Bond Duration Calculation And Sensitivity Workshop

Why Bond Duration Problems Require Careful Setup

A bond-duration calculation is not just a weighted average. The price depends on the bond’s cash flows and yield, while the duration measure depends on the timing convention and the way that yield is quoted. A portfolio adds another step: component durations must be weighted by market value, not by face amount or by the number of bonds held.

The method in this tutorial is aligned with the SOA Exam FM (Financial Mathematics) syllabus. It assumes fixed promised cash flows, a yield that is unchanged across the bond’s payment dates, and a small parallel change in yield when using duration to estimate a price change. It does not model a change in expected cash flows or a change in the shape of the yield curve.

Definition: For fixed cash flows $C_k$ paid at times $t_k$, the bond price $P$ is the sum of their present values at the bond’s yield. Macaulay duration $D_{\mathrm{Mac}}$ is the present-value-weighted average payment time. Under periodic compounding, modified duration $D_{\mathrm{mod}}$ measures the approximate proportional price sensitivity to a change in the annual quoted yield.

Price, Macaulay Duration, and Modified Duration

Suppose payments occur every $1/m$ years and the annual nominal yield convertible $m$ times per year is $i^{(m)}$. The yield per payment period is $j=i^{(m)}/m$. If payment $C_k$ occurs after $k$ periods, its payment time is $t_k=k/m$ years. Discount every payment using the periodic yield $j$:

P=\sum_{k=1}^{n}C_k(1+j)^{-k}, \qquad D_{\mathrm{Mac}}=\frac{\sum_{k=1}^{n}t_k C_k(1+j)^{-k}}{P}, \qquad D_{\mathrm{mod}}=\frac{D_{\mathrm{Mac}}}{1+j}.

The duration is measured in years because each $t_k$ is in years. When payments are annual and the yield is an effective annual rate $i$, take $m=1$, so $j=i$. When the yield is nominal with a different payment frequency, first convert it to the rate per payment period; do not discount with the unconverted annual nominal rate.

Why Modified Duration Estimates Price Sensitivity

The duration formula follows by differentiating the present-value price with respect to the quoted annual nominal yield, while holding the cash flows and payment dates fixed. Because $j=i^{(m)}/m$, the derivative of each discount factor includes both the payment-period count and the conversion from annual yield to periodic yield:

\frac{dP}{d i^{(m)}}= -\sum_{k=1}^{n}\frac{k}{m}C_k(1+j)^{-k-1} = -\frac{1}{1+j}\sum_{k=1}^{n}t_k C_k(1+j)^{-k}.

The sum on the right is $P D_{\mathrm{Mac}}$, by the definition of Macaulay duration. Therefore,

\frac{1}{P}\frac{dP}{d i^{(m)}}=-\frac{D_{\mathrm{Mac}}}{1+j}=-D_{\mathrm{mod}}.

For a small yield change $\Delta i^{(m)}$, the first-order approximation is consequently

\frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta i^{(m)}, \qquad \Delta P\approx-PD_{\mathrm{mod}}\Delta i^{(m)}.

A yield increase gives a negative estimated price change; a yield decrease gives a positive one. This is a local estimate, not an exact repricing formula. Its accuracy is best for small yield changes; price curvature is not included here.

Worked Example: Annual Coupon Bond Price and Duration

A hypothetical bond has a face amount of $1,000, pays an annual coupon of 6% of face, and matures in three years. Its yield is 5% effective annually. Calculate its price, Macaulay duration, modified duration, and estimated dollar price change if the yield increases by 10 basis points. Assume all promised payments are made.

Method. List each cash flow, discount at the effective annual yield, and use the resulting present values both for price and for the duration weights. The yield change is $10/10{,}000=0.001$, not 0.01.

The annual coupon is $0.06(1{,}000)=\$60$. Cash flows are $60 at years 1 and 2, and $1,060 at year 3. Discounting at $i=0.05$ gives:

\begin{aligned} PV_1&=\frac{60}{1.05}=57.1429,\\ PV_2&=\frac{60}{1.05^2}=54.4218,\\ PV_3&=\frac{1{,}060}{1.05^3}=915.6679. \end{aligned}

Thus the price is $P=57.1429+54.4218+915.6679=\$1{,}027.2326$. The time-weighted present-value sum is

$$ 1(57.1429)+2(54.4218)+3(915.6679)=2{,}912.9902. $$

Therefore $D_{\mathrm{Mac}}=2{,}912.9902/1{,}027.2326=2.8358$ years, and $D_{\mathrm{mod}}=2.8358/1.05=2.7008$ years. For the 10-basis-point increase, the estimated dollar change is

\Delta P\approx-(1{,}027.2326)(2.7008)(0.001)=-\$2.77.

The estimated new price is about $\$1{,}024.46$. This is an estimate rather than an exact repricing. The calculation avoids the common mistake of applying the 5% yield directly to duration without converting Macaulay duration to modified duration.

Converting a Nominal Yield Before Calculating Duration

When the yield is nominal convertible $m$ times per year, both the discounting and modified-duration conversion must use the periodic yield $j=i^{(m)}/m$. Payment times in the Macaulay calculation remain expressed in years. Keeping those two units distinct prevents a frequent error: using period numbers as though they were years.

Worked Example: Semiannual Payments and a Nominal Yield

A hypothetical bond has a face amount of $5,000, pays an 8% annual coupon rate in semiannual installments, and matures in two years. Its yield is 6% nominal convertible semiannually. Calculate the price and both duration measures, then estimate the price change for a 25-basis-point increase in the quoted nominal yield. Assume fixed cash flows and semiannual compounding for the yield.

Method. Convert the quoted yield to a semiannual rate, discount four payments, and express payment times as 0.5, 1, 1.5, and 2 years. The coupon each half-year is $5{,}000(0.08/2)=\$200$; the final payment is $5,200. The periodic yield is $j=0.06/2=0.03$.

\begin{aligned} PV_1&=200(1.03)^{-1}=194.1748,\\ PV_2&=200(1.03)^{-2}=188.5192,\\ PV_3&=200(1.03)^{-3}=183.0283,\\ PV_4&=5{,}200(1.03)^{-4}=4{,}620.1327. \end{aligned}

Adding the present values gives $P=\$5{,}185.8549$. Weighting them by payment times in years gives

$$ 0.5(194.1748)+1(188.5192)+1.5(183.0283)+2(4{,}620.1327)=9{,}800.4144. $$

Hence $D_{\mathrm{Mac}}=9{,}800.4144/5{,}185.8549=1.8898$ years. The modified duration for a change in the annual nominal yield is $D_{\mathrm{mod}}=1.8898/(1.03)=1.8348$ years. A 25-basis-point increase is $\Delta i^{(2)}=0.0025$, so

\Delta P\approx-(5{,}185.8549)(1.8348)(0.0025)=-\$23.78.

The estimated price is about $5,162.07. The key unit check is that the 25-basis-point change is in the annual quoted yield, while the denominator in the duration conversion uses the semiannual periodic yield $0.03$.

Portfolio Duration Uses Market-Value Weights

For a portfolio of bonds with component market values $P_r$, total market value $P_{\mathrm{port}}=\sum_r P_r$, and component modified durations $D_{\mathrm{mod},r}$, the portfolio modified duration for a common small yield shift is the market-value-weighted average:

D_{\mathrm{mod,port}}= \sum_r\frac{P_r}{P_{\mathrm{port}}}D_{\mathrm{mod},r}.

To justify the weights, apply the duration estimate to each component: $\Delta P_r\approx-P_rD_{\mathrm{mod},r}\Delta y$. Adding these changes gives $\Delta P_{\mathrm{port}}\approx-\Delta y\sum_r P_rD_{\mathrm{mod},r}$. Dividing by total portfolio value gives the stated weighted average. This result assumes the same small absolute yield shift applies to each component and that each component’s cash flows remain fixed. For a yield shift that differs by bond or maturity, one portfolio duration and one common-shift estimate may not describe the change.

Worked Example: Market-Value-Weighted Portfolio Sensitivity

A hypothetical portfolio holds two fixed-cash-flow bond positions. Position A has market value $12,000 and modified duration 1.5 years; position B has market value $28,000 and modified duration 4.0 years. Estimate the portfolio’s dollar value change if yields decline in parallel by 10 basis points. Assume the stated durations apply to this shift.

Method. First calculate the market-value weights. Total value is $12{,}000+28{,}000=\$40{,}000$, so the weights are $12{,}000/40{,}000=0.30$ and $28{,}000/40{,}000=0.70$. The portfolio modified duration is

D_{\mathrm{mod,port}}=(0.30)(1.5)+(0.70)(4.0)=0.45+2.80=3.25\text{ years}.

A 10-basis-point decline means $\Delta y=-0.001$. Therefore,

\Delta P_{\mathrm{port}}\approx-(40{,}000)(3.25)(-0.001)=+\$130.00.

The positive sign is consistent with the inverse relationship between bond prices and yields. The common mistake is to lose a decimal place in converting basis points: using $0.0001$ for a 10-basis-point change would incorrectly produce a $13 estimate. The correct shift is $10/10{,}000=0.001$.

Common Traps and a Practical Checklist

Bond-duration questions often combine several conventions in a single calculation. Before substituting numbers, identify what the quoted yield means, when each payment occurs, and whether the requested sensitivity is proportional or in dollars. In particular:

Use present values as weights. Macaulay duration weights each payment time by its present value, not by its undiscounted cash flow.
Convert nominal yields first. For nominal yield $i^{(m)}$, use $i^{(m)}/m$ per payment period when payment frequency is $m$.
Keep years and periods distinct. A payment $k$ periods away occurs at $k/m$ years, not $k$ years.
Use modified duration for yield sensitivity. Macaulay duration alone is not the proportional price sensitivity under periodic compounding.
Convert basis points correctly. One basis point is $0.0001$ in rate form; 25 basis points is $0.0025$.
Weight portfolio duration by market value. Face amounts are not the weights in the portfolio price-change calculation.

A useful final check is sign and scale. If a fixed-cash-flow bond’s yield rises, its estimated price change should be negative. If the portfolio is worth tens of thousands of dollars, has duration of several years, and yields move by a few basis points, the estimated dollar change should be consistent with multiplying those quantities—not smaller by an unexplained factor of ten.

Key takeaway: Price each cash flow at the correct periodic yield, calculate duration using payment times in years, convert Macaulay duration to modified duration, and use market-value weights for portfolios. Then apply $\Delta P\approx-PD_{\mathrm{mod}}\Delta y$, with the yield change written as a decimal.

Check Your Understanding

For each problem, assume fixed promised cash flows and use the first-order duration estimate where requested.

A bond position is worth $25,000, has Macaulay duration 4.2 years, and yields 5% effective annually. Find modified duration and estimate the dollar price change for a 20-basis-point yield increase. (Answer: Modified duration $4.0$ years; estimated change $-\$200.00$.)
A portfolio has two positions: $18,000 with modified duration 1.5 years and $42,000 with modified duration 4.25 years. Find portfolio modified duration and estimate the dollar change for a parallel 10-basis-point yield decline. (Answer: Portfolio modified duration $3.425$ years; estimated change $+\$205.50$.)
A bond has present-value cash-flow amounts of $150 at year 1, $250 at year 2, and $600 at year 4. Its annual effective yield is 4%. Calculate its price, Macaulay duration, modified duration, and estimated price change for a 30-basis-point increase. (Answer: Price $\$1{,}000$; Macaulay duration $3.05$ years; modified duration $2.9327$ years; estimated change $-\$8.80$.)
A bond has Macaulay duration 5.4 years and a 10% nominal yield convertible semiannually. Its market value is $40,000. Estimate its price change for a 15-basis-point increase in that quoted yield. (Answer: Modified duration $5.1429$ years; estimated change $-\$308.57$.)
A portfolio contains positions worth $12,000 and $28,000, with modified durations 1.5 and 4.0 years, respectively. Estimate the dollar change for a parallel 10-basis-point yield decline. (Answer: Portfolio modified duration $3.25$ years; estimated change $+\$130.00$.)

What You'll Learn