 ## The Price-Yield Relationship

Duration is a measure of the average (cash-weighted) term-to-maturity of a bond. The are two types of duration, Macaulay duration and modified duration. Macaulay duration is useful in immunization, where a portfolio of bonds is constructed to fund a known liability. Modified duration is an extension of Macaulay duration and is a useful measure of the sensitivity of a bond's price (the present value of its cash flows) to interest rate movements.

## Macaulay Duration

The calculation of Macaulay Duration is shown below: ## Modified Duration

Modified duration is a measure of the price sensitivity of a bond to interest rate movements. It is calculated as shown below:

Modified Duration = Macaulay Duration /( 1 + y/n), where y = yield to maturity and n = number of discounting periods in year ( 2 for semi - ann pay bonds )

Then, % Price Change = -1 * Modified Duration * Yield Change

Modified duration indicates the percentage change in the price of a bond for a given change in yield. The percentage change applies to the price of the bond including accrued interest. In the section showing a bonds price as the present value of its cash flows, the bond shown was priced initially at par (100), when the YTM was 7.5%, with a Macaulay Duration of 4.26 years. The bond was repriced for an increase and decrease in rates of 2.5%. The Modified Duration for this bond will be: Dmod = -1 * 4.26 / (1 + .075/2) = 4.106 years. Therefore, a change in the yield of +/- 2.5% should result in a % change in the price of the bond of: -/+ 4.106 * .025 = +/- 0.10265 (+/- 10.265 %). Since the bond was initially priced at par, the estimated prices are : \$110.27 at 5.00% and \$89.74 at 10.00%. The actual prices were: \$110.94 at 5.00% and \$90.35 at 10.00%. The discrepancy between the estimated change in the bond price and the actual change is due to the convexity of the bond, which must be included in the price change calculation when the yield change is large. However, modified duration is still a good indication of the potential price volatility of a bond. ## Convexity

The previous percentage price change calculation was inaccurate because it failed to account for the convexity of the bond. Convexity is a measure of the amount of "whip" in the bond's price yield curve (see above) and is so named because of the convex shape of the curve. Because of the shape of the price yield curve, for a given change in yield down or up, the gain in price for a drop in yield will be greater than the fall in price due to an equal rise in yields. This slight "upside capture, downside protection" is what convexity accounts for. Mathematically Dmod is the first derivative of price with respect to yield and convexity is the second (or convexity is the first derivative of modified duration) derivative of price with respect to yield. An easier way to think of it is that convexity is the rate of change of duration with yield, and accounts for the fact that as the yield decreases, the slope of the price - yield curve, and duration, will increase. Similarly, as the yield increases, the slope of the curve will decrease, as will the duration. By using convexity in the yield change calculation, a much closer approximation is achieved (an exact calculation would require many more terms and is not useful).

Using convexity (C) and Dmod then: % Price Chg. = -1 * D mod * Yield Chg. + C/2 * Yield Chg * Yield Chg.

Using the previous example, convexity can be calculated and results in the expected price change being: \$111.02 at 5.00% and \$90.49 at 10.00% The actual prices were: \$110.94 at 5.00% and \$90.35 at 10.00% 