Duration and convexity for Fixed-Income Securities RES9850 Real Estate Capital Market Professor Rui Yao

Duration and convexity: Outline I. Macaulay duration II. Modified duration III. Examples IV. The uses and limits of duration V. Duration intuition VI. Convexity 11/12/2018 Professor Yao

A Quick Note Fixed income securities’ prices are sensitive to changes in interest rates This sensitivity tends to be greater for longer term bonds But duration is a better measure of term than maturity Duration for 30-year zero = 30 Duration for 30-year coupon with coupon payment < 30 A 30-year mortgage has duration less than a 30-year bond with similar yield Amortization Prepayment option 11/12/2018 Professor Yao

I. (Macauly) duration Weighted average term to maturity Measure of average maturity of the bond’s promised cash flows Duration formula: where: is the share of time t CF in the bond price and t is measured in years 11/12/2018 Professor Yao

Duration - The expanded equation For an annual coupon bond Duration is shorter than maturity for all bonds except zero coupon bonds Duration of a zero-coupon bond is equal to its maturity 11/12/2018 Professor Yao

IV An Example – page 1 Consider a 3-year 10% coupon bond selling at $107.87 to yield 7%. Coupon payments are made annually. 11/12/2018 Professor Yao

II. Modified duration (D*m) Direct measure of price sensitivity to interest rate changes Can be used to estimate percentage price volatility of a bond 11/12/2018 Professor Yao

Derivation of modified duration So D*m measures the sensitivity of the % change in bond price to changes in yield 11/12/2018 Professor Yao

An Example – page 2 Modified duration of this bond: If yields increase to 7.10%, how does the bond price change? The percentage price change of this bond is given by: = –2.5661  .1% = –.2566% 11/12/2018 Professor Yao

An Example – page 3 What is the predicted change in dollar terms? New predicted price: $107.87 – .2768 = $107.5932 Actual dollar price (using PV equation): $107.5966 Good approximation! 11/12/2018 Professor Yao

Summary: Steps for finding the predicted price change Step 1: Find Macaulay duration of bond. Step 2: Find modified duration of bond. Step 3: Recall that when interest rates change, the change in a bond’s price can be related to the change in yield according to the rule: Find percentage price change of bond Find predicted dollar price change in bond Add predicted dollar price change to original price of bond  Predicted new price of bond 11/12/2018 Professor Yao

V. Check your intuition How does each of these changes affect duration? Decreasing the coupon rate. verify this with a 10-year bond with coupon rate from 5% to 15%, and ytm of 10% Decreasing the yield-to-maturity. verify this property with a 10-year bond with coupon rate of 10%, and ytm from 5% to 15% Increasing the time to maturity. verify this property with a par bond with a coupon rate of 10%, and term from 5 to 15 years 11/12/2018 Professor Yao

V. Dollar Duration We have derived the following relationship between duration and price changes (bond returns): Hence Note the term on the RHS of the equation above measures the (absolute value of) slope of the yield-price curve, which is also called dollar duration We can then predict price changes using dollar duration: 11/12/2018 Professor Yao

Duration with intra-year compounding In practice, lots of bonds do not pay annual coupon and we need to change the formula a bit to account for it Some calculus (note: each step in summation is 1/m year so there are m*T terms in total) So dollar duration Duration 11/12/2018 Professor Yao

V. Effective Duration– Numerical Approximation Instead of calculating modified duration based on weighing the time of cash flow with the present value share of the CF, and then modify by dividing by (1+y), we can numerically approximate the modified duration from the slope of price-yield chart: The slope of the yield-price curve is the dollar duration 11/12/2018 Professor Yao

V. Effective Duration – Numerical Approximation We can approximate the slope of the graph / dollar duration by averaging the forward and backward slope (“central difference method”) The duration is then The modified duration then can be estimated as This approach directly uses the idea that the duration measures price sensitivity to interest rate Can duration be negative? 11/12/2018 Professor Yao

VI. Duration and Convexity – Numerical Approximation price P- Po P+ yield y- y+ y° 11/12/2018 Professor Yao

VI. Convexity Duration is the first order approximation for percentage change in bond prices for a one percent change in yield to maturity For a fixed rate non-callable bond, duration underestimates change when yield falls and overestimates when yield rises The difference is captured by convexity Convexity is typically positive for bond Is this good or bad? Mortgage is a difficult product to evaluate due to embedded call options Duration tends to become shorter when interest becomes lower as borrower prepays mortgage Duration becomes longer when interest rate becomes higher as borrower holds on to his mortgage Negative convexity Opposite to the case of a typical bond 11/12/2018 Professor Yao

V. Convexity The forecast of price response using dollar duration is Essentially it is a linear projection using the slope measured at yo However as soon as you move away from yo the slope will change The rate of slope change is captured by dollar convexity A little calculus yields (take first order derivative of dollar duration with respect to yield) 11/12/2018 Professor Yao

V. Convexity – Numerical Approximation What is the predicted dollar duration at y+ using dollar duration from yo and convexity measure at yo? So the average of slopes at y+ and yo , which gives a better approximation of changes in prices when yield changes, is 11/12/2018 Professor Yao

V. Convexity The forecast of price response using duration and convexity In percentage term The term is referred to as convexity 11/12/2018 Professor Yao

V. Convexity with intra-year coupons Dollar convexity with intra-year coupons Convexity with intra-year coupons 11/12/2018 Professor Yao

V. Effective Dollar Convexity – Numerical Approximation Instead of analytical formula, in practice $ convexity is frequently approximated using numerical methods based on price-yield relations Numerical approximation is very useful when cash flow size and timing are uncertain due to built-in options in the bond payments 11/12/2018 Professor Yao

V. Homework 30 year T-bond has a yield to maturity of 3.0% and price at par, and coupon is paid annually. 1. Find out analytically the following measure at 3.0%: A. duration B. modified duration C. dollar duration D. dollar convexity E. convexity 2. Also calculate B, C, D, and E using numerical approximations using a step (delta y) of 1 basis point. How accurate is the approximation compared with analytical solutions from part 1? 3. Use dollar duration measure to predict price when yields change from 1% to 5% at 0.5% interval. 4. Use both duration and convexity to predict bond price when yields change from 1% to 5% at 0.5% interval. note: you can use either the analytical duration/convexity or their numerical approximation for question 3 and 4. 5. What do you conclude when comparing results from 3 and 4? Which one is more accurate and why? 11/12/2018 Professor Yao