FINC4101 Investment Analysis

1. FINC4101 Investment Analysis Instructor: Dr. Leng Ling Topic: Term Structure Analysis

2. Learning objectives • Define term structure of interest rates and yield curve. • Explain the purpose of theories of the term structure • Describe the expectations theory and discuss its implications. • Show how different expectations of future short-term interest rates can lead to different yield curves. • Use the expectations theory to infer future short-term interest rates (forward rates). • Describe the liquidity preference theory. • Define the liquidity premium.

3. Term structure of interest rates • Term structure of interest rates: • Relationship between yields to maturity and term to maturity across bonds. • Yield curve: The graphical representation of the term structure.

4. Term structure/ yield curve has 4 different ‘looks’ • Term structure/ yield curve can take on one of the following shapes: • Rising/ upward sloping (most common) • Inverted/ downward sloping/ falling • Hump-shaped, rising and then falling • Flat

5. Term structure/ yield curve has 4 different ‘looks’

6. Expectations Theory (1) • Yields to maturity are determined by expectations of future short-term interest rates. • In other words, the shape of the term structure/ yield curve depends on the expected short-term interest rates in the future.

7. Expectations Theory (2) • Depending on what investors expect the future short-term rate to be: • Higher than current short-term rate, or • Lower than current short-term rate, or • Same as current short-term rate We get different yield curves.

8. Expectations Theory (3) • market expecting increases in future short-term interest rates leads to upward sloping yield curve. • market expecting decreases in future short-term interest rates leads to downward sloping yield curve. • market expecting no change in future short-term interest rates leads to flat yield curve.

9. How does the expectations theory work? • Current 1-year interest rate is 8%. • Suppose everyone in the market expects that one year from now, the 1-year interest rate will rise to 10%. • How does this expectation determine the current 2-year interest rate? • All interest rates are quoted on an annual basis. • In this situation, investors can choose one of two possible strategies.

10. Investment strategies • For simplicity, assume • All bonds are zero-coupon bonds with $1000 face value. • You can invest in fractional amount of bonds • Rollover: • Buy 1-year bond now. • When it matures, buy another 1-year bond next year and hold till it matures. • Buy-and-hold: • Buy a 2-year bond now and hold it till it matures in year 2. • The expectations theory says that the two strategies should produce the same expected total returns.

11. Deriving the 2-year interest rate (1) • Suppose you have $1 to invest. • With the rollover strategy, at the end of year 1, you get: $1 x (1 + current 1-year interest rate) = 1 x (1.08) = 1.08 • Suppose at the end of year 1, the 1-year interest rate is 10% as expected. • You invest $1.08 (total proceeds) in a 1-year bond promising 10%. • After two years, you get: 1.08 x (1.10) = $1.188

12. Deriving the 2-year interest rate (2) • Let y2 be the 2-year interest rate. • According to the expectations theory, investing $1 in a 2-year bond with an interest rate of y2 must also produce a total return of $1.188. That is,

13. Compare 1-year and 2-year rates • 1-year interest rate: 8% • 2-year interest rate: 8.995% • Conclusion: longer maturity bonds have higher interest rates/ yields to maturity. • The term structure is upward sloping! * If we observe 1-year rate 8% and 2-year rate 8.995%, then market expects that, one year from now, the 1-year rate will increase from current 8% to 10%.

14. Verify for yourself • If the market expects future 1-year interest rate to fall to 6%: • 2-year interest rate is 6.995%. • Yield curve is downward sloping. • If market expects future 1-year interest rate to stay at 8%: • 2-year interest rate is 8%. • Yield curve is flat.

15. Inferring expected future rates (1) • Expectations theory says that the expected total returns from the buy-and-hold strategy and the rollover strategy are the same. • We can use this assumption to infer expected future interest rates, which are called “forward rates”. • Forward rate: The inferred short-term interest rate for a future period that makes the expected total return of a long-term bond equal to that of rolling over short-term bonds.

16. Inferring expected future rates (2) • In general, we obtain the forward rate by equating the return on an n-period zero-coupon bond with that of an (n – 1)-period zero-coupon bond rolled over into a one-year bond in year n: (1 + yn)n = (1 + yn-1)n-1(1 + fn) YTM of n-period zero coupon bond YTM of (n-1)-period zero coupon bond Forward rate for period n.

17. Inferring expected future rates (3) • Thus, the forward rate formula is

18. Applying the forward rate formula • Two-year maturity bonds offer yield-to-maturity of 6%, and three-year bonds have yields of 7%. What is the forward rate for the third year? • Verify that the forward rate is 9.03%

19. The current yield curve for default-free zero-coupon bonds is as follows: Assume that the zero-coupon bonds have a face value of $1000. Problems (1)

20. Problems (2) • What are the implied one-year forward rates? • Assume that the pure expectations hypothesis is correct. If market expectations are accurate, one year from now what will be the yields to maturity on one- and two-year zero coupon bonds? • If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the first year? What if you purchase a three-year zero-coupon bond?

21. Liquidity Preference Theory • Investors prefer to hold short-term bonds because short-term bonds have more ‘liquidity’ than long-term bonds. • Short-term bonds offer greater price certainty and trade in more active markets. • Investors like these attributes. • Because long-term bonds are less liquid by comparison, investors demand a liquidity premium for holding long-term bonds. • Liquidity premium: extra expected return demanded by investors as compensation for the lower liquidity of long-term bonds.

22. Liquidity Preference Theory • The theory says that investors demand a liquidity premium on long-term bonds. • Thus, long-term bonds have higher interest rates than short-term bonds. • This gives rise to an upward sloping yield curve. • Even if future short-term interest rates are expected to remain unchanged, yield curve will be upward sloping because of the liquidity premium.

23. Liquidity premium • The spread between the forward rate of interest and the expected short-term rate: Liquidity premium = forward rate – expected short-term rate Re-arranging the formula, we get Forward rate = expected short-term rate + liquidity premium So, forward rate > expected short-term rate

24. More problems (1) • The yield to maturity on one-year zero coupon bonds is 8%. The yield to maturity on two-year zero-coupon bonds is 9%. a) What is the forward rate for the second year?

25. More problems (2) b) If you believe the expectations theory, what is your best guess as to the expected short-term interest rate next year? c) If you believe the liquidity preference theory, is your best guess as to next year’s short-term interest rate higher or lower than in (b)?

26. More problems (3) • Which of the following statements is true? • The expectations theory indicates a flat yield curve if anticipated future short-term rates exceed current short-term rates. • The basic conclusion of the expectations theory is that the long-term rate is equal to the anticipated short-term rate. • The liquidity preference theory indicates that, all other things being equal, longer maturities will have higher yields. • The liquidity preference theory states that a rising yield curve implies that the market anticipates increases in interest rates.

27. Summary • Term structure of interest rates. • Theories of the term structure try to explain the shape of the term structure/ yield curve. • Pure Expectations theory and its implications. • Calculate forward rates using observed interest rates of different maturities. • Liquidity Preference theory and its implications. • Relationship between forward rate, expected short-term rate and liquidity premium.

28. Practice 8 • Chapter 10: 41(a)(b), 43.