Yield Rates, Duration, Convexity, Spot Rates, Forward Rates, and Immunization

A pension fund matches PV of assets to a $50 million liability but ignores duration and convexity. A 200 bp rate spike leaves them $2.1 million short.

The yield rate (IRR) is $i$ such that:

$$\sum _{t}C{F}_t\cdot v^t=0,v=1/(1+i)$$

Unique when cash flows change sign once. Multiple sign changes can produce multiple yields.

The CF worksheet plus IRR returns the yield rate directly: enter the cash flows as CF0, C01, C02, and so on, then compute IRR. NPV (enter a rate as I, compute NPV) values an irregular stream at a flat rate; spot-rate discounting must be entered cash flow by cash flow. Duration and convexity are weighted sums you build by hand from the PV column, the calculator gives you the PVs quickly, not the duration itself.

PV-weighted average time to receipt:

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

Measured in years. For a zero maturing at $n$: duration = $n$. For a coupon bond: duration < $n$.

Common mistakes

• Forgetting to divide by $(1+i)$. Using D-mac directly in the additive price formula gives $-3.7232\times 0.015=-5.585\%$ instead of -5.319%. Trap: -5.585%. D-mac belongs in the multiplicative Macaulay approximation $P(i_0){\left[(1+i_0)/(1+i)\right]}^{D_{\text{mac}}}$ (lesson 5b), never in the additive formula.
• Wrong convexity formula. Macaulay uses $t^2$. Modified uses $t(t+1)$ with $v^{t+2}$. Mixing them distorts the second-order correction.
• Confusing spot and forward rates. $s_3=4\%$ means a 3-year zero yields 4% annually. $f_{2,3}$ is the implied rate from year 2 to 3. These are different numbers.

Bottom line

• Macaulay duration = PV-weighted average time (years). Modified duration = Macaulay / $(1+i)$.
• First-order: $\Delta P/P\approx -D_{\text{mod}}\cdot \Delta i$. Second-order adds $+\frac{1}{2}{C}_{\text{mod}}(\Delta i{)}^2$.
• Spot rate $s_t$ prices a zero at maturity $t$. Forward rate links adjacent spots via no-arbitrage.
• Redington immunization: PV match, duration match, and convexity dominance; protects against small parallel shifts only.

Exam shortcut

"Approximate price change", identify first vs. second order. Write the formula before plugging in to prevent decimal errors. "DMC": Duration first-order, Modified from Macaulay (divide by $1+i$), Convexity second-order. Immunization: "R-F-C". Redington (small), Full (any, single), Cash flow (any, any). "Spot is Solo, Forward is Future."

The full lesson (about 2,028 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 5a

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