Lecture 13: Term structure of interest rate

1. Lecture 13:Term structure of interest rate Mishkin Ch 6 – part B page 134-147

2. Review • How do demand and supply curves and shift of curves determine interest rate? • Theory of asset demand: bond market • Liquidity preference framework: money market • Why interest rates are different? • Risk structure of interest rate • Term structure of interest rate

3. Expected return and risk • Risk-return tradeoff. • Then the ‘average’ of return, i.e. expected return is measured by ‘mean’ of R: • risk is measured by variance or standard deviation

4. Risk structure of interest rate

5. Term structure of interest rate

6. Term structure of interest rates • Bonds with identical default risk, liquidity, and tax characteristics may still have different interest rates because the time remaining to maturity is different. • Yield curve is a plot of the yield on bonds with differing terms to maturity but the same risk, liquidity and tax considerations. • Shape of the yield curve: • Usually upward-sloping (long-term i > short-term i ) sometimes ‘inverted’ (long-term i < short-term i ) • Flatimplies short- and long-term rates are similar.

7. Facts about term structure of interest rates • Interest rates on bonds of different maturities move together over time. • When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward and be inverted – ‘mean reversion’. • Yield curves almost always slope upward.

8. Three candidate theories • Expectations theory explains the first two facts but not the third. • Segmented markets theory explains fact three but not the first two. • Liquidity premium theory combines the two theories to explain all three facts.

9. Expectations theory • Assume buyers of bonds do not prefer bonds of one maturity over another -perfect substitutes expected returns of bonds with different maturity should equal. • Conclusion: The interest rate on a long-term bond will equal average of the short-term interest rates that people expect to occur over the life of the long-term bond.

10. Example • Suppose one-year interest rate over the next five years are expected to be: 5%, 6%, 7%, 8% and 9% • Then, interest rate on the two-year bond: • (5% + 6%)/2 = 5. 5% • Interest rate on the five-year bond: • (5% + 6% + 7% + 8% + 9%)/5 = 7% • Interest rates on one to five-year bonds: • 5%, 5.5%, 6%, 6.5%, and 7%

11. Expectations theory - derivation • start with $1 • If invest in one long-term bonds, the net rate of return would be … • If invest in two successive short-term bonds, the net rate of return would be … • Two ways to invest your $1 should yield the same return, otherwise one kind of bond will not be held.

12. Investing in a long-term bond back

13. Investing in two short-term bonds back

14. Expectations theory - derivation

15. Explanation power of expectation theory • Explains why interest rates on bonds with different maturities move together over time (fact 1) current short-term i rise  rise in future  long-term i • Explains why yield curves tend to slope up when short-term rates are low and slope down when short-term rates are high (fact 2) current short-term i low  increase to normal level in the future  long term i would be high • Cannot explain why yield curves usually slope upward (fact 3).

16. Segmented markets theory • Assume: bonds of different maturities are not substitutes, investors have preferences for bonds of one maturity over another. • If investors generally prefer bonds with shorter maturities that have less interest-rate risk, then this explains why yield curves usually slope upward (fact 3). • Can’t explain facts 1 and 2.

17. Liquidity premium theory • Assume bonds of different maturities are substitutes but not perfect substitutes. • The interest rate on a long-term bond will equal the average of short-term interest rates expected to occur over the life of the long-term bond plus a liquidity premium that responds to supply and demand conditions for that bond

18. Liquidity premium theory

19. Example • Suppose one-year interest rate over the next five years are 5%, 6%, 7%, 8%, 9%, liquidity premiums for one to five-year bonds are 0%, 0.25%, 0.5%, 0.75%, 1.0% • Then, interest rate on the two-year bond: (5% + 6%)/2 + 0.25% = 5.75% • Interest rate on the five-year bond: (5% + 6% + 7% + 8% + 9%)/5 + 1.0% = 8% • Interest rates on one to five-year bonds: 5%, 5.75%, 6.5%, 7.25% and 8%.

20. Comparing with the expectations theory, liquidity premium theory implies a more steeply upward sloped yield curve. Gap = liquidity premium

21. Interpret the yield curve using liquidity premium theory • You can figure out what the market is predicting about future short-term interest rates by looking at the slope of the yield curve.

22. Explanatory power of liquidity premium theory • Fact 1: interest rates on different maturity bonds move together over time; • explained by the first term – expected average part. • Fact 2: yield curves tend to slope upward when short-term rates are low and to be inverted when short-term rates are high; • explained by the liquidity premium term in the first case and by a low expected average in the second case. • Fact 3: yield curves typically slope upward; • explained by a larger liquidity premium as the term to maturity lengthens.