Title: Understanding Duration — What It Is, How It Works, and Practical Steps to Use It
Source: Investopedia — https://www.investopedia.com/terms/d/duration.asp
Overview
Duration is a core fixed‑income concept that measures both (a) the weighted average time it takes to receive a bond’s cash flows and (b) the sensitivity of a bond’s price to changes in interest rates. Higher duration → greater sensitivity (more interest‑rate risk). Duration is not the same as time to maturity, although both are expressed in years.
Key takeaways
– Macaulay duration = the weighted average time (in years) to receive a bond’s cash flows.
– Modified duration = an interest‑rate sensitivity measure: the approximate percentage change in bond price for a 1% (100 basis‑point) change in yield.
– Effective duration = used for bonds with embedded options (calls/puts) and measures price sensitivity considering possible cash‑flow changes.
– Duration decreases when coupon rates rise (more early cash flows), and increases with longer maturities and lower coupons.
– Duration is an approximation; convexity measures the curvature / second‑order price response to yield changes.
How duration works in investing (intuition)
– A bond that returns more cash earlier (higher coupon, shorter maturity) has lower duration because you get your money back sooner.
– A zero‑coupon bond has duration equal to its time to maturity (all cash returned at the end).
– Modified duration translates Macaulay’s time measure into a percent price change rule: for a given Δyield, approximate %Δprice ≈ −(Modified duration) × Δyield. Example: a 5‑year duration implies roughly a 5% price fall if yields rise 1%.
Types of duration
– Macaulay duration: weighted average time to cash flows (years). Useful for concept and calculations.
– Modified duration: Macaulay / (1 + yield per period); gives approximate percentage price sensitivity to yield changes.
– Effective duration: used when cash flows may change with rates (e.g., callable bonds, mortgage‑backed securities).
– Dollar duration (DV01): the dollar change in price for a 1 basis‑point change in yield (useful for hedging).
– Key‑rate duration: sensitivity to changes at specific points on the yield curve.
– Portfolio duration: the market‑value weighted average duration of all holdings in a portfolio.
Macaulay duration — formula and explanation
Macaulay duration = Σ [t × PV(CF_t)] / Price, where:
– t = time in years until cash flow t,
– PV(CF_t) = present value of cash flow t (discounted at the bond’s yield),
– Price = sum of all PV(CF_t).
Interpretation: Macaulay duration is the weighted average year in which the bondholder receives cash flow (weights = PV of each cash flow).
Macaulay duration — calculation example (summary)
Bond: face = 100, 10% coupon, semiannual payments ($5 every 6 months), maturity = 3 years, YTM = 6% annually (3% per period).
Steps:
1. List cash flows and discount each at the period rate (3%). For periods 1–5: $5; period 6: $105.
2. Sum the PVs → that is the bond price.
3. For each period, compute (time in years) × (PV of that cash flow). Sum those products.
4. Divide the sum from step 3 by the bond price → Macaulay duration in years.
Result from the example: Macaulay duration ≈ 2.684 years. (This matches the worked example in the source.)
Modified duration — formula and use
Modified duration = Macaulay duration / (1 + y/k), where y = annual YTM, k = coupon periods per year.
– For semiannual coupons, divide y by 2 (k = 2).
Modified duration ≈ the percentage change in price for a 1% (= 0.01) change in yield (act as a linear estimate).
Percent change ≈ −Modified duration × Δyield (in decimal form).
Example (continuing above): Modified ≈ 2.684 / (1 + 0.06/2) ≈ 2.608. That means roughly a 2.608% price change for a 1% change in yield. If you quote price per $100 par, a 2.608% change = $2.61 per $100 par.
Convexity (brief)
Duration is a linear approximation. Price/yield relationship is curved (convex). Convexity measures the degree of curvature and improves accuracy for larger yield moves. When managing risk, combine duration and convexity for better estimates.
Why is bond price sensitivity called “duration”?
– Historically: “Macaulay duration” is named after Frederick R. Macaulay (1938), who formulated the weighted average time concept.
– Conceptually: duration captures a “length” or “effective maturity” of cash flows and, via modification, becomes a sensitivity measure (hence the same word is used for both concepts).
What else duration tells you
– Relative interest‑rate risk among bonds (higher duration → higher sensitivity).
– The trade‑off between price risk and reinvestment risk: longer duration → more price risk and less reinvestment uncertainty; shorter duration → less price risk but more reinvestment risk.
– For immunization: duration of liabilities ≈ duration of assets to reduce interest‑rate exposure.
– In portfolio management: target duration expresses the portfolio’s exposure to rate moves (e.g., conservative portfolio = low duration).
Strategies using duration
– Duration matching / immunization: set asset duration ≈ liability duration to lock in a target outcome regardless of small rate swings. Common in pension/fixed‑income liability management.
– Laddering: buying bonds with staggered maturities reduces portfolio duration and reinvestment risk.
– Barbell vs. Bullet: barbell uses short and long maturities (can adjust duration while exploiting yield curve movements); bullet concentrates maturities at a point.
– Active duration management: increase duration (buy long‑dated or low‑coupon bonds) when you expect rates to fall; decrease duration (buy short or higher coupon, or use derivatives) when you expect rates to rise.
– Hedging: use futures, swaps, or options to shift portfolio duration quickly.
Explain Like I’m 5 (ELI5)
Imagine you lend someone money in several small chunks over time. If you get more of your money back sooner, you’re less worried if the price of money (interest rates) changes. Duration tells you, on average, how long until you get your money back. If that “average time” is long, your loan’s value will wiggle more when interest rates wiggle.
Practical steps — how to find and use duration (step‑by‑step)
1. Gather bond data: coupon schedule, maturity, yield to maturity, payment frequency, and current market price.
2. For a single bond:
a. If you want Macaulay duration: compute PV of each cash flow at YTM, multiply each PV by its time in years, sum and divide by price.
b. For modified duration: divide Macaulay by (1 + yield per period).
c. For callable/putable bonds, use effective duration (option‑adjusted models or a pricing tool).
3. For a bond fund: use the fund’s published effective duration (available in fund factsheets or from your broker).
4. For a portfolio: compute market‑value weighted average of individual bond durations to get portfolio duration.
5. Translate to price sensitivity: %Δprice ≈ −(Modified duration) × Δyield. Multiply %Δprice by portfolio value to estimate dollar exposure.
6. Hedge or adjust: use bonds, futures, swaps, or options to change portfolio duration to meet your target. Monitor convexity for large expected yield moves.
7. Rebalance periodically: duration shifts over time (bonds age, yields move, trades occur).
Tools and shortcuts
– Excel functions: DURATION (Macaulay), MDURATION (modified) — requires settlement, maturity, coupon, YTM, frequency.
– Online bond calculators and broker tools will compute Macaulay, modified, effective duration and convexity.
– Most mutual funds and ETFs publish effective duration in daily/weekly factsheets.
Practical examples of everyday use
– Conservative investor: prefers low portfolio duration (e.g., ≤2 years) to limit losses if rates rise.
– Retirement plan sponsor: matches asset duration to liability duration to immunize funded status.
– Tactical manager: increases duration near the start of expected rate cuts to benefit from price gains; shortens duration when rate hikes are expected.
– Hedging: A portfolio manager with $100M in bonds and duration 6 has dollar duration ≈ 6% × $100M = $6M per 1 unit of yield (i.e., 1 = 100%? — better to convert into DV01 for 1 bp: DV01 ≈ (Modified duration × portfolio value) × 0.0001).
Common pitfalls
– Using Macaulay when bond has embedded options (use effective duration instead).
– Relying only on duration for large yield changes (need convexity).
– Forgetting that duration is affected by yield levels: as yields change, modified duration and prices change too.
– Confusing duration (sensitivity) with time to maturity.
The bottom line
Duration is a fundamental measure of interest‑rate risk and an “effective maturity” concept for bonds. Use Macaulay to understand timing of cash flows, modified/effective duration to estimate price sensitivity, and convexity to improve estimates for larger yield moves. Investors and institutions use duration to size, hedge, and immunize portfolios against interest‑rate changes.
Further reading / source
– Investopedia: “Duration” — https://www.investopedia.com/terms/d/duration.asp
If you want, I can:
– Walk through the full Macaulay calculation for the 3‑year example step‑by‑step with numeric PVs, or
– Show Excel formulas and a sample spreadsheet to compute Macaulay, modified, effective duration, and convexity for a bond or a portfolio. Which would be most helpful?