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

2. 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

3. 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

4. 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

5. 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

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

7. 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

8. 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

9. 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:
= –  .1% = –.2566%
11/12/2018
Professor Yao

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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

19. 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

20. 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

21. 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

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

23. 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

24. 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