Duration and Convexity Calculations, Yield Curve Applications

Simple-average duration gives 6.7; PV-weighted gives 7.4. That difference translates to a $3.6 million error in a regulatory stress test.

Systematic approach:

• Compute $v = 1/(1+i)$.
• For each cash flow: $PV_t = CF_t \cdot v^t$.
• For each: $t \cdot PV_t$.
• Sum PVs, sum weighted PVs, divide.

$$D_{\text{mac}} = \frac{\sum_t t \cdot CF_t \cdot v^t}{\sum_t CF_t \cdot v^t}$$

For a level annuity:

$$D_{\text{mac}} = \frac{(Ia)_{\overline{n\vert}}}{a_{\overline{n\vert}}}$$

$$C_{\text{mod}} = \frac{\sum_t t(t+1) \cdot CF_t \cdot v^{t+2}}{P} = \frac{1}{(1+i)^2} \cdot \frac{\sum_t t(t+1) \cdot CF_t \cdot v^t}{P}$$

The $v^{t+2}$ is essential. Do not use $v^t$.

HIGH-FREQUENCY: The exam presents a handful of cash flows and asks for a Macaulay or modified duration, a convexity, or a first-order price estimate. It does not ask for the second-order (duration plus convexity) approximation: outcome 5(b) lists only the two first-order forms, and the required reading from study note FM-24-17 is Sections 1-4, while the second-order form is its Section 6.

Common mistakes

• Simple averages for portfolio duration. Bond A: PV $800, duration 4. Bond B: PV $200, duration 10. Correct: $(800 \times 4 + 200 \times 10)/1000 = 5.2$. Simple average: $(4+10)/2 = 7.0$. Trap: 7.0.
• Swapping numerator/denominator in forward formula. Correct: bigger accumulation factor on top. Writing $(1+s_t)^t/(1+s_{t+1})^{t+1}$ gives the reciprocal minus 1, a negative rate on a normal curve. Negative forward = check your fraction.
• Wrong power for C-mod. $v^{t+2}$, not $v^t$. For $t = 4$: power is 6, not 4. Off by $(1+i)^2$ for every term.

Bottom line

• Portfolio duration/convexity = PV-weighted averages, never simple averages.
• First-order Macaulay approximation is multiplicative: $P(i) \approx P(i_0)\left[(1+i_0)/(1+i)\right]^{D_{\text{mac}}}$. A different number than the additive $-D_{\text{mod}}\Delta i$.
• Additive equivalence: $-D_{\text{mod}}\Delta i$ equals $-D_{\text{mac}}\Delta i/(1+i)$ because $D_{\text{mod}} = D_{\text{mac}}/(1+i)$.
• Perpetuity durations: level immediate $D_{\text{mac}} = (1+i)/i$, $D_{\text{mod}} = 1/i$, from $D_{\text{mod}} = -P'(i)/P$; the due version is one year shorter.

Exam shortcut

Build an "accumulation factor column" first: $(1+s_1)^1, (1+s_2)^2, (1+s_3)^3, \ldots$. Every forward rate is a ratio of adjacent entries. Write this column in the margin. Forward: "Big over Small minus 1." C-mod: "t plus 2" power. Bootstrapping: "Short to Long." "Macaulay multiplies, Modified adds." Perpetuity: "Mac is $1/d$, Mod is $1/i$."

The full lesson (about 4,211 words, 28 min read) adds 9 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 5b

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