FINC4101 Investment Analysis

1. FINC4101 Investment Analysis Instructor: Dr. Leng Ling Topic: Interest Rate Risk

2. Learning objectives • Define interest rate risk for bonds. • Recognize maturity as a major determinant of interest rate risk. • Compute Macaulay’s duration and modified duration. • Identify the determinants of Macaulay’s duration.

3. Portfolio Theory Foreign Exchange Asset Pricing FI400 Equity Derivatives Fixed Income Market Efficiency Concept Map

4. Interest rate risk Why are changes in interest rates a risk for bonds? • Interest rate risk: Sensitivity of a bond’s price to changes in market interest rate (i.e., yield to maturity). • In our discussion, interest rate risk and interest rate sensitivity are synonymous. Changes in bond price (capital gain/loss) Uncertainty in return from bond investment Changes in interest rates

5. Bond Pricing Relationships (1) D C B A

6. Bond Pricing Relationships (2) For all 4 bonds, we observe that: • Bond prices and yields are inversely related. • As YTM increases, bond price falls • As YTM decreases, bond price rises • Price-yield relationship is convex. • An increase in YTM results in a smaller price change than a decrease in YTM of equal magnitude. • This property is called “convexity”.

7. Bond Pricing Relationships (3) Comparing bonds A and B, we see that: • Interest rate sensitivity increases with maturity. • Prices of long-maturity bonds tend to be more sensitive to interest rate changes than prices of short-maturity bonds. • Interest rate sensitivity increases at a decreasing rate as maturity increases.

8. Bond Pricing Relationships (4) Comparing bonds B and C, we see that, • Interest rate sensitivity is inversely related to coupon rate. • Prices of high-coupon bonds are less sensitive to changes in interest rates than prices of low-coupon bonds. Comparing bonds C and D, we see that, • Interest rate sensitivity is inversely related to the YTM at which the bond currently is selling.

9. Maturity and interest rate risk • A bond’s maturity is a major determinant of interest rate risk. • Maturity alone is not enough to measure interest rate risk. • Cash flows received before maturity affects the relationship between maturity and interest rate sensitivity. • Consider the following…

10. Coupon bond vs. zero coupon bond • Prices of 8% annual coupon bonds • Prices of zero-coupon bonds

11. Measuring interest rate risk • An acceptable measure of interest rate risk must account for: • Time to maturity. • Cash flows which are paid out during the life of the bond. • Such a measure is Macaulay’s duration.

12. Macaulay’s duration (1) • A measure of interest rate sensitivity. • Macaulay’s duration is the weighted average of the times to each coupon or principal payment made by the bond. • Often referred to as ‘duration’. • The weight for each payment time is the PV of the payment divided by the bond price.

13. Macaulay’s duration (2) Time until 4th cash flow Weight of 4th cash flow Cash flow paid at time t

14. Examples: 3-year bond, 8%, annual coupon payments, YTM = 10%

15. Macaulay’s duration measures interest rate sensitivity Higher duration is associated with larger percentage price changes, i.e., greater interest rate sensitivity.

16. Macaulay’s duration for semi-annual payment coupon bond • For a semi-annual coupon bond, use the same formula as for an annual coupon bond, but: • Each period is a semi-annual period • CFt : cash flow for semi-annual period t • Yt : semi-annual YTM • Result: semi-annual Macaulay duration. Annual Macaulay duration = Semi-annual Macaulay duration/2

17. Example: 3-year bond, 8%, semi-annual coupon payments, YTM = 10% Semi-annual duration Annualized duration

18. Modified duration (1) • Turns out that we can transform Macaulay’s duration into a measure that estimates percentage price change for a given change in YTM. • This measure is called Modified duration. • Directly gauges the impact of interest rate risk because it approximates the change in price when yield changes (increases or decreases).

19. Modified duration (2) • Modified duration, D* YTM (in decimal)

20. Using Modified duration to estimate percentage price change Dollar change in bond price Change in YTM in decimal Initial price Modified duration

21. Using Modified duration to estimate dollar price change • For a given change in yield, , the predicted price =

22. Interpretation • As Macaulay’s duration ↑, interest rate sensitivity ↑ , vice versa. • As modified duration ↑, interest rate sensitivity ↑ , vice versa.

23. Problems (1) • A nine-year bond has a yield of 10% and a (Macaulay) duration of 7.194 years. If the bond’s yield increases by 50 basis points, what is the percentage change in the bond’s price. • Note: • 1 basis point = 0.0001 or 0.01% • 100 basis points = 1%

24. Problems (2) • A 30-year maturity bond making annual coupon payments with a coupon a rate of 12% has a duration of 11.54 years. The bond currently sells at a yield to maturity of 8%. If the yield to maturity falls to 7%, compute: • The percentage price change. • The estimated dollar price change. • The predicted price.

25. Problems (3) • A six-year 6.1% semi-annual coupon bond has a yield to maturity of 10% and a semi-annual Macaulay duration of 10.014. • What is the annual Macaulay duration? • What is the modified duration on a semi-annual basis? • What is the modified duration on an annual basis? • If the yield increases by 25 basis points, what is the percentage price change using the annual modified duration?

26. Determinants of duration (1) • Rule 1: The duration of a zero-coupon bond equals its time to maturity. • Rule 2: Holding time to maturity and yield to maturity constant, a bond’s duration and interest rate sensitivity are higher when the coupon rate is lower.

27. Determinants of duration (2) • Rule 3: Holding the coupon rate constant, a bond’s duration and interest rate sensitivity generally increase with time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par. • Rule 4: Holding other factors constant, the duration and interest rate sensitivity of a coupon bond are higher when the bond’s yield to maturity is lower.

28. Determinants of duration (3) • Rule 5: The duration of a perpetuity is

29. Graphical summary Rule 1 Rule 4 Rule 2 Rule 3

30. Conceptual problems (1) • Rank the following bonds in order of descending duration.

31. Conceptual problems (2) • Which set of conditions will result in a bond with the greatest price volatility? • A high coupon and a short maturity • A high coupon and a long maturity • A low coupon and a short maturity • A low coupon and a long maturity

32. Conceptual problems (3) • An investor who expects declining interest rates would be likely to purchase a bond that has a ________ coupon and a ________ term to maturity. • Low, long • High, short • High, long • Zero, long

33. Conceptual problems (4) • With a zero-coupon bond: • Duration equals the weighted average term to maturity. • Term to maturity equals duration. • Weighted average term to maturity equals the term to maturity. • All of the above.

34. Summary • Interest rate fluctuations are a key risk to bonds. • Maturity is a major determinant of interest rate risk. • Compute Macaulay’s duration and modified duration. • Determinants of Macaulay’s duration.

35. Practice 7 • Chapter 11: 4,8,9,12,14.

36. Homework 7 1. A 10% semi-annual coupon bond currently sold at $950 will mature after 20 years. Investors expect that the firm will be able to make good on the remaining interest payments but that at the maturity date bondholders will receive full $1000 par value with 0.5 probability and 70% of par value with 0.5 probability. Compute the expected bond equivalent YTM. 2. Suppose you are a money manager who manages a portfolio of two bonds. A bond is a 5% semi-annual coupon bond with 5 years to maturity. Its par value is $1000. The YTM is 7%. There are 10,000 unit of A bonds hold in the portfolio. B bond is a 3-year 6% semi-annual coupon bond of $500 par value and its YTM is 8%. You hold 5,000 unit of B bond. (You must keep 5 decimal places in all steps, eg. $5.67867, otherwise you get 0) (1) What is the duration of A and B bond? (you can use excel) (2) If the YTM for both bonds increase by 50 basis points, what is the change in the portfolio value? You have to use modified duration to solve this question.