Macaulay duration and modified duration are mainly used to calculate the durations of bonds. The Macaulay duration calculates the weighted average time before a bondholder would receive the bond’s cash flows. Conversely, modified duration measures the price sensitivity of a bond when there is a change in the yield to maturity.
The Macaulay Duration
The Macaulay duration is calculated by multiplying the time period by the periodic coupon payment and dividing the resulting value by 1 plus the periodic yield raised to the time to maturity. Next, the value is calculated for each period and added together. Then, the resulting value is added to the total number of periods multiplied by the par value, divided by 1, plus the periodic yield raised to the total number of periods. Then the value is divided by the current bond price.
A bond’s price is calculated by multiplying the cash flow by 1, minus 1, divided by 1, plus the yield to maturity, raised to the number of periods divided by the required yield. The resulting value is added to the par value, or maturity value, of the bond divided by 1, plus the yield to maturity raised to the number of total number of periods.
For example, assume the Macaulay duration of a five-year bond with a maturity value of $5,000 and a coupon rate of 6% is 4.87 years ((1*60) / (1+0.06) + (2*60) / (1 + 0.06) ^ 2 + (3*60) / (1 + 0.06) ^ 3 + (4*60) / (1 + 0.06) ^ 4 + (5*60) / (1 + 0.06) ^ 5 + (5*5000) / (1 + 0.06) ^ 5) / (60*((1- (1 + 0.06) ^ -5) / (0.06)) + (5000 / (1 + 0.06) ^ 5)).
The modified duration for this bond, with a yield to maturity of 6% for one coupon period, is 4.59 years (4.87/(1+0.06/1). Therefore, if the yield to maturity increases from 6% to 7%, the duration of the bond will decrease by 0.28 year (4.87 – 4.59).
The formula to calculate the percentage change in the price of the bond is the change in yield multiplied by the negative value of the modified duration multiplied by 100%. This resulting percentage change in the bond, for a 1% yield increase, is calculated to be -4.59% (0.01*- 4.59* 100%).
The Modified Duration
The modified duration is an adjusted version of the Macaulay duration, which accounts for changing yield to maturities. The formula for the modified duration is the value of the Macaulay duration divided by 1, plus the yield to maturity, divided by the number of coupon periods per year. The modified duration determines the changes in a bond’s duration and price for each percentage change in the yield to maturity.
For example, assume a six-year bond has a par value of $1,000 and an annual coupon rate of 8%. The Macaulay duration is calculated to be 4.99 years ((1*80) / (1 + 0.08) + (2*80) / (1 + 0.08) ^ 2 + (3*80) / (1 + 0.08) ^ 3 + (4*80) / (1 + 0.08) ^ 4 + (5*80) / (1 + 0.08) ^ 5 + (6*80) / (1 + 0.08) ^ 6 + (6*1000) / (1 + 0.08) ^ 6) / (80*(1- (1 + 0.08) ^ -6) / 0.08 + 1000 / (1 + 0.08) ^ 6).
The modified duration for this bond, with a yield to maturity of 8% for one coupon period, is 4.62 years (4.99 / (1 + 0.08 / 1). Therefore, if the yield to maturity increases from 8% to 9%, the duration of the bond will decrease by 0.37 year (4.99 – 4.62).
The formula to calculate the percentage change in the price of the bond is the change in yield multiplied by the negative value of the modified duration multiplied by 100%. This resulting percentage change in the bond, for an interest rate increase from 8% to 9%, is calculated to be -4.62% (0.01* – 4.62* 100%).
Therefore, if interest rates rise 1% overnight, the price of the bond is expected to drop 4.62%.