Convexity of Cash Flows and Bonds Explained

Understand Convexity

Key rules to remember

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.

How to solve Convexity questions

1. 1Write down the cash flows A_t, their times t, and the rate i. Note whether the question uses an effective rate or force of interest.
2. 2Calculate v = 1 ÷ (1 + i) and the present value of each cash flow, A_t v^t. Add them to get V.
3. 3Calculate the first derivative sum Σ t A_t v^t. Divide by V and then by (1 + i) to get D_mod.
4. 4Calculate Σ t(t+1) A_t v^t. Multiply by v² and divide by V to get C. This equals Σ t(t+1) A_t v^(t+2) ÷ V.
5. 5If asked to estimate a price change for a rate move h, compute −D_mod × h + ½ × C × h². Multiply by V for the money change.
6. 6If asked for the new price, add the change to V. State clearly that it is an estimate.
7. 7If a comparison is needed (Redington), compute V'' for both assets and liabilities and check which is larger. Write the conclusion in words.

Quickest way: Table method with PV weights

Common mistakes in Convexity

• Using Σ t² A_t v^t ÷ V as the convexity when the question works with the effective rate i.

  Both are called convexity in some texts, and the δ version looks simpler.

  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.

  You work with PV weights and stop at the weighted average.

  Fix: Always multiply the weighted average of t(t+1) by v². Write C = [Σ t(t+1) PV ÷ V] × v².

• Leaving out the ½ in the second-order term.

  You treat it as a simple addition of two terms.

  Fix: Remember the Taylor series: the second-order term is ½ h² V''. Write it with the ½ first.

• Putting h in percent, such as 1 instead of 0.01.

  Rates are quoted in percent, and the formula needs decimals.

  Fix: Convert h to a decimal before squaring. A 1% rise is h = 0.01 and h² = 0.0001.

• Treating the estimate as exact.

  The numbers look precise after several decimal places.

  Fix: Say 'approximately'. The duration and convexity estimate still ignores higher-order terms, so compare it to exact PV when asked.

• Assuming higher convexity is good for any position.

  Students remember convexity as 'good' for a bondholder.

  Fix: It helps the holder of the asset when rates move. In immunisation you need asset convexity greater than liability convexity, not just high convexity in the assets.

Worked examples