How to Calculate Bond Duration and Price Sensitivity: A Step-by-Step Interview Walkthrough

Short answer: price the bond by discounting every coupon and the principal at the market yield, weight each present value by the year it arrives to get Macaulay duration, divide by one plus the yield to get modified duration, then multiply that by the yield move to estimate the price change. Five steps, no calculator tricks required — and the whole sequence fits inside a two-minute interview answer.

This walkthrough uses a single five-year corporate bond and carries the same numbers through every step, so nothing has to be re-derived. If you would rather attempt it before reading the answer, the identical setup is available as the practice case Bond Basics: Duration, Yield, Price. For the conceptual background — why the price moves at all — the companion piece on discounting logic covers the same present value machinery applied to a company.

The setup

Two of these inputs do different jobs and are constantly confused. The coupon rate sizes the cash flow: 5.0% of $1,000 is $50 a year, and that never changes. The market yield prices that cash flow: it is the return investors currently demand for this credit and this maturity, and it moves daily. Discounting the coupons at the coupon rate is the single most common arithmetic error in this question, and it always produces exactly $1,000, which should be your tell that something went wrong.

Step 1: Price the bond

Price = Sum of [ Coupon / (1 + y)^t ] for t = 1 to N, plus Face Value / (1 + y)^N

The discounting table

Note the year 5 cash flow: $1,050, not $50. The final coupon and the principal repayment arrive together. Dropping the $1,000 principal is the second classic mistake, and it is fatal — it removes 78% of the bond's value and destroys the duration calculation that follows.

The shortcut for doing this without a spreadsheet

Discounting five cash flows by hand is slow. Split the bond into an annuity plus a lump sum instead:

Price = Coupon × Annuity Factor + Face Value × Discount Factor

Annuity Factor = (1 − (1 + y)^−N) / y = (1 − 0.7473) / 0.06 = 4.2124

Price = $50 × 4.2124 + $1,000 × 0.7473 = $210.62 + $747.26 = $957.88

If you can hold 1.06^5 ≈ 1.338 in your head, everything else follows. The reciprocal is 0.747, the annuity factor is (1 − 0.747) / 0.06 ≈ 4.21, and you are one multiplication from the answer. This kind of decomposition is worth practising because it recurs constantly — the same annuity-plus-lump-sum split appears whenever you value a fixed stream, including in terminal value calculations.

Sanity check before moving on

Coupon rate 5.0% is below the market yield of 6.0%, so the bond must trade at a discount to $1,000. It does: $957.88. If your answer came out above par, you have discounted at the wrong rate. Doing this check out loud in an interview costs three seconds and demonstrates that you understand the result rather than just producing it.

Step 2: Calculate Macaulay duration

Macaulay Duration = Sum of [ t × PV(Cash Flow at t) ] / Price

Weighting each present value by its year

Macaulay Duration = $4,343.64 / $957.88 = 4.53 years

Reuse the present values from Step 1 rather than recomputing them. The whole calculation is one extra column.

Why year 5 dominates, and what that tells you

Year 5 contributes $3,923 of the $4,344 total — roughly 90% of the weighting. That is not an accident of these particular numbers: it is because the principal repayment is bundled into the final cash flow. Understanding this makes three follow-up questions answerable instantly:

• Why is duration 4.53 and not 5? The four intermediate coupons return cash early and pull the weighted average forward. Without them the answer would be exactly 5.
• What is the duration of a zero-coupon bond? Exactly its maturity, because 100% of present value sits in the final year. A five-year zero has a Macaulay duration of 5.00 years.
• What happens if the coupon is higher? Duration falls. A 10% coupon on the same bond returns more cash early, shifting weight forward and reducing rate sensitivity.

Step 3: Convert to modified duration

Modified Duration = Macaulay Duration / (1 + y) = 4.53 / 1.06 = 4.28

One division, but it changes what the number means. Macaulay duration is measured in years and answers "how long until I am repaid, on average?". Modified duration has no units and answers "what percentage of value do I lose per percentage point of yield increase?". When an interviewer asks how much the price moves, the answer is 4.28 — and candidates who say 4.53 have answered a question that was not asked.

Step 4: Estimate the price change

Estimated % Price Change = − Modified Duration × Change in Yield

For a 100 basis point rise in yields, a change of +1.0% (+0.010):

Estimated % Price Change = −4.28 × 0.010 = −4.28%

Estimated $ Price Change = −4.28% × $957.88 = −$40.98

Estimated New Price = $957.88 − $40.98 = $916.90

Scaling to DV01

Desks rarely think in whole percentage points. DV01 — the dollar value of one basis point — is the same sensitivity divided by 100, or roughly $0.41 per bond here. It is the unit hedges are sized in, because it lets a trader offset the risk of one instrument with a completely different one. If you can produce a DV01 unprompted after giving the duration answer, you are speaking the language of the desk rather than the textbook.

Step 5: Check the estimate against an actual reprice

Rerun Step 1 at 7.0% and at 5.0% and compare:

Note the useful coincidence in the first row: at a 5.0% yield the coupon rate and the yield are identical, so the bond prices at exactly $1,000. That gives you a free check on your arithmetic, and it is worth pointing out in an interview.

The convexity adjustment

The duration estimate is wrong by about $1.10 in both directions, and always in the holder's favour. That gap is convexity, and it can be quantified directly from the three prices you already have:

Convexity = (P(y − d) + P(y + d) − 2 × P(y)) / (P(y) × d^2)

Convexity = ($1,000.00 + $918.00 − 2 × $957.88) / ($957.88 × 0.010^2) = $2.24 / 0.0958 = 23.4

Convexity Adjustment = 0.5 × 23.4 × 0.010^2 = 0.117% of price = +$1.12

Duration plus convexity gives −$40.98 + $1.12 = −$39.86 against an actual move of −$39.88 — two cents out on a $958 bond. That is the point worth making: duration alone is a linear approximation to a curved relationship, and the curvature is systematically favourable on a plain vanilla bond. On a callable bond or a mortgage-backed security the curvature runs the other way, which is why those instruments must offer a higher yield to compensate.

The semi-annual variant

Most corporate and government bonds pay twice a year, and interviewers sometimes switch to that mid-question to see whether you understood the mechanics or memorised a template. Three adjustments handle it:

Halve the coupon, halve the yield, double the periods. The duration that comes out is measured in half-year periods, so divide by two to express it in years. And modified duration divides by (1 + y/2) rather than (1 + y). The price barely moves — around $957.35 versus $957.88 — because receiving coupons slightly earlier is worth very little. Saying that out loud, rather than grinding through ten periods, is usually the better answer.

Three "yields" that are not the same thing

A frequent trap is the interviewer switching yield definitions mid-question. Three measures get loosely called "the yield", and only one of them prices the bond:

Current yield sits between the other two on a discount bond, and that is not a coincidence. It captures the coupon income but ignores the $42 capital gain the holder earns as the price pulls back to par over the remaining five years. Yield to maturity captures both, which is why it is the only one of the three that can be used as a discount rate or compared meaningfully against another bond. On a premium bond the ordering reverses: current yield sits above yield to maturity, because the holder is absorbing a slow capital loss down to par.

Working backwards from a price to a yield

Sometimes the question is inverted: here is the price, what is the yield? There is no closed-form solution for yield to maturity, so the honest answer is that a spreadsheet solves it iteratively — but you can bracket it fast enough to sound credible. Start from the coupon rate and use the discount-or-premium rule to establish direction: a bond priced at $958 against $1,000 par must yield above its 5% coupon. Then estimate the size of the gap using the capital gain: roughly $42 of gain spread over five years is about $8.40 a year, on a $958 investment that is roughly 0.9% a year on top of the 5.2% current yield, landing you at about 6.1%. The true answer is 6.0%.

That approximation is close enough to quote, and the reasoning behind it — income plus amortised capital gain — is what the interviewer is actually listening for. It also makes clear why yield and price are two ways of saying the same thing: fix one and the other is fully determined.

Mistakes that cost candidates the question

How to deliver this in sixty seconds

Interviewers are testing structure as much as arithmetic. A clean verbal answer runs: "I'd price the bond by discounting the five coupons and the principal at the 6% market yield, which gives about $958 — a discount, because the 5% coupon is below market. Macaulay duration is the present-value-weighted average time to repayment, around 4.5 years; dividing by 1.06 gives a modified duration of about 4.3, so a 100 basis point rise costs roughly 4.3%, or $41. That's a linear approximation though — convexity means the actual loss is a bit smaller, around $40, and the gain if yields fell would be a bit larger than 4.3%."

That is the complete answer, including the follow-up, in under a minute. State the assumption, give the number, then flag the limitation.

Where these numbers show up next

The market yield you just used to price a bond is the same figure that belongs in the cost of debt line of a WACC calculation — not the coupon on debt issued in a different rate environment. The mechanics are worked through in the step-by-step WACC walkthrough and practised in WACC with Leverage, where unlevering and relevering beta sits alongside the debt cost.

On the leveraged finance side, the interest rate assumption drives every line of an LBO Debt Schedule and sets the ceiling in Debt Capacity, because a borrower's capacity is ultimately a function of what it costs to service the debt. The instrument choices themselves — fixed versus floating, senior versus mezzanine, call protection — are covered in LBO debt structures explained. And on the issuance side, the rate environment is what opens and closes the windows described in the IPO process guide.

Work the arithmetic yourself once and it stops being a formula to remember. The full case, with the follow-up questions on zero-coupon duration, premium bonds and hedging, is at Bond Basics: Duration, Yield, Price.