In this case one can measure the logarithmic derivative with respect to yield: When the yield is expressed continuously compounded, Macaulay duration and modified duration are numerically equal. To see this, if we take the derivative of price or present value, expression (2), with respect to the continuously compounded yield we see that: In other words, for yields expressed continuously compounded, . where: indexes the cash flows, is the time in years until the th payment will be received, is the present value of all cash payments from the asset. Periodically compounded In financial markets, yields are usually expressed periodically compounded (say annually or semi-annually) instead of continuously compounded. Then expression (2) becomes: To find modified duration, when we take the derivative of the value with respect to the periodically compounded yield we find Rearranging (dividing both sides by -V) gives: which is the well-known relationship between modified duration and Macaulay duration: where: indexes the cash flows, is the compounding frequency per year (1 for annual, 2 for semi-annual, 12 for monthly, 52 for weekly, etc.), is the cash flow of the th payment from an asset, is the time in years until the th payment will be received (e.g. a two-year semi-annual would be represented by a index of 0.5, 1.0, 1.5, and 2.0), is the yield to maturity for an asset, periodically compounded is the present value of all cash payments from the asset. This gives the well-known relation between Macaulay duration and modified duration quoted above. It should be remembered that, even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay duration is a weighted average time until repayment (measured in units of time such as years) while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield. Units Macaulay duration is measured in years. Modified duration is measured as the percent change in price per one unit (percentage point) change in yield per year (for example yield going from 8% per year (y = 0.08) to 9% per year (y = 0.09)). This will give modified duration a numerical value close to the Macaulay duration (and equal when rates are continuously compounded). Formally, modified duration is a semi-elasticity, the percent change in price for a unit change in yield, rather than an elasticity, which is