The duration of a bond measures sensitivity of a bond’s full price (inclusive of accrued interest) to changes in interest rates if all other factors are held constant. Duration is the period for which if we hold a bond, the increase in reinvestment income would offset by the decrease in price if interest rates increase, and vice versa. In other words, it is the time needed to balance out the coupon reinvestment risk and market price risk. It can also be interpreted as the approximate time for which a bond must be held so as to realize the yield to maturity envisioned at the time of purchase even interest rate changes (for once).
There are two categories of duration: yield duration (including Macaulay duration, modified duration, money duration and price value of a basis point (PVBS) and curve duration (the effective duration).
Macaulay duration
Macaulay duration equals the time to receipt of bond cash flows weighted at the proportion of the present value of the relevant cash flow to the bond’s full price.
For example, if a bond paying 10% coupon and yielding 9% has two years till maturity, we need to take the following steps to work out its Macaulay duration:
- Step 1: Find bond cash flows and find time the expected date of receipt of each cash flow.
- Step 2: Calculate the present value of each cash flow and divide it by full price to find out the share of each cash flow in the price.
- Step 3: Multiply the time till cash flow by the relevant share of the cash flow in the bond’s full price.
- Step 4: Sum all the values obtained for all cash flows to get the Macaulay duration.
We can also use a plug-and-chug formula given below:
\[ Macaulay\ Duration=\frac{1+r}{r}-\frac{1+r+\left[n\times(c-r)\right]}{c\times\left[{(1+r)}^n+r\right]}-(t/T) \]This equation gives us the Macaulay duration per period. For example, if coupon rate c, yield to maturity r, and number of periods n are expressed in quarters, the Macaulay duration shall be in quarter too and must be divided by the number of periods per year to arrive at the annual Macaulay duration.
Modified duration
Modified duration equals Macaulay duration divided by 1 + required yield per period. It gives us the estimated change in the price of a bond in response to a 1% change in yield.
\[ ModDur=\frac{MacDur}{1+r} \]If the annual yield to maturity is 6% and the bond pays coupon semiannually, we need to divide by 1 + 3%.
The expected percentage change in full price (%∆P) in response to a change in yield (∆Y) is given by the product of annual modified duration and change in yield:
\[ \%∆P^F = -AnnModDur ×∆Y \]The negative sign shows that when yield increases, price decreases, and vice versa.
The modified duration can also be estimated using the following equation:
\[ AppModDur=\frac{PV_–PV_+}{2×∆Y×P_0} \]This equation approximates the slope of a line tangent to the price-yield curve by calculating price when yield decreases PV− and price when yield increases PV+. P0 represents the original price of the bond. As the difference between the two bond prices approaches zero, approximate modified duration equals modified duration. Approximate Macaulay duration equals approximate modified duration (AppModDur) multiplied by 1+ r. The approximation formulas produce annualized durations.
Effective duration
Effective duration is a curve duration statistic calculated using a formula similar to the formula used for approximate duration:
\[ EffDur=\frac{PV_–PV_+}{2×∆Curve×P_0} \]The difference lies in the denominator. The effective duration uses a parallel shift in the benchmark yield curve such as the government par curve instead of a change in yield to maturity of the bond.
Key rate duration
While the effective duration measures bond sensitivity to a parallel shift in the benchmark yield curve, the key-rate duration measures the sensitivity of a bond’s price to a change in the shape of the yield curve i.e. steepening or flattening. A key-rate duration is calculated changing the benchmark yield curve at specific maturity while keeping it constant at all other maturities.