• Interest rate sensitivity measures how much a fixed‑income security’s price will change when market interest rates move. Higher sensitivity = larger price swings.
– Duration is the standard way to quantify sensitivity. Common measures: Macaulay duration, modified duration, effective duration, and key‑rate duration.
– Approximate percent price change ≈ −(duration) × (change in yield in decimals). For large rate moves you also need convexity (second‑order effect).
– Investors manage sensitivity by changing portfolio duration, using ladder/barbell structures, or hedging with derivatives (futures, swaps, options).
What is interest rate sensitivity?
Interest rate sensitivity is the relationship between changes in market interest rates and the resulting change in the price (market value) of a fixed‑income asset. Because bond prices and yields move in opposite directions, a rise in rates generally reduces bond prices and vice versa. Duration converts that relationship into a single number that estimates the expected percentage price change per unit change in yield.
Why it matters
– If you expect to hold a bond to maturity, interim price swings may be irrelevant; if you may sell in the secondary market or have liabilities that require funding earlier, sensitivity matters.
– Portfolio managers use duration to match assets to liabilities, control interest‑rate risk, and design hedging strategies.
– Different bonds with the same maturity can have very different sensitivities because of coupon size, yield, embedded options, or amortization.
Main types of duration (interest‑rate sensitivity measures)
1. Macaulay duration
– Definition: The weighted average time (in years) until the bond’s cash flows are received, with weights equal to the present value of each cash flow divided by the bond price.
– Interpretation: For a zero‑coupon bond, Macaulay duration = maturity. For coupon bonds it is less than maturity.
– Use: Fundamental measure; input to modified duration.
2. Modified duration
– Definition: Macaulay duration adjusted for the bond’s yield. It estimates the approximate percentage price change for a small parallel shift in yield.
– Formula: modified duration ≈ Macaulay duration / (1 + y/k), where y is yield per annum and k is coupon periods per year (k = 1 if annual).
– Rule of thumb: %ΔPrice ≈ −(modified duration) × Δy (Δy in decimals).
3. Effective duration
– Definition: A duration measure that captures the sensitivity of bonds with embedded options (calls, puts, prepayment risk). It uses modeled price changes from small parallel yield shifts and thus incorporates expected changes in cash flows.
– Use: Preferred for mortgage‑backed securities, callable/putable bonds, and funds with optionable holdings.
4. Key‑rate duration (partial duration)
– Definition: Measures sensitivity to yield changes at a specific point (key rate) on the yield curve (e.g., 2‑yr, 5‑yr, 10‑yr).
– Use: Helps analyze non‑parallel shifts in the yield curve and where a portfolio is exposed along the curve.
Convexity
– Convexity is the second‑order term in the price–yield relationship. Duration gives a linear approximation; convexity improves accuracy for larger yield changes.
– Bonds with higher convexity fall less (or rise more) than predicted by duration alone for large rate moves.
Simple numeric examples
Example A — Using modified duration
– A bond or fund has modified/effective duration = 11. If yields rise 1.0% (0.01 in decimals), estimated percent change ≈ −11 × 0.01 = −11% (approximate).
Example B — Short duration bond
– Corporate bond duration = 2.5. If yields fall 0.5% (0.005), estimated percent change ≈ −2.5 × (−0.005) = +0.0125 = +1.25%.
Worked calculation — 2‑year annual coupon bond
– Face = 100, coupon = 5% (5 paid annually), YTM = 5% (price = 100).
– PV(cash flows): Year 1 = 5/1.05 = 4.7619; Year 2 = 105/1.05^2 = 95.2381; Price = 100.
– Macaulay duration = (1×4.7619 + 2×95.2381)/100 = 1.9524 years.
– Modified duration = 1.9524 / (1 + 0.05) = 1.8594.
– If yields increase 1%: estimated price change ≈ −1.8594%.
Practical steps for investors and portfolio managers
1. Define objectives and constraints
• Determine investment horizon, liability dates, desired income, and acceptable volatility.
• Decide whether you want to actively take duration risk or immunize liabilities.
2. Measure current sensitivity
• Calculate portfolio weighted average duration (use modified or effective duration depending on holdings).
• Estimate key‑rate durations to see where along the curve you are exposed.
• Use vendor tools (Bloomberg, portfolio platforms, or fund factsheets) if you do not model manually.
3. Stress test scenarios
• Test parallel shifts (±25 bps, ±50 bps, ±100 bps), steepener/flatteners, and non‑parallel shifts using key‑rate shifts.
• Include credit spread widening or tightening and liquidity stress (price impact).
4. Decide on a target duration
• Liability matching: set portfolio duration ≈ duration of liabilities (immunization).
• Tactical view: shorten duration if you expect rising rates, lengthen if you expect falling rates.
• Risk budget: set maximum dollar or percent loss for a given rate move.
5. Implement changes and hedges
• Change holdings: buy short‑duration or floating‑rate notes to reduce sensitivity; buy long‑duration securities to increase it.
• Structure: ladder (evenly spaced maturities), barbell (short + long), or bullet (clustered near a date) to shape exposures.
• Derivatives: use Treasury futures, interest rate swaps, or options to adjust duration more precisely and efficiently.
• Example: selling futures reduces duration; buying futures increases it.
• Swaps: pay fixed/receive floating to shorten duration; receive fixed/pay floating to lengthen duration.
• Consider funds/ETFs that target specific duration profiles.
6. Monitor and rebalance
• Recalculate duration after coupons, principal payments, and market moves.
• Rebalance after large rate moves or when cashflows shift portfolio duration away from target.
7. Consider costs and risks
• Hedging costs, margin requirements, basis risk (futures vs. underlying holdings), and counterparty risk for swaps.
• Embedded option risk: prepayment in mortgages causes negative convexity and makes effective duration path‑dependent.
• Reinvestment risk: coupons received and reinvested at different rates change realized return.
Practical examples of managing sensitivity
– To reduce sensitivity quickly: sell long‑dated bonds, buy short notes or floating‑rate notes, or enter into pay‑fixed interest rate swaps (receive floating).
– To increase sensitivity cheaply: buy longer maturity or lower‑coupon bonds, or receive fixed in a swap.
– Immunization for defined liability: match asset duration to liability duration and manage convexity and cashflow timing through frequent rebalancing.
Limitations and caveats
– Duration is an approximation best for small, parallel yield moves. For large moves or non‑parallel shifts use convexity and key‑rate analysis.
– Duration ignores credit spread changes; a bond’s price can move for reasons other than Treasury yield changes (issuer credit, liquidity).
– Effective duration requires modeling assumptions about how cash flows change (e.g., prepayment speeds).
– Transaction costs and taxes can make frequent adjustments costly.
Useful resources and further reading
– Investopedia — Interest Rate Sensitivity (source article):
– Morningstar — What Is Duration?: (search “What Is Duration? Morningstar”) [Morningstar article on duration]
– CFA Institute — Understanding Fixed‑Income Risk and Return:
– BlackRock — What is Bond Duration?
– Calculate the duration for a specific bond or your portfolio if you provide cash flows, coupon, yield, and holdings.
– Produce a short stress‑test table (price change under ±25/50/100 bps) for a given duration.