IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Duration, convexity and immunisation: formula sheet

Full chapter guide

Key formulas

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.
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.
Present value: V(i) = Σ A_t v^t, where v = 1 ÷ (1 + i); A_t is the cash flow at time t. Use the same i throughout.
First derivative: V'(i) = −Σ t A_t v^(t+1); Gives modified duration: D_mod = −V' ÷ V.
Second derivative: V''(i) = Σ t(t+1) A_t v^(t+2); The numerator of convexity.
Convexity (with respect to i): C = V''(i) ÷ V(i); This is the second derivative with respect to the effective rate i, divided by V. It is not the standard (Core Reading) convexity, which is taken with respect to δ. Use it with the price change estimate below, and match the formula to the variable the question uses.
Standard convexity (Core Reading, with respect to force of interest): V''(δ) ÷ V = Σ t² A_t v^t ÷ V; This is the standard Core Reading convexity. It differs from C. It is the discounted mean of t², and it is the version used when working with δ. Check which variable the question uses.
Price change estimate: ΔV ÷ V ≈ −D_mod × h + ½ × C × h²; h is the change in the effective rate i. Needs C defined with respect to i.
Zero-coupon bond redeemed at time n: C = n(n + 1) v²; Single cash flow, so the PV cancels. Useful as a check.
Redington condition (convexity): V_A''(i₀) > V_L''(i₀); Third Redington condition. It comes with equal PVs and equal V' values at the starting rate.
Condition 1: equal present values: V_A(i₀) = V_L(i₀); Assets and liabilities have the same present value at the current rate i₀.
Condition 2: equal slopes: V_A′(i₀) = V_L′(i₀); Derivatives taken with respect to the interest rate (or force of interest). With condition 1, this is the same as equal DMT.
Condition 3: convexity: V_A″(i₀) > V_L″(i₀); Strict inequality. With condition 1, this means asset convexity is greater than liability convexity.
Discounted mean term: DMT = Σ t · CF_t · vᵗ ÷ Σ CF_t · vᵗ; The present-value-weighted average time of the cash flows. Use v = 1 ÷ (1 + i₀).
Convexity: c = Σ t² · CF_t · vᵗ ÷ Σ CF_t · vᵗ; Equals V″(δ) ÷ V(δ) when differentiating with respect to the force of interest δ. The common divisor V means the comparison is valid once present values are equal.
Volatility (modified duration): ν = −V′(i) ÷ V(i) = DMT ÷ (1 + i); Equal DMT with equal present values gives equal volatility.
Two-bond matching of one liability: x + y = V_L and t₁x + t₂y = t_L(x + y); x and y are the present values spent on bonds maturing at t₁ and t₂. Then check that t₁ < t_L < t₂.
Surplus function: S(δ) = V_A(δ) − V_L(δ); V_A and V_L are the present values of asset and liability cashflows at force of interest δ. Immunisation concerns the behaviour of S near the current δ.
Condition (i): equal present values: V_A(δ₀) = V_L(δ₀); Assets must be worth the same as liabilities at the current rate.
Condition (ii): equal discounted mean terms: V_A'(δ₀) = V_L'(δ₀), i.e. DMT_A = DMT_L; Valid because condition (i) holds, so equal first derivatives mean equal DMTs. DMT = Σ t·PV(t) ÷ Σ PV(t).
Condition (iii): convexity: V_A''(δ₀) > V_L''(δ₀); With equal present values this means Σ t²·PV_A(t) ÷ V_A > Σ t²·PV_L(t) ÷ V_L. Asset cashflows are more spread out than liability cashflows.
Single liability test: DMT_A = t and Σ t_k²·PV_k ÷ V_A > t²; Here t is the liability date and PV_k are present values of asset payments at times t_k.
Two zero-coupon assets: w·t₁ + (1 − w)·t₂ = t, with t₁ < t < t₂; w is the proportion of the present value in the earlier bond. Then the nominal amount at time t_k is PV_k × (1 + i)^(t_k).

Quick revision

Discounted mean term = Σ t·v^t·cash flow at t ÷ Σ v^t·cash flow at t, the present-value-weighted average time.
Macaulay duration is the discounted mean term; state the interest rate you use.
Volatility = −(1 ÷ V) × dV/di, where V is the present value at interest rate i.
Modified duration equals Macaulay duration ÷ (1 + i) when interest is effective annual and cash flows are discrete.
Convexity = (1 ÷ V) × d²V/di², using the same V and i as in duration.
A first-order estimate of the change in V is −V × volatility × Δi; the second-order term adds ½ × V × convexity × (Δi)².
Redington condition 1: present value of assets equals present value of liabilities at the current rate.
Redington condition 2: the duration of assets equals the duration of liabilities, so the first derivative of the surplus is zero.
Redington condition 3: convexity of assets exceeds convexity of liabilities, so the surplus has a local minimum at the current rate.
The three conditions protect against small changes in the interest rate only.
Full immunisation requires the asset cash flows to surround the liability cash flows, at least in the classic one-liability case.
Practical limits: non-parallel yield curve shifts, rebalancing needs, transaction costs and uncertain liability cash flows.

Common mistakes

Weighting times by the cash flows instead of the present values 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 Fix: At maturity the cash flow is coupon plus redemption. Write this as one combined amount, for example 106, not 6.
Using the Macaulay duration as the volatility without dividing by (1 + i). 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. Fix: Write ΔV ≈ −V × ν × Δi each time. Then check that an increase in i gives a fall in V.
Using Σ t² A_t v^t ÷ V as the convexity when the question works with the effective rate i. Fix: For derivatives with respect to i, use t(t+1) and v^(t+2). Use t² only when differentiating with respect to δ.
Forgetting the factor v² when going from Σ t(t+1) PV ÷ V to C. Fix: Always multiply the weighted average of t(t+1) by v². Write C = [Σ t(t+1) PV ÷ V] × v².
Writing the second condition as equal durations of the cash flows without equal present values. Fix: Always state condition 1 first. DMT equality and equal slopes are equivalent only when present values are equal.
Putting the nominal amounts, not the present values, into the mean-term equation. Fix: Let x and y be present values, solve, then convert back to payments by accumulating at i₀ for the bond's term.
Stopping after matching present value and mean term and not checking convexity. Fix: Always compute Σ PV·t² for assets and compare with the liability. Without it, the surplus could have a maximum instead of a minimum.
Choosing an asset that matures exactly at the liability date and calling it immunised. Fix: That is exact matching, not immunisation. For immunisation the asset cashflows must be spread both before and after the liability date so that convexity is larger.

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.
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.