Actuarial Mathematics for Modelling · Duration, convexity and immunisation

Volatility and Modified Duration of Cash Flows

Updated 11 October 2026 · Fact-checked

Volatility is the proportional fall in present value for a small rise in interest rate: ν(i) = −V′(i) ÷ V(i). For an effective annual rate i it equals the discounted mean term divided by (1 + i). To estimate a price change, use ΔV ≈ −V × ν × Δi.

Understand Volatility and Modified Duration

Start with a set of future payments. Their present value V depends on the interest rate i. If i rises, each payment is discounted more heavily, so V falls. The question is: by how much?

The size of the fall in rupees depends on how big V is. So we measure the fall as a proportion of V. This is the volatility, also called modified duration. It is defined as ν(i) = −(1 ÷ V) × dV/di. The minus sign makes it a positive number for a normal set of positive cash flows.

Now link it to Macaulay duration. If V = Σ c_t v^t with v = 1/(1 + i), then dV/di = −Σ t c_t v^(t+1). Divide by −V and you get ν = v × Σ t c_t v^t ÷ V. The fraction is the discounted mean term (Macaulay duration), so ν = v × DMT = DMT ÷ (1 + i).

If you work with the force of interest δ instead, V = Σ c_t e^(−δt). Then −(1 ÷ V) × dV/dδ equals the DMT exactly. So the DMT is the volatility with respect to δ, and the modified duration is the volatility with respect to the effective annual rate i.

Volatility gives a first-order (straight-line) estimate. For small changes in i it works well. For larger changes the true price curve bends, and you need convexity to improve the estimate.

Key rules to remember

Present value: V(i) = Σ c_t × v^t, where v = 1 ÷ (1 + i); c_t is the cash flow at time t. Use the rate given in the question.
Volatility (modified duration): ν(i) = −V′(i) ÷ V(i) = −(1 ÷ V) × dV/di; Proportional sensitivity of V to i. Positive for positive cash flows.
Discounted mean term (Macaulay duration): DMT = Σ t × c_t × v^t ÷ Σ c_t × v^t; Time-weighted average of payment times, with weights equal to present values.
Link between volatility and DMT: ν(i) = v × DMT = DMT ÷ (1 + i); For an effective annual rate i. With nominal rate i^(m) convertible m-thly, divide the DMT by (1 + i^(m) ÷ m) when the change is in i^(m).
Sensitivity to force of interest: −(1 ÷ V) × dV/dδ = DMT; Here V is written as a function of δ, with V = Σ c_t e^(−δt).
First-order price change estimate: ΔV ≈ −V × ν × Δi, or ΔV ÷ V ≈ −ν × Δi; Valid for small Δi. Estimate overstates the fall and understates the rise when convexity is positive.
Second-order estimate: ΔV ÷ V ≈ −ν × Δi + ½ × C × (Δi)², where C = V″(i) ÷ V(i); C is convexity. Use it when the question gives it or asks for a better estimate.

How to solve Volatility and Modified Duration questions

Use this method for any question on volatility, modified duration or estimating a price change.

1Write down the cash flows c_t and the times t. Note whether the rate is effective annual, nominal or a force of interest.
2Calculate the present value V at the given rate. Keep at least five or six decimals in the discount factors.
3Calculate Σ t × c_t × v^t, using the same discount factors.
4Divide by V to get the DMT. This is the Macaulay duration.
5Convert to volatility: ν = DMT ÷ (1 + i) for an effective annual rate. If the question uses δ, the DMT is already the sensitivity.
6Find the change in rate Δi, with the correct sign (increase is positive).
7Estimate the change using ΔV ≈ −V × ν × Δi. Add the convexity term only if asked or if the change is large.
8State the answer with sign, units and meaning: a fall or rise in value, in rupees or percent.

Quickest way: Shortcut from DMT to price change

When to use it: Use when the question gives you V and the DMT, or when a multiple-choice question asks for an approximate price change.

Compute ν = DMT ÷ (1 + i). Do not forget this division.
Percentage change in V ≈ −ν × Δi. For example, ν = 7.5 and Δi = +0.5% gives about −3.75%.
Multiply by V only if the answer is wanted in rupees.
Check the sign: a rate rise must lower V.
Rule out options that use DMT directly instead of ν, since they will be too large by a factor of (1 + i).

Common mistakes in Volatility and Modified Duration

Using the Macaulay duration as the volatility without dividing by (1 + i).
The two terms are often treated as the same thing, and both are measured in years.
Fix: For an effective annual rate, always compute ν = DMT ÷ (1 + i). Only the sensitivity to δ equals the DMT.
Forgetting the minus sign and predicting that value rises when the rate rises.
Volatility is stated as a positive number, so the sign is dropped in the estimate.
Fix: Write ΔV ≈ −V × ν × Δi each time. Then check that an increase in i gives a fall in V.
Entering Δi as a whole number, such as 0.5 instead of 0.005.
Rate changes are quoted in percent, but the formula needs a decimal.
Fix: Convert every rate to a decimal before substituting. A change of 0.5% is 0.005.
Using present-value weights incorrectly in the DMT, for example weighting times by the cash flows alone.
Students remember a simple average of times and forget that the weights are discounted values.
Fix: Compute each term as t × c_t × v^t and divide by V. Check that the DMT lies between the earliest and latest payment times.
Treating the first-order estimate as exact.
The formula looks like a precise relationship.
Fix: Say it is an approximation for small changes in i. For a large change, recompute V at the new rate or add the convexity term.
Using the same duration for a nominal rate without checking which rate changes.
The conversion factor depends on which rate is being shifted.
Fix: Identify the rate that moves. For i^(m) changing, divide the DMT by (1 + i^(m) ÷ m). State your assumption in the answer.

Worked examples

Example 1

A portfolio pays ₹10,000 at the end of year 2 and ₹10,000 at the end of year 4. The effective annual interest rate is 5%. (a) Calculate the present value, the discounted mean term and the volatility. (b) Estimate the change in present value if the rate rises to 5.5%, and compare it with the exact change.

Show the solution

v = 1 ÷ 1.05 = 0.952381. v² = 0.907029. v⁴ = 0.822702.
V = 10,000 × (0.907029 + 0.822702) = 10,000 × 1.729731 = ₹17,297.31.
Σ t c_t v^t = 2 × 9,070.29 + 4 × 8,227.02 = 18,140.58 + 32,908.08 = 51,048.66.
DMT = 51,048.66 ÷ 17,297.31 = 2.9512 years (approximately).
Volatility ν = 2.9512 ÷ 1.05 = 2.8107.
Δi = +0.005. ΔV ≈ −17,297.31 × 2.8107 × 0.005 = −₹243.09 (approximately).
Exact check: at 5.5%, v² = 1 ÷ 1.113025 = 0.898452 and v⁴ = 0.807216, so V = 10,000 × 1.705668 = ₹17,056.68.
Exact change = 17,056.68 − 17,297.31 = −₹240.63.

Answer: V = ₹17,297.31, DMT ≈ 2.95 years, volatility ≈ 2.81. The estimated change is about −₹243.09, against an exact change of about −₹240.63. The estimate slightly overstates the fall, as expected with positive convexity.

Example 2

An insurer's liability cash flows have present value ₹50,00,000 at an effective annual rate of 6%. Their discounted mean term is 8 years. (a) Find the volatility. (b) Estimate the new present value if the rate falls to 5.5%.

Show the solution

Volatility ν = DMT ÷ (1 + i) = 8 ÷ 1.06 = 7.5472.
Δi = 5.5% − 6% = −0.005.
Proportional change ≈ −ν × Δi = −7.5472 × (−0.005) = +0.037736, which is about +3.77%.
ΔV ≈ 50,00,000 × 0.037736 = +₹1,88,679 (approximately).
New V ≈ 50,00,000 + 1,88,679 = ₹51,88,679.

Answer: Volatility ≈ 7.55. The liability value rises by about ₹1,88,679 to about ₹51,88,679, a rise of about 3.77%. This is a first-order estimate, so the exact value will be slightly higher because of positive convexity.

Exam tips

Check the wording. If a question says modified duration, volatility or effective duration, work out whether it expects ν = DMT ÷ (1 + i) and say so in your answer.
In written answers, show the formula ν = −V′ ÷ V first, then the numbers. Method marks are given for the formula and for the working, not only for the result.
In computer-based questions, build the cash flow table with discount factors, then V, then Σ t c_t v^t. Keep the discount factors unrounded in cells so the DMT is accurate.
State that the estimate is a first-order approximation valid for small changes in interest rate, and mention convexity as the refinement. This is often worth a mark.
In multiple-choice questions, estimate first with ν × Δi. It usually removes two options in seconds.

Practice questions from Duration, convexity and immunisation

Volatility and Modified Duration in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Volatility and Modified Duration: frequently asked questions

What is the difference between modified duration and Macaulay duration?

Macaulay duration (the discounted mean term) is the present-value-weighted average time of the payments, measured in years. Modified duration, or volatility, is the DMT divided by (1 + i) for an effective annual rate. It measures the proportional change in value per unit change in the rate.

How do I estimate a price change using volatility?

Use ΔV ≈ −V × ν × Δi. Convert Δi to a decimal and keep its sign. A rate rise gives a fall in value, and a rate fall gives a rise.

Is volatility the same as the standard deviation of returns?

No. In this topic, volatility means the proportional sensitivity of present value to the interest rate. It is not a statistical measure of variability, so do not confuse it with the term used in option pricing.

Why does the volatility estimate become less accurate for large rate changes?

The relationship between V and i is curved, not a straight line. Volatility gives the slope at the starting rate. For large changes, add the convexity term or recompute V at the new rate.