do not arise for the standard total modified duration because of the dependence of the key rate durations on the specific type of the yield curve used to value the instruments (see Coleman, 2011 ). Formulas For a standard bond with fixed, semi-annual payments the bond duration closed-form formula is: FV = par value C = coupon payment per period (half-year) i = discount rate per period (half-year) a = fraction of a period remaining until next coupon payment m = number of full coupon periods until maturity P = bond price (present value of cash flows discounted with rate i) For a bond with coupon frequency but an integer number of periods (so that there is no fractional payment period), the formula simplifies to: where y = Yield (per year, in percent), c = Coupon (per year, in decimal form), m = Number of coupon periods. Example Consider a 2-year bond with face value of $100, a 20% semi-annual coupon, and a yield of 4% semi-annually compounded. The total PV will be: The Macaulay duration is then . The simple formula above gives (y/k =.04/2=.02, c/k = 20/2 = 10): The modified duration, measured as percentage change in price per one percentage point change in yield, is: (% change in price per 1 percentage point change in yield) The DV01, measured as dollar change in price for a $100 nominal bond for a one percentage point change in yield, is ($ per 1 percentage point change in yield) where the division by 100 is because modified duration is the percentage change. Step-by-step example Consider a bond with a $1000 face value, 5% coupon rate and 6.5% annual yield, with maturity in 5 years. The steps to compute duration are the following: 1. Estimate the bond value The coupons will be $50 in years 1, 2, 3 and 4. Then, on year 5, the bond will pay coupon and principal, for a total of $1050. Discounting to present value at 6.5%, the bond value is $937.66. The detail is the following: Year 1: $50 / (1 + 6.5%) ^ 1 = 46.95 Year 2: $50 / (1 + 6.5%) ^ 2 = 44.08 Year 3: $50 / (1 + 6.5%) ^ 3 = 41.39 Year 4: $50 / (1 + 6.5%) ^ 4 = 38.87 Year 5: $1050 / (1 + 6.5%) ^ 5 = 766.37 2. Multiply the time