Macaulay duration and revamped duration are chiefly used to calculate the durations of bonds. The Macaulay duration calculates the weighted average time prior to a bondholder would receive the bond’s cash flows. Conversely, modified duration measures the price sensitivity of a chains when there is a change in the yield to maturity.
The Macaulay Duration
The Macaulay duration is calculated by multiplying the time time 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 planned for each period and added together. Then, the resulting value is added to the total number of periods multiplied by the par value, disjoined by 1, plus the periodic yield raised to the total number of periods. Then the value is divided by the current manacles price.
A bond’s price is calculated by multiplying the cash flow by 1, minus 1, divided by 1, extra 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 full growth value, of the bond divided by 1, plus the yield to maturity raised to the number of total number of periods.
For model, 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 altered 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). The case, 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 way 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 portion 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 rendering of the Macaulay duration, which accounts for changing yield to maturities. The formula for the modified duration is the value of the Macaulay duration rank by 1, plus the yield to maturity, divided by the number of coupon periods per year. The modified duration determines the substitutions 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 grade 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 give up the fight 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 thongs 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 return multiplied by the negative value of the modified duration multiplied by 100%. This resulting percentage change in the bond, for an prevail upon 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 outlay of the bond is expected to drop 4.62%.
The Modified Duration and Interest Rate Swaps
Modified duration could be extended to determine the amount of years it would take an interest rate swap to repay the price paid for the swap. An interest calculate swap is the exchange of one set of cash flows for another and is based on interest rate specifications between the parties.
The modified duration is intended by dividing the dollar value of a one basis point change of an interest rate swap leg, or series of cash flows, by the grant value of the series of cash flows. The value is then multiplied by 10,000. The modified duration for each series of spondulicks flows can also be calculated by dividing the dollar value of a basis point change of the series of cash flows by the notional value and the market value. The fraction is then multiplied by 10,000.
The modified duration of both legs must be calculated to compute the remoulded duration of the
Comparing the Macaulay Duration and the Modified Duration
Since the Macaulay duration measures the weighted average every now an investor must hold a bond until the present value of the bond’s cash flows is equal to the amount pay off for the bond, it is often used by bond managers looking manage bond portfolio risk with immunization procedures.
In contrast, the modified duration identifies how much the duration changes for each percentage change in the yield, while plan how much a change in the interest rates impact the price of a bond. Thus, the modified duration can provide a risk estimation to bond investors by approximating how much the price of a bond could decline with an increase in interest rates. It’s impressive to note that bond prices and interest rates have an inverse relationship with each other.