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Actuarial Mathematics for Modelling · Duration, convexity and immunisation

Discounted Mean Term and Macaulay Duration Explained

Updated 11 October 2026 · Fact-checked

The discounted mean term (Macaulay duration) is the average time to receive a cash flow stream, where each time t is weighted by the present value of the payment at t. DMT = Σ t·v^t·C_t ÷ Σ v^t·C_t. Discount each cash flow, multiply by t, add up, then divide by the total present value.

Understand Discounted Mean Term and Macaulay Duration

Imagine a set of payments spread over several years. You want one number that says how long, on average, you wait for the money. A simple average of the payment times ignores the amounts and the discounting. The discounted mean term (DMT), also called Macaulay duration, fixes this.

Each payment time t gets a weight equal to the present value of the payment at that time, as a share of the total present value. A large payment, or an early payment that is barely discounted, gets more weight. The DMT is the weighted average of the times, measured in years.

The DMT depends on the interest rate you use to discount. A higher interest rate shrinks the present value of late payments more, so the DMT falls. A bond with big coupons has a shorter DMT than a zero-coupon bond of the same term, because more of its value arrives early.

The DMT is a time, not a percentage. For a single payment at time n, the DMT is exactly n. For any stream with several positive payments, the DMT lies between the first and last payment times.

Do not confuse it with volatility (modified duration). The DMT is a weighted average time. Volatility measures the proportional fall in present value for a small rise in interest rate. The two are linked, and the link is in the formulas below.

Key rules to remember

Discounted mean term (discrete cash flows)
DMT = Σ t·C_t·v^t ÷ Σ C_t·v^t, where v = 1 ÷ (1 + i)
C_t is the payment at time t. The denominator is the present value of the whole stream. Use the same i in both sums.
Discounted mean term (continuous payments)
DMT = ∫ t·ρ(t)·v^t dt ÷ ∫ ρ(t)·v^t dt
ρ(t) is the payment rate at time t. Use this form when the question gives a continuous payment rate.
Link with force of interest
DMT = −(1 ÷ PV) × d(PV)/dδ
Differentiating PV with respect to δ brings down −t from e^(−δt), so DMT is minus the proportional change in PV per unit change in δ.
Volatility and DMT
Volatility ν = −(1 ÷ PV) × d(PV)/di = DMT ÷ (1 + i)
This holds when i is the effective annual rate. Volatility is also called modified duration.
Zero-coupon bond
DMT = n
A single payment at time n has a DMT of n years.
Par bond shortcut
DMT = ä_n at rate i
Only valid for a bond with annual coupons, redeemed at par, with coupon rate equal to the yield i. Here ä_n is the annuity-due factor.
Level annuity-immediate
DMT = (Ia)_n ÷ a_n
For payments of 1 at times 1, 2, ..., n. Both functions are at the same rate i.

How to solve Discounted Mean Term and Macaulay Duration questions

Use this method for any DMT question on bonds, loans or general cash flows. A table keeps the working tidy and lets you earn method marks even if you slip on arithmetic.

  1. 1Write down the rate i to use (for example the bond's yield) and compute v = 1 ÷ (1 + i).
  2. 2List every cash flow C_t with its time t in years. For a bond, include each coupon and the redemption payment at the final date.
  3. 3Compute the present value of each payment, C_t·v^t, in a column.
  4. 4Add the present values to get the total PV. This is your denominator.
  5. 5Multiply each present value by its time t to get t·C_t·v^t, then add these up. This is your numerator.
  6. 6Divide the numerator by the denominator to get the DMT in years.
  7. 7Sense-check: the answer must lie between the first and last payment times. If you need volatility, divide by (1 + i).

Quickest way: Shortcuts for standard cash flow patterns

When to use it: Use when the question is a zero-coupon bond, a bond with coupon rate equal to yield, a level annuity, or a stream that is a simple mix of these.

  1. For one payment at time n, write DMT = n straight away.
  2. For a par bond with annual coupons and coupon rate equal to the yield, use DMT = ä_n at i.
  3. For a level annuity-immediate, use (Ia)_n ÷ a_n.
  4. For a portfolio of two streams, the DMT is the PV-weighted average of the individual DMTs. Find each PV and each DMT, then weight.
  5. Always check the shortcut's conditions. If the coupon rate differs from the yield, go back to the full table.

Common mistakes in Discounted Mean Term and Macaulay Duration

  • Weighting times by the cash flows instead of the present values

    Students compute Σ t·C_t ÷ Σ C_t, which is the undiscounted mean term.

    Fix: Always include v^t. Numerator is Σ t·C_t·v^t and the denominator is Σ C_t·v^t.

  • Forgetting the redemption payment in the final time period

    The table shows only coupons, so the last row is written as a coupon only.

    Fix: At maturity the cash flow is coupon plus redemption. Write this as one combined amount, for example 106, not 6.

  • Dividing by the face value instead of the total present value

    Students assume the bond price is 100.

    Fix: The denominator is the PV of the cash flows at the given i. It equals 100 only if coupon rate equals yield and redemption is at par.

  • Treating DMT and volatility as the same thing

    Both are called duration in some books, and they have close values.

    Fix: Volatility = DMT ÷ (1 + i) for an effective annual rate. Read the question to see which one is asked for.

  • Using the par bond shortcut when conditions are not met

    The result ä_n is quick, so students apply it to every bond.

    Fix: Check that coupons are annual, redemption is at par, and coupon rate equals the yield. Otherwise build the table.

  • Giving an answer outside the range of payment times

    An arithmetic slip in the numerator or denominator goes unnoticed.

    Fix: The DMT cannot be below the first payment time or above the last. Check this before moving on.

Worked examples

Example 1

A 3-year bond has a face value of ₹100, pays annual coupons of 6% and is redeemed at par at the end of year 3. Calculate the discounted mean term and the volatility at an effective annual yield of 5%.

Show the solution
  1. v = 1 ÷ 1.05 = 0.952381. Also v² = 0.907029 and v³ = 0.863838.
  2. Cash flows: 6 at t = 1, 6 at t = 2, and 106 at t = 3.
  3. Present values: 6 × 0.952381 = 5.7143; 6 × 0.907029 = 5.4422; 106 × 0.863838 = 91.5669.
  4. Total PV = 5.7143 + 5.4422 + 91.5669 = 102.7233.
  5. Weighted by time: 1 × 5.7143 = 5.7143; 2 × 5.4422 = 10.8844; 3 × 91.5669 = 274.7006. Sum = 291.2993.
  6. DMT = 291.2993 ÷ 102.7233 = 2.836 years. It lies between 1 and 3, as it should.
  7. Volatility = DMT ÷ (1 + i) = 2.8358 ÷ 1.05 = 2.701.

Answer: DMT ≈ 2.84 years; volatility ≈ 2.70.

Example 2

An insurer must pay a liability of ₹10,000 at the end of year 2 and ₹20,000 at the end of year 5. Calculate the discounted mean term of the liabilities at an effective annual interest rate of 8%.

Show the solution
  1. v = 1 ÷ 1.08. Then v² = 1 ÷ 1.1664 = 0.857339 and v⁵ = 1 ÷ 1.469328 = 0.680583.
  2. PV of first payment = 10,000 × 0.857339 = ₹8,573.39.
  3. PV of second payment = 20,000 × 0.680583 = ₹13,611.66.
  4. Total PV = 8,573.39 + 13,611.66 = ₹22,185.05.
  5. Numerator = 2 × 8,573.39 + 5 × 13,611.66 = 17,146.78 + 68,058.30 = 85,205.08.
  6. DMT = 85,205.08 ÷ 22,185.05 = 3.84 years.
  7. Check: 3.84 lies between 2 and 5, and is closer to 5 because the later payment has the larger present value.

Answer: DMT ≈ 3.84 years.

Exam tips

  • Set out a four-column table: t, C_t, C_t·v^t, t·C_t·v^t. Examiners award marks for method, and a table makes errors easy to spot.
  • State the interest rate used and write v to at least six decimal places. Rounding too early shifts the answer.
  • Read whether the question asks for DMT or volatility. Give the unit (years) for DMT.
  • In Redington immunisation questions, the first condition is PV of assets = PV of liabilities. The second condition is that the DMT (volatility) of assets equals that of liabilities at the same i. Compute both with the same table layout.
  • In Paper B (computer-based) you can compute this in R or Excel with SUMPRODUCT or sum(t*C*v^t)/sum(C*v^t). Show the formula and the result.

Practice questions from Duration, convexity and immunisation

Discounted Mean Term and Macaulay Duration: frequently asked questions

What is the formula for discounted mean term?

DMT = Σ t·C_t·v^t ÷ Σ C_t·v^t. It is the present-value-weighted average of the payment times. Use the same interest rate in the numerator and denominator.

What is the difference between discounted mean term and volatility?

DMT is an average time in years. Volatility is the proportional fall in present value for a small rise in the interest rate. With an effective annual rate i, volatility = DMT ÷ (1 + i).

How do I calculate the Macaulay duration of a bond?

List the coupons and the redemption payment with their times. Discount each at the yield, multiply each present value by its time, add up, and divide by the total present value. The result is in years.

Is the Macaulay duration of a coupon bond less than its term?

Yes, when all payments are positive and there is at least one coupon before maturity. Early coupons pull the weighted average time below the final payment date. A zero-coupon bond has a DMT equal to its term.

Does the DMT change if the interest rate changes?

Yes. A higher rate reduces the present value of late payments more, so the weights move towards earlier times and the DMT falls. Always use the rate given in the question.