Finance-Theory · Unit 7 · Video 5 · Interactive Practice

Duration: How Far a Bond's Price Moves When Its Yield Moves

IKey Formulas

Formula	Name	What you need
$D_m = \sum_{k=1}^{T} k \cdot \omega_k, \quad \omega_k = \dfrac{C_k/(1+y)^k}{P}$	Macaulay duration: the present-value-weighted average payment date	Cash flows $C_k$ , yield $y$ per period, price $P = \sum_k C_k/(1+y)^k$ ; the answer is in periods
$\dfrac{1}{P}\dfrac{\partial P}{\partial y} = -\dfrac{D_m}{1+y} = -D_m^*$	Modified duration: the percentage fall in price per unit rise in yield	$D_m$ in periods and $y$ per period
$\Delta P \approx -D_m^* \cdot P \cdot \Delta y$	Price change along the tangent; $D_m^* \cdot P$ is the price risk	$D_m^*$ and $\Delta y$ in the same time unit
$\text{Annual } D_m = \dfrac{\sum_k k \cdot \omega_k}{q}, \quad \text{Annual } D_m^* = \dfrac{\text{Annual } D_m}{1 + y/q}$	Converting periods to years	$q$ payments a year; $y$ the quoted annual yield, so $y/q$ per period

Key Insight: duration is a time. The four-year 7% T-note has $D = 7.13$ half-years, about $3.57$ years, because about 79% of its value arrives at period 8. Its modified duration, $7.13/1.03 = 6.92$ half-years, says a rise of $0.1\%$ in the half-year yield cuts its price by about $6.92 \times 0.1\% \approx 0.69\%$ .

IIThe Balance Point of the Present Values

Macaulay duration is where the time axis balances when each payment's present value stands on it as a bar.

Annual coupon rate 7.0%

💡 Discounting at exactly 3% per half-year, the eight present values sum to $103.51 rather than the quoted $103.50 that the video's table shows as its total, and period 3's $k \cdot \text{PV}$ comes to $9.61$ where the table shows $9.60$ ; either way $D = 7.13$ half-years.

IIIDuration Is the Slope of the Price–Yield Curve

At today's yield the tangent has slope $-D_m^* \cdot P$ ; the convex curve bends above that line.

Δy per half-year +10 bp

IVSame Time Unit for $D$ and $\Delta y$

Pair a duration with a yield change measured in the same time unit, or the price move comes out doubled.

Rise in the quoted annual yield 10 bp

VQuiz Questions

Problem 1 · Macaulay Duration of a Two-Period Bond

Given: a bond pays $10 at the end of period 1 and $110 at the end of period 2, and the yield is $y = 10\%$ per period. Find its Macaulay duration $D_m$ in periods.

✅ Correct! About 91% of the value arrives at period 2, so the balance point sits just short of the maturity.

❌ Not quite.

1.50

weights the two dates equally. The weights are present values: $90.91 of the $100 price arrives at period 2.

❌ Not quite.

1.74 = 1.91/1.10

is the modified duration. The question asks for Macaulay duration, before dividing by

1+y

❌ Not quite. Only a zero-coupon bond has duration equal to its maturity. The $10 coupon at period 1 pulls the balance point earlier.

❌ Not quite. Weight each date by its present value's share of the price.

Show solution

Present values at $10\%$ per period:

\text{PV}_1 = \frac{10}{1.10} = 9.09, \qquad \text{PV}_2 = \frac{110}{1.10^2} = 90.91, \qquad P = 9.09 + 90.91 = 100.00

Weights $\omega_1 = 9.09/100 = 0.0909$ and $\omega_2 = 90.91/100 = 0.9091$ , which add to 1. Then

D_m = 1 \times 0.0909 + 2 \times 0.9091 = 0.0909 + 1.8182 = 1.9091 \approx 1.91 \text{ periods.}

Problem 2 · The T-Note's Annual Modified Duration

Given: the video's four-year 7% T-note pays semi-annually, has quoted yield $6\%$ ( $3\%$ per half-year), and has Macaulay duration $D = 7.13$ half-years. Find its annual modified duration.

✅ Correct!

7.13/2.06 = 3.46

years, which is the half-year modified duration

6.92

divided by

q = 2

❌ Not quite.

7.13/2 = 3.57

is the annual Macaulay duration. Modified duration still divides by

1 + y/q = 1.03

❌ Not quite.

7.13/1.03 = 6.92

is correct but it is in half-years, not years. Divide by

q = 2

❌ Not quite.

7.13/(2 \times 1.06) = 3.36

divides by one plus the annual yield. The per-period yield is

y/q = 3\%

, so divide by

1.03

❌ Not quite. Convert the periods to years and divide by

1 + y/q

Show solution

With $q = 2$ payments a year and quoted yield $y = 0.06$ :

\text{Annual } D_m = \frac{7.13}{2} \approx 3.57 \text{ years}, \qquad \text{Annual } D_m^* = \frac{7.13/2}{1 + 0.06/2} = \frac{7.13}{2.06} = 3.46 \text{ years.}

The same number comes from the half-year modified duration: $6.92/2 = 3.46$ years.

Problem 3 · A Zero-Coupon Bond Takes a 20 bp Rise

Given: a 10-year zero-coupon bond, annual compounding, yields $y = 5\%$ . Its yield rises 20 basis points, to $5.20\%$ . Find its modified duration and the approximate percentage change in its price.

Modified duration $D_m^*$

Approximate price change

✅ Correct!

D_m^* = 10/1.05 = 9.52

years, so

\Delta P/P \approx -9.52 \times 0.20\% = -1.90\%

❌ Not quite.

10

years is the Macaulay duration, which for a zero equals its maturity. Modified duration divides it by

1 + y = 1.05

❌ Check the modified duration. Divide the Macaulay duration by

1 + y

: neither multiply nor subtract the yield.

❌ Not quite.

-10 \times 0.20\%

uses Macaulay duration. The percentage price change is set by the modified duration

9.52

❌ Check the size of $\Delta y$ . 20 basis points is

0.20\% = 0.002

, not

0.02

❌ Check the sign. Price and yield move in opposite directions: a rise in yield lowers the price.

❌ Check the price change. Use

\Delta P/P \approx -D_m^* \cdot \Delta y

with

\Delta y = 0.002

Show solution

A zero-coupon bond has a single payment, so $D_m = T = 10$ years (annual periods). Then

D_m^* = \frac{D_m}{1+y} = \frac{10}{1.05} = 9.52 \text{ years}

\frac{\Delta P}{P} \approx -D_m^* \cdot \Delta y = -9.52 \times 0.002 = -0.0190 = -1.90\%.

Repricing exactly, $\left(\tfrac{1.05}{1.052}\right)^{10} - 1 = -1.88\%$ : the price falls slightly less than the tangent says, because the price–yield curve is convex.

Problem 4 · Duration of a Two-Bond Portfolio

Given: a portfolio holds $40,000 (market value) of bond A with Macaulay duration 2 years and $60,000 of bond B with Macaulay duration 7 years, both priced at the same annual yield. Find the portfolio's Macaulay duration.

✅ Correct!

0.4 \times 2 + 0.6 \times 7 = 5.0

years: the portfolio's duration is the value-weighted average of its bonds' durations.

❌ Not quite.

0.6 \times 2 + 0.4 \times 7 = 4.0

swaps the weights. Bond B is 60% of the value.

❌ Not quite.

4.5

weights the two bonds equally, but the weights are shares of the portfolio's value,

0.4

and

0.6

❌ Not quite. Duration is an average date, not a total: combining bonds cannot push it beyond the longest bond's 7 years.

❌ Not quite. Weight each bond's duration by its share of the portfolio's value.

Show solution

Collect the portfolio's cash flows in each period and weight each date by its present value over the portfolio's value $V = P_A + P_B$ . Because both bonds are discounted at the same yield, the sum splits bond by bond:

D_{\text{port}} = \frac{\sum_k k \cdot \text{PV}_{A,k} + \sum_k k \cdot \text{PV}_{B,k}}{V} = \frac{P_A D_A + P_B D_B}{V}.

With weights $40{,}000/100{,}000 = 0.4$ and $60{,}000/100{,}000 = 0.6$ :

D_{\text{port}} = 0.4 \times 2 + 0.6 \times 7 = 0.8 + 4.2 = 5.0 \text{ years.}

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Duration: How Far a Bond's Price Moves When Its Yield Moves

IKey Formulas

IIThe Balance Point of the Present Values

IIIDuration Is the Slope of the Price–Yield Curve

IVSame Time Unit for DDD and Δy\Delta yΔy

VQuiz Questions

IVSame Time Unit for $D$ and $\Delta y$