Chapter 3 Term Structure of Interest Rates: Empirical Properties and Classical Theories

1. FIXED-INCOME SECURITIES Chapter 3 Term Structure of Interest Rates: Empirical Properties and Classical Theories

2. Outline • Types of TS • Shapes of the TS • Dynamics of the TS • Stylized Facts • Theories of the TS

3. Types of Term Structures • The term structure of interest rates is the series of interest rates ordered by term-to-maturity at a given time • The nature of interest rate determines the nature of the term structure • The term structure of yields to maturity • The term structure of zero coupon rates • The term structure of forward rates • TS shapes • Quasi-flat • Increasing • Decreasing • Humped How is the curve today?

4. Quasi-Flat Quasi-Flat

5. Increasing Increasing

6. Decreasing Decreasing (or inverted)

7. Humped (1) Humped (decreasing then increasing)

8. Humped (2) Humped (increasing then decreasing)

9. Dynamics of the Term Structure • The term structure of interest rates changes in response to • Wide economic shocks • Market-specific events • Example • On 10/31/01, Treasury announces that there will not be any further issuance of 30 year bonds • Price of existing 30 year bonds is pushed up (buying pressure) • 30 year rate is pushed down

10. Example – US YTM TS

11. Stylized Facts (1) : Mean Reversion • Mean reversion: high (low) values tend to be followed by low (high) values • Example: 10 Y swap rate versus Dow Chemical

12. Stylized Facts (2) : Correlation • Rates with different maturities are • Positively correlated one another • Not perfectly correlated though (more than one factor) • Correlation decreases with difference in maturity • Example: France (1995-2000)

13. Stylized Facts (3) • The evolution of the interest rate curve can be split into three standard movements • Shift movements (changes in level), which account for 70 to 80% of observed movements on average • A twist movement (changes in slope), which accounts for 15 to 30% of observed movements on average • A butterfly movement (changes in curvature), which accounts for 1 to 5% of observed movements on average • That 3 factors account for more than 90% of the changes in the TS is valid • Whatever the time period • Whatever the market

14. Shift Movements

15. Twist Movements

16. Butterfly Movements

17. Theories of the Term Structure • Studying the TS boils down to wondering about the preferences of participants' for curve maturities • Investors • Borrowers • Indeed, if they were indifferent in terms of maturity • Interest rate curves would be invariably flat • Notion of TS would be meaningless • Market participants' preferences can be guided • By their expectations • By the nature of their liability or asset • By the level of the risk premiums they require

18. Theories of the Term Structure • Term structure theories attempt to account for the relationship between interest rates and their residual maturity • They fall within the following categories • Pure expectations • Pure risk premium • Liquidity premium • Preferred habitat • Market segmentation • To these main types, we can add • The biased expectations theory, that combines the first two theories

19. Theories of the Term Structure • Remember: 1+R0,t = [(1+ R0,1)(1+ F1,2)(1+ F2,3)…(1+ Ft-1,t)]1/t • The pure expectations theory postulates that forward rates exclusively represent future short term rates as expected by the market • The pure risk premium theory postulates that forward rates exclusively represent the risk premium required by the market to hold longer term bonds • The market segmentation theory postulates that • Each of the two main market investor categories is invariably located on a given curve portion (short, long) • As a result, short and long curve segments are perfectly impermeable

20. Pure Expectations • TS reflects market expectations of future short-term rates • An increasing (resp. flat, resp. decreasing) structure means that the market expects an increase (resp. a stagnation, resp. a decrease) in future short-term rates • Example: from a flat curve to an increasing curve • The current TS is flat at 5% • Investors expect a 100bp increase in rates within one year • For simplicity, assume that the short (resp.long) segment of the curve is the one-year (resp. two-year) maturity • Then, under these conditions, the interest rate curve will not remain flat but will be increasing • Why?

21. Pure Expectations (Cont’) • Consider a long-term investor (2-year horizon) • His objective is maximizing his return on the period • Either invests in a long 2-year security or invests in a short 1-year security, then reinvests in one year the proceeds in another 1-year security • Before interest rates adjust at the 6% level • First option returns an annual return of 5% over two years • Second option returns 5% the first year and, according to his expectations, 6% the second year, i.e., 5.5% on average per year over two years • Investor will thus buy short bonds (one year) rather than long bonds (two years) • Similar behavior for the short-term investor (return on 2-y bond after 1 year is 4.05%=(5+105/1.06-100)/100 < 5% (return on 1 year bond) • As a result • The price of the one-year bond will increase (its yield will decrease) • The price of the two-year bond will decrease (its yield will increase) • The curve will steepen

22. Pure Expectations (Cont’) • In summary, market participants behave collectively to let the relative appeal of one maturity compared to the others disappear • In other words, they neutralize initial preferences for some curve maturities • The pure expectations theory has an important limitation • Investors behave in accordance with their expectations for the unique purpose of maximizing their investment return • They are risk neutral - They do not take into account the fact that their expectations may be wrong • The pure risk premium theory includes this contingency

23. Pure Risk Premium • Indeed, if forward rates were perfect predictors of future rates, future bond prices would be known with certainty • Unfortunately, it is not the case • Future interest rates are unknown (re-investment risk) • Future bond prices are unknown (market risk) • Example: an investor with a 3 years horizon • May invest in a 3-year zero coupon bond and holding it until maturity • May invest in a 5-year zero coupon bond and selling it in 3 years • May invest in a 10-year zero coupon bond and selling it in 3 years • What would you prefer? • Return of the first investment is known ex-ante with certainty • Not the case for the 2nd and 3rd • We don’t know the price of these instruments in three years

24. Pure Risk Premium (con’t) • However, we know something about their risk (volatility) • A bond price risk measured by price volatility • Tends to increase with maturity (P’(r)>0) • In a decreasing proportion (P’’(r)<0) • Assume interest rates increase to 6% • The long bond price will fall to 5/1.06+ 105/1.062 = $ 98.17 • The short one will fall to 105/1.06 = $99.06. • Decrease in 2-year bond price nearly twice as big as decrease in 1-year bond price • Pure risk premium theory: TS reflects risk premium required by the market for holding long bonds • The two versions of this theory differ about the shape of the risk premium

25. Pure Risk Premium - Liquidity • Risk premium increases with maturity in a decreasing proportion • Formally 1+Ro,t = [(1+ R0,1) (1+ E(R1,2)+ L2)(1+ E(R2,3)+ L3)… (1+ E(Rt-1,t)+ Lt)]1/t • Lk is the liquidity (actually risk) premium required by the market to invest in a bond maturing in k years 0 = L1 < L2 < L3 < ...< Lt L2 -L1 > L3 -L2 > L4 -L3 > ... > Lt -Lt-1 • Hence, an investor will be interested in holding all the longer bonds as their return contains a high risk premium, offsetting their higher volatility • Implies that a “normal” TS is increasing

26. Preferred Habitat • Postulates that risk premium is not uniformly increasing • Indeed, investors have a preferred investment horizon dictated by the nature of their liabilities (shorter is not always better) • Nevertheless, depending on bond supply and demand on specific segments • Some lenders and borrowers are ready to move away from their preferred habitat • Provided that they receive a risk premium that offsets their price or reinvestment risk aversion • Thus, all curves shapes can be accounted for

27. Market Segmentation • Extreme version of pure risk premium theory • Investors never move away from preferred habitat (infinite risk premia) • Commercial banks invest on a short/medium term basis • Life-insurance companies and pension funds invest on a long term basis • Shape of the curve determined • By supply and demand on short and long-term bond markets • Insurance comp., pension funds are structural buyers of long-term bonds • Commercial banks' behavior is more volatile: banks prefer to lend money directly to corporations and individuals than invest in bond securities • Their demand for short-term bonds is influenced by business conditions • During growth periods, sell bonds to meet corporations' and individuals' demand for loans => relative increase in short-term yields • During slow-down periods, corporations and individuals pay back their loans, thus increasing bank funds; then banks invest in short-term bonds => relative decrease in short-term yields compared to long-term yields

28. Biased Expectations Theory and Stochastic Approach • All these theories are not mutually exclusive • Biased expectations theory is an integrated approach • Combines pure expectations theory and risk premium theory • Postulates that TS reflects market expectations of future interest rates as well as permanent liquidity premia that vary over time • Thus, all curve shapes can be accounted for • Stochastic Approach • Uncertainty about future interest rates is not implicit in current TS • Difficult to correctly anticipate future interest rates driven by surprise effects • TS modeled as a predictable term plus a stochastic process • This theory represents an alternative to traditional theories generally used for pricing and hedging contingent claims

29. Synthesis – Explanation of TS Shapes