(time to payment), which is 1.78 years in this case. For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the : (2) (3) where: indexes the cash flows, is the present value of the th cash payment from an asset, is the cash flow of the th payment from an asset, is the yield to maturity (continuously compounded) for an asset, is the time in years until the th payment will be received, is the present value of all cash payments from the asset until maturity. Macaulay gave two alternative measures: Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and Expression (3) which uses the bond's yield to maturity to calculate discount factors. The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield , not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration. Duration versus Weighted Average Life Similarities in both values and definitions of Macaulay duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5-year fixed-rate interest-only bond would have a Weighted Average Life of 5, and a Macaulay duration that should be very close. Mortgages behave similarly. The differences between the two are as follows: Macaulay duration only measures fixed period cash flows, Weighted Average Life factors in all principal cash flows whether they be in fixed or floating. Thus for Fixed Period Hybrid ARM mortgages, for modeling purposes, the entire fixed period ends on the date of the last fixed payment or the month prior to reset. Macaulay duration discounts all cash flows at the corresponding cost of capital. Weighted Average Life does not discount. Macaulay duration uses both principal and interest when weighting cash flows. Weighted Average Life only uses principal. Modified duration In contrast to Macaulay duration, modified duration (sometimes abbreviated MD) is a price sensitivity measure, defined as the percentage derivative of price with respect to yield (the logarithmic derivative of bond price with respect to yield). Modified duration applies when a bond or other asset is considered as a function of yield.