1. Yields & Prices: Continued
Chapter 11

2. Learning Objectives
Understand interest rate risk and the key bond pricing relation
Compute and understand the valuation implications of:
Duration
Modified Duration, and
Convexity of a bond portfolio
Construct immunized bond portfolios

3. Prices and Yields
Remember: Yield changes have a larger impact on longer maturity bonds
All else equal price changes are larger the lower the coupon rate
SO: The longer the maturity and the lower the coupon rate the greater the price fluctuation when interest rates change

4. Bond Prices as a Function of Change in YtM

5. Rate Changes and Bond Prices
Known as interest rate risk
Consider three bond
A: 8% Coupon Annual, 4 Years till maturity
B: 8% Coupon Annual, 10 Years till maturity
A: 4% Coupon Annual, 4 Years till maturity
Calculate the change in the price of each bond if:
Interest rates fall from 8% to 6%
Interest rates rise from 8% to 10%

6. Measuring Interest Rate Risk
We can measure a bond’s interest rate risk with DURATION
Duration: Measures a bond’s effective maturity
Can tells us the effective average maturity of a portfolio of bonds
The weighted average of the time until each payment is received
Weights are proportional to the payment’s PV
Duration is shorter than maturity for coupon bonds
Duration is equal to maturity for zeros.

7. Duration Calculation
CFt = Cash flow at time t
y = YTM

8. Duration Example
What is the duration of a 2 year 12% annual bond? The YTM is 10%.
Price?
Duration?
T (Years) CF
P.V. Wt
t*Wt 1 2
Price = 1,034.71
T CF PV Wt
/ = 0.105
1, / =
Duration = 1* * =

9. Duration as a Risk Measure
When yields change the resulting price change is proportional to Duration
Practitioners generally Modify Duration
Modified Duration = D* = D/(1+y)

10. Duration Example 2 Two bonds have a duration of 1.8852 years
8% 2-year bond with YTM=10%
Zero coupon bond maturing in years
Semiannual compounding
Duration in semi annual periods
yrs x 2 = semiannual periods
Modified D = /(1+0.05) = periods
What happens if interest rates increase by 0.01%?
= x 0.01% = %
Bond prices will fall by %
Bonds with equal D have the same interest rate sensitivity.

11. Duration Determinants
The duration of a zero-coupon bond equals its time to maturity
Holding maturity constant, a bond’s duration is higher when the coupon rate is lower
Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity
Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
The duration of a level perpetuity is equal to:
(1 + y) / y

12. Duration & Maturity

13. Portfolio Duration Example
You are managing a $1 million portfolio. Your target duration is 10 years. You can choose from two bonds: a zero-coupon bond with a maturity of 5 years and a perpetuity, each currently yielding 5%.
How much of each bond will you hold in your portfolio? (Hint: Start with the perpetuity’s duration)
How do these fractions change next year if target duration is now 9 years?

14. Immunization Basics: Match the duration of the assets and liabilities
A strategy to shield the net worth of a bond
Control interest rate risk
Widely used by pension funds, insurance companies, and banks
Basics: Match the duration of the assets and liabilities
As a result, value of assets will track the value of liabilities whether rates rise or fall

15. Immunization Example We need $14,693.28 in five years (Liability)
Received $10,000 and guaranteed an 8% return
We can invest $10,000 in a 6yr 8% (an) bond (Asset)
Duration of the obligation and asset is 5 years
Cashflows
Yr 5 8%
Yr 5 7%
Yr 5 9% 1
800
1,088.39
1,048.64
1,129.27 2
1,007.77
980.03
1,036.02 3
933.12
915.92
950.48 4
864.00
856.00
872.00 5
800.00 6
10,800
10,000.00
10,093.46
9,908.26
14,693.28
14,694.05
14,696.03

16. Tuition
You have tuition expense of $18,000 per semester (assume semi-annual) for the next two years. Bonds currently yield 8%.
What is the duration of your obligation?
What is the duration of a zero that would immunize you, and its future redemption value?
What happens to your net position if yields increase to 9%?
Difference between obligation and asset
Time Cash PV W T*W
To immunize the obligation buy a 0 with semi-annual periods till maturity
The PV of the 0 must be 65, so 65,338.11*(1.04)^ = 71, (FV)
If rates increase to 9%
Obligation PV becomes 64,575.46
0 PV becomes 64,574.54
Net postion loss $0.92

17. Breaking Down Interest Effects
When interest rates change it affects the bond investor in two ways
Affects the price of the bond (Price Risk)
Negative relation
Affects the investment opportunities available for coupon payments (Reinvestment Risk)
Positive relation
When a portfolio is immunized the Price risk and reinvestment rate risk exactly cancel out

18. Immunization Example 2
Suppose you are managing a pension’s obligation to make perpetual $2M payments. The YTM on all bonds is 16%.
5 yr 12% (annual) bonds have a 4yr duration
20 yr 6% (annual) bonds have an 11yr duration
What are the weights of your immunized portfolio?
What is the par value of your holdings in the 20-year bond?

19. Immunization Example 3
Your pension plan will pay you $10,000 per year for 10 years. The first payment will be in 5 years. The pension fund wants to immunize its position. The current interest rate is 10%
What is the duration of its obligations to you?
If the plan uses 5-year and 20-year zero coupon bonds to construct the immunized position, how much money ought to be placed in each position?

20. Rebalancing
An bond’s duration will change as yields changes, rebalancing is the practice of altering our weights in the portfolio to keep the durations matched

21. Immunization Alternative
Cash Flow Matching
Match the cash flows from the fixed income assets with obligation
Automatic immunization
Dedication is cash flow matching over multiple periods
Not widely used because of constraints associated with bond choices

22. Actual vs Duration Approx Price Change 30 yr, 8% Coupon, 8% YTM

23. The Real Price Yield Relation
Bond prices are not linearly related to yields
Duration is a good approximation only for small yields changes
Convexity is the measure of the curvature in the price-yield relation
Bonds with greater convexity have more curvature in the price-yield relationship.
Convexity Correction

24. Convexity of Two Bonds
Which bond is more Convex?

25. Why do Investors Like Convexity?
Bonds with greater curvature gain more in price when yields fall than they lose when yields rise.
This asymmetry becomes more attractive as interest rates become more volatile
Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal.

26. Convexity Example
A 12% (Annual) 30-year bond has a duration of years and convexity of The bond currently sells at a yield to maturity of 8%.
Find the bond price changes if YTM falls to 7% or rises to 9%.
What is the price change according to the duration rule, and the duration-with-convexity rule
Price at 8% N=30, I/Y = 8, PV=?, PMT=120, FV=1000 PV = 1,450.31
Price at 7% N=30, I/Y = 7, PV=?, PMT=120, FV=1000 PV = 1, (11.73%)
Price at 9% N=30, I/Y = 9, PV=?, PMT=120, FV=1000 PV = 1, (-9.8%)
Modified Duration = 11.54/1.08 =
Duration Rule 7%= *-0.01 =
Duration with Convexity 7%= * (1/2)*(192.4*(-0.012))) = = 11.65%
Duration Rule 9%= *0.01 =
Duration with Convexity 9%= * (1/2)*(192.4*(-0.012))) = = %

27. Convexity Example 2
A year zero-coupon bond has a YTM=8% (effective annual) has convexity of and modified duration of years.
A 30-year, 6% coupon (annual) bond also has YTM=8% has nearly identical duration = 11.79, but higher convexity=231.2.
YTM of both bonds increases to 9%. What is the percentage loss on each bond? What percentage loss is predicted by duration with convexity rule?
What if YTM decreases to 7%?
Given the above results, what is the attraction of convexity?
Zero Price at 8% N=12.75, I/Y = 8, PV=?, PMT=0, FV=1000 PV =
Coupon Price at 8% N=30, I/Y = 8, PV=?, PMT=60, FV=1000 PV =
Zero Price at 9% N=12.75, I/Y = 9, PV=?, PMT=0, FV=1000 PV = (11.09%)
Coupon Price at 9% N=30, I/Y = 9, PV=?, PMT=60, FV=1000 PV =
Zero Price at 7% N=12.75, I/Y = 7, PV=?, PMT=0, FV=1000 PV =
Coupon Price at 7% N=30, I/Y = 7, PV=?, PMT=60, FV=1000 PV =
Zero: Duration with Convexity 9%= * (1/2)*(150.3*(-0.012))) = = %
Coupon: Duration with Convexity 9%= * (1/2)*(231.2*(-0.012))) = = %
Zero: Duration with Convexity 9%= * (1/2)*(150.3*(-0.012))) = = %
Coupon: Duration with Convexity 9%= * (1/2)*(231.2*(-0.012))) = = %

28. Active Bond Management
There are two ways to make money
Interest rate forecasting
Anticipating changes in the whole market
Identifying relative mispricings
However, you must be right and first
If everyone already knows it, then its already priced

29. Active Bond Strategies
Substitution Swap
Switch one bond for a nearly identical (mispriced)
Intermarket Spread Swap
Switching two bonds from different market segments (mispriced)
Rate Anticipation Swap
Changing between bond duration (Rate Forecasting)
Pure Yield Swap
Moving into longer duration bonds for the higher rate