio interest-rate sensitivity?

Lesson summary

1Market prices are informative but not automatically correct — they are current sentiment, not truth.
2Identical future cash flows should have identical prices. That is the law of one price.
3The law of one price does not require a full equilibrium model — only that someone prefers more money to less.
4Arbitrage means buying the cheap cash-flow stream and selling the expensive one: no money down, no future net obligation, positive value today.
5Short-selling restrictions and transaction costs can prevent arbitrage from closing price gaps.
6Multiple coupon bonds create an overdetermined system of equations. No solution can mean mispricing.
7Bond prices and yields move inversely.
8Macaulay duration is the present-value-weighted average time to receive cash flows.
9Modified duration approximates the percentage price change for a small yield move.
10Convexity captures curvature and improves approximations for larger yield moves.
11Portfolio duration and convexity are value-weighted averages of component bonds.

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