Bond Duration

Bond duration is a measure of the sensitivity of a bond's price to changes in interest rates, and also of the weighted average time until the bond's cash flows are received. It is a central concept in fixed-income analysis and in the management of interest rate risk.

Macaulay Duration

Macaulay duration is the weighted average of the times at which the bond's cash flows are received, where the weights are the present values of the cash flows relative to the bond's price. It is expressed in years and provides a measure of the effective maturity of the bond.

The duration of a bond depends on its maturity, its coupon rate, and its yield. Longer maturities increase duration, higher coupons reduce it, and higher yields reduce it. A zero-coupon bond has a duration equal to its maturity, since all its cash flow is received at maturity.

Modified Duration

Modified duration measures the percentage change in a bond's price for a one per cent change in its yield, and it is computed from Macaulay duration by dividing by one plus the yield per period. It provides a direct measure of interest rate sensitivity: a bond with a modified duration of five experiences a price change of about five per cent for a one per cent change in yield.

Modified duration is a first-order approximation, valid for small changes in yield. For larger changes, the approximation becomes less accurate, and convexity, the second-order term, is used to improve the estimate. Convexity measures the curvature of the price-yield relationship and is positive for most bonds.

Uses in Risk Management

Duration is used to measure and manage interest rate risk. By matching the duration of assets and liabilities, an institution can immunise its balance sheet against small changes in interest rates, protecting its net worth. Duration gaps, the difference between the durations of assets and liabilities, indicate the exposure to interest rate changes.

Duration is also used in portfolio management, where it determines the sensitivity of a portfolio to interest rate movements. Managers adjust the duration of a portfolio in response to their expectations about rates, and they use duration to compare the risk of different bonds and portfolios.

Limitations

The limitations of duration include its assumption of a parallel shift in the yield curve, since it does not capture the effects of changes in the shape of the curve. It is also a local approximation, valid only for small changes in yield, and it assumes that the bond's cash flows do not change when yields change, which may not hold for bonds with embedded options.

For bonds with embedded options, such as callable or putable bonds, effective duration is used, which accounts for the effect of the option on the bond's cash flows. The analysis of such bonds requires option-adjusted techniques that incorporate the behaviour of the embedded option.

Convexity

Convexity measures the curvature of the relationship between a bond's price and its yield. It captures the change in duration as yields change, and it improves the accuracy of the estimate of price changes for larger movements in yields.

For a given change in yield, a bond with greater convexity experiences a smaller price decline when yields rise and a larger price increase when yields fall. Convexity is therefore a desirable property, and investors may pay a premium for it.

Duration of Portfolios

The duration of a portfolio is the weighted average of the durations of its constituent bonds, with the weights being the shares of the portfolio's value. The duration of the portfolio measures its sensitivity to changes in interest rates.

Portfolio duration is used in the management of interest rate risk. Managers adjust the duration of a portfolio in response to their expectations about rates, and they use duration to compare the risk of different portfolios and to design strategies that match the sensitivity of assets and liabilities.

Immunization

Immunisation is a strategy that seeks to protect a portfolio against interest rate changes by matching the duration of assets and liabilities. A duration-matched portfolio is protected against small parallel shifts in the yield curve.

The strategy requires continuous rebalancing, since durations change as time passes and as rates change. It also has limitations, including its inability to protect against changes in the shape of the curve and against large shifts, where convexity matters.

Applications in Banking

In banking, duration analysis is used to measure and manage the interest rate risk of the balance sheet. The duration of assets, principally loans and securities, and the duration of liabilities, principally deposits and borrowings, determine the sensitivity of the bank's equity to changes in rates.

The analysis informs the management of the net interest margin and the assessment of capital adequacy under different interest rate scenarios. It is a central element of asset liability management and of the supervision of interest rate risk.

See also

• Interest Rate Spread
• Financial Asset
• Rate of Return