Duration

The duration of a bond measures how sensitive its price is to changes in interest rates. More formally, duration is the weighted average time until the bondholder receives all cash flows (coupons and principal). A bond with a 5-year duration loses approximately 5% in value when interest rates rise 1%; it gains approximately 5% when rates fall 1%.

Macaulay duration and the weighted average concept

Macaulay duration is the weighted average time until you receive all cash flows. For a 10-year bond paying semi-annual coupons, Macaulay duration might be 7 years — meaning on average, you receive your cash flows in 7 years (not 10).

The weighting reflects the timing of cash flows. Early coupons (received soon) have low weight; the principal repayment (received at year 10) has high weight.

For a zero-coupon bond, duration equals maturity (all cash flow received at maturity). For a high-coupon bond, duration is much shorter than maturity (many coupons returned early).

Modified duration and price sensitivity

Modified duration converts Macaulay duration into a price sensitivity measure. It answers the question: “If yields rise 1%, how much does the bond price fall?”

The formula is: Price change (%) ≈ -modified duration × yield change

Example: A bond with modified duration of 5 will:

• Lose 5% in value if yields rise 1%
• Gain 5% in value if yields fall 1%
• Lose 10% in value if yields rise 2%

This makes duration the primary measure for understanding interest-rate risk in bond portfolios.

Duration vs. maturity

Duration and maturity are different:

• Maturity = the date the bond matures
• Duration = the weighted average time to cash flows

A 10-year bond with a 4% coupon has duration of approximately 8.5 years. A 10-year bond with an 8% coupon has duration of approximately 7 years (shorter, because coupons are received earlier).

This matters for interest-rate risk. A 10-year 8% coupon bond is less sensitive to rate changes than a 10-year 4% coupon bond, despite having the same maturity.

Duration of portfolios

For a portfolio of bonds, the portfolio duration is the weighted average of the bond durations. A $100M portfolio with 50% in 5-year duration bonds and 50% in 10-year duration bonds has portfolio duration of 7.5 years.

This is useful for liability matching. An insurance company with liabilities due in 10 years can buy bonds with 10-year duration, ensuring that portfolio returns match the liability timeframe.

Negative duration

Some securities have negative duration — their prices rise when yields rise. Inverse-floating-rate bonds are an example. These are rare and used primarily for hedging.

Duration and convexity

Duration measures the linear relationship between price and yield. But the relationship is curved — bonds have positive convexity, meaning they gain more in price when rates fall than they lose when rates rise (by the same amount).

For large rate moves, convexity becomes important. For small moves, duration dominates.

Effective duration for complex bonds

For callable bonds and other complex securities, effective duration accounts for the option-adjusted characteristics. A callable bond has shorter effective duration than a straight bond because the call option limits upside when rates fall.

Portfolio management use

Bond managers use duration actively:

• Rising-rate environment — Reduce portfolio duration (buy shorter-duration bonds) to minimize losses
• Falling-rate environment — Increase portfolio duration (buy longer-duration bonds) to maximize gains
• Liability matching — Match portfolio duration to liability duration to immunize the portfolio

Key rate duration

For a portfolio manager, “key rate duration” measures sensitivity to specific parts of the yield curve. A portfolio might have 5-year key rate duration of +2 and 10-year key rate duration of -1, reflecting different sensitivities across the curve.

See also