Risks of return Default risk Does the issuer do it’s obligations? Interest rate volatility risk How much is the interest rate volatile? Reinvestment risk What is the rate of periodical payments return? Liquidity risk Is there an active secondary market?
Price volatility in option-free bonds There is a reverse relationship between yield to maturity and price. Price Yield to maturity
Factors affecting interest rate volatility Coupon rate All other factors constant, the lower the coupon rate, the greater the price volatility. maturity All other factors constant, the longer the maturity, the greater the price volatility. Yield to maturity All other factors constant, the higher the yield level, the lower the price volatility.
Percentage price change for Four Hypothetical Bonds Initial yield for all four bonds is 6% Percentage price change 9% 20-year9% 5-year6% 20-year6% 5-yearNew yield 25.048.5727.368.984.00% 11.534.1712.554.385.00% 5.542.066.022.165.50% 1.070.411.170.435.90% 0.110.040.120.045.99% -0.11-0.04-0.12-0.046.01% -1.06-0.41-1.15-0.436.10% -5.13-2.01-5.55-2.116.50% -9.89-3.97-10.68-4.167.00% -18.40-7.75-19.798.118.00%
Duration Duration is a measure of interest rate volatility risk: Duration is the measure of fixed income securities price sensitivity versus interest rate changes. Duration encompasses the three factors (coupon, maturity and yield level) that affects bond’s price volatility.
Duration Duration is a proxy for maturity: Duration is a proxy better than maturity and may be considered as effective maturity of fixed income securities. Duration is standardized weighted average of bond’s term to maturity where the weights are the present value of the cash flows.
Duration is elasticity Duration is a proxy that shows bond’s percentage price change when yield changes.
Price equation of an option-free bond P: price C: periodical coupon interest Y: yield to maturity M: maturity value (face value) N: number of periods P: price C: periodical coupon interest Y: yield to maturity M: maturity value (face value) N: number of periods
First derivative of price equation The first derivative of price equation shows the approximate change in price when small change in yield occurs.
Example 1: Duration calculation Duration for a 9% 5-year bond selling to yield 6% with semiannual coupon payments and face value of 100$ is: PV× tPresent valueCash flowPeriod 4.3689 4.5 1 8.48344.24174.5 2 12.35444.11814.5 3 15.99283.99824.5 4 19.40873.88174.5 5 22.61213.76874.5 6 25.61243.65894.5 7 28.41873.55234.5 8 31.03993.44894.5 9 777.578177.7578104.5 10 945.8694112.7953Total 8.38Macaulay duration (in half years) 4.19Macaulay duration (in years) 4.07Modified duration (in years)
Example 2: Using duration to approximate price change Duration for a 9% 20-year bond selling to yield 6% with semiannual coupon payments and face value of 100$ is: If yields increase instantaneously from 6% to 6.1%, the percentage price change is: If yields decrease instantaneously from 6% to 5.9%, the percentage price change is:
Example 3: When duration does not work well? For the previous example, the real and approximate price change when yields change are as follows: difference Approximate price change (based on duration) Real price change (based on price equation) Yield change (in percent) 0.06 -1.66-1.600.1 0.04 +1.66+1.70-0.1 2.92-21.32 -18.402.0 3.72+221.32 +25.04-2.0
Reason of duration inadequacy Duration does not capture the effect of convexity of a bond on it’s price performance. Price Yield Underestimation Overestimation
Improvement in price change approximation Taylor series for price equation:
Convexity calculation Convexity is the second derivative of the price equation divided by the price.
Example 4: convexity calculation Convexity for a 9% 5-year bond selling to yield 6% with semiannual coupon payments and face value of 100$ is: PV × t × (t+1)PVCash flowPeriod 8.73784.36894.5 1 25.45024.24174.5 2 49.41724.11814.5 3 79.9643.99824.5 4 116.4513.88174.5 5 158.28543.76874.5 6 204.89843.65894.5 7 255.76563.55234.5 8 310.4013.44894.5 9 8553.35877.7578104.5 10 9762.729112.7953Total 40.792Convexity (in half years ) 10.198Convexity (in years)
Example 5: Using convexity to approximate price change Convexity for a 9% 20-year bond selling to yield 6% with semiannual coupon payments and face value of 100$ is: If yields increase instantaneously from 6% to 8%, the percentage price change is: If yields decrease instantaneously from 6% to 4%, the percentage price change is:
Using duration and convexity simultaneously Estimated percentage price change
Example 6: Comparing approximate price change using duration and convexity and real price change difference Approximate price change (based on convexity) Real price change (based on price equation) Yield change (in percent) -0.36-18.04 -18.402 0.44+24.60 +25.04-2