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: $P{V}_t=C{F}_t\cdot v^t$.
• For each: $t\cdot P{V}_t$.
• Sum PVs, sum weighted PVs, divide.

$$D_{\text{mac}}=\frac{{\sum }_{t}t\cdot C{F}_t\cdot v^t}{{\sum }_{t}C{F}_t\cdot v^t}$$

For a level annuity:

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

$$C_{\text{mod}}=\frac{{\sum }_{t}t(t+1)\cdot C{F}_t\cdot v^{t+2}}{P}=\frac{1}{(1+i{)}^2}\cdot \frac{{\sum }_{t}t(t+1)\cdot C{F}_t\cdot v^t}{P}$$

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

HIGH-FREQUENCY: The exam often presents 4-5 cash flows and asks for the second-order approximation, requiring both duration and convexity.

$$D_{\text{mod}}^{\text{port}}=\sum _j{w}_j{D}_{\text{mod},j},w_j=P{V}_j/\sum _{m}P{V}_m$$

Use PV weights, not face-value or par-value weights.

The yield curve plots the spot rate against term. It is normal (upward sloping) when long rates exceed short rates, inverted (downward sloping) when short rates are higher, and flat...

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 $D_{\text{mod}}=-P^{\prime }(i)/P$; the due version is one year shorter.

Exam shortcut

Build an "accumulation factor column" first: $(1+s_1{)}^1,(1+s_2{)}^2,(1+s_3{)}^3,\dots$. 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 3,451 words, 23 min read) adds 5 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 5b

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