Duration and convexity for Fixed-Income Securities

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 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 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 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 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. 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 Professor Yao

Derivation of modified duration • So D*m measures the sensitivity of the % change in bond price to changes in yield 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% 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! 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 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 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: 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 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 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? Professor Yao

VI. Duration and Convexity – Numerical Approximation price P- Po P+ yield y+ y- y° 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 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) 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 Professor Yao

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

V. Convexity with intra-year coupons • Dollar convexity with intra-year coupons • Convexity with intra-year coupons 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 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? Professor Yao

Duration and convexity for Fixed-Income Securities