What is effective duration — and how do you use it?
Effective duration measures a bond’s price sensitivity to small changes in interest rates when the bond’s cash flows may change as rates move (for example, bonds with embedded options such as calls, puts, or prepayment features). It generalizes the idea of modified duration to securities whose future payments are interest‑rate dependent. Effective duration tells you the approximate percentage change in a bond’s price for a small parallel shift in the yield curve, accounting for the option‑driven changes in expected cash flows.
Why it matters
– For option‑free bonds, modified duration is usually sufficient. But when cash flows can change (callable bonds, mortgage pools, step‑up coupons), modified duration understates or misstates interest‑rate sensitivity.
– Effective duration incorporates how expected cash flows will reprice or be exercised as rates change and is therefore the preferred measure for embedded‑option securities.
– It’s used for risk management (hedging, stress testing), portfolio construction (duration matching), and pricing decisions.
Key features
– Expressed in years (or in percentage change per 1.00 = 100 basis points).
– Always less than or equal to the bond’s maturity (because duration measures weighted timing of cash flows).
– Typically computed using small up/down yield shocks and the resulting model prices.
– Good for small, parallel yield shifts; for large moves or non‑parallel shifts, combine with convexity, key‑rate durations, or full scenario analysis.
The formula (practical)
Effective duration ≈ [P(↓) − P(↑)] / [2 × P(0) × Δy]
Where:
– P(0) = base (current) clean price of the bond.
– P(↓) = price when the yield is decreased by Δy.
– P(↑) = price when the yield is increased by Δy.
– Δy = the absolute change in yield used for the shock (in decimal form; e.g., 0.001 for 10 basis points).
A worked example (step‑by‑step)
Assume:
– Current price P(0) = 100.00 (par = 100).
– Base yield = 6.00%.
– Choose Δy = 0.001 (10 basis points = 0.10%).
– Price if yield falls 10 bps: P(↓) = 101.00.
– Price if yield rises 10 bps: P(↑) = 99.25.
Step 1 — compute numerator:
P(↓) − P(↑) = 101.00 − 99.25 = 1.75
Step 2 — compute denominator:
2 × P(0) × Δy = 2 × 100 × 0.001 = 0.20
Step 3 — compute effective duration:
Effective duration = 1.75 / 0.20 = 8.75
Interpretation: A 100‑basis‑point (1.00%) parallel rise in yield is estimated to reduce the bond’s price by ≈ 8.75% (and a 1% fall to increase price by ≈ 8.75%), ignoring convexity. For more accurate estimates for larger moves, include effective convexity.
Practical steps to compute effective duration in real life
1. Choose a pricing model appropriate for the security:
– Option‑free plain bonds: present value of fixed cash flows is sufficient.
– Bonds with embedded options (callable, putable, mortgages): use an option model — e.g., binomial interest‑rate trees, Monte Carlo, or option‑adjusted spread (OAS) frameworks — so the model will repriced cash flows when yields change.
2. Select the yield shock Δy:
– Common choices: ±5 bps, ±10 bps, ±25 bps. Smaller shocks give a better local linear approximation; ensure the pricing model remains stable.
3. Reprice the bond at yield = yield0 + Δy → P(↑) and yield = yield0 − Δy → P(↓):
– For option bonds, the exercise decisions (calls, prepayments) must be re‑solved under each yield scenario.
4. Plug into effective duration formula and compute.
5. (Optional) Compute effective convexity for a better approximation for larger yield moves:
– Effective convexity ≈ [P(↓) + P(↑) − 2P(0)] / [P(0) × (Δy)^2]
– Use both duration and convexity to approximate price change: ΔP/P ≈ −D × Δy + 0.5 × Conv × (Δy)^2.
6. Use the duration for risk metrics:
– DV01 or PVBP = (Effective duration × P(0) × Δy) / 100 (or directly compute price difference for 1 bp).
– Aggregate position risk by weighting individual effective durations by market value.
Excel and quick implementation tips
– For option‑free cash flows: compute P(0), P(↑), P(↓) with PV formulas. For periodic coupon bonds, you can use Excel’s PRICE or NPV functions.
– For callable/putable: Excel alone is not sufficient unless you include an interest‑rate tree or use add‑ins that support option models.
– Example Excel formula for a single price at yield y (annual coupons): =SUM(Coupon/(1+y)^t) + Face/(1+y)^T. Compute three prices (y, y±Δy) and plug into formula above.
Simple Python outline (option‑free bond)
– Compute PV at y, y+Δy, y−Δy using vectorized discounting and apply formula. For embedded options, use a specialized library or implement a short‑rate tree.
Limitations and cautions
– Local approximation: Effective duration assumes small, parallel shifts in yields. For large or non‑parallel moves, duration alone is insufficient.
– Model risk: For callable/putable securities, results depend on the interest‑rate model and assumptions about volatility, prepayment behavior, and issuer strategy. Different models / OAS assumptions yield different effective durations.
– Convexity and negative convexity: Many option‑embedded securities (e.g., mortgage pools) exhibit negative convexity at times — rising yields increase duration and falling yields shorten it — making single‑number duration measures less informative unless accompanied by convexity or scenario analysis.
– Nonparallel curve moves: Use key‑rate durations or full re‑pricing under multiple curve shifts to capture term structure changes.
When to use effective duration vs modified duration
– Use modified duration for option‑free, fixed cash‑flow bonds (it’s faster and analytic).
– Use effective duration when cash flows can change with rates (callable, putable, mortgage‑backed securities). If an embedded option is “deeply out of the money” (unlikely to be exercised at current rates), effective duration may be close to modified duration.
Practical applications
– Risk management: aggregate effective duration across holdings to measure portfolio interest‑rate exposure and to construct hedges (e.g., duration‑matching with Treasuries or swaps).
– Pricing and trading: traders use effective duration to size hedges and estimate P/L for small rate moves.
– Performance attribution: separate returns due to yield changes versus option exercise behavior/prepayment.
Summary checklist (quick)
– Determine if cash flows are rate‑dependent → if yes, use effective duration.
– Choose Δy (small shocks) and an appropriate pricing model.
– Reprice under yield up/down and apply formula.
– Complement with convexity, key‑rate durations, and scenario analysis.
– Document model assumptions (volatility, prepayment speeds, tree parameters) — results can be sensitive to these inputs.
Source
– Investopedia: “Effective Duration” — https://www.investopedia.com/terms/e/effectiveduration.asp
If you want, I can:
– compute effective duration for a specific bond if you give coupon, maturity, current yield, par, and whether it’s callable/putable; or
– provide an Excel template or a short Python script (option‑free or simple tree) to compute effective duration.