Analysis under Certainty

1. Analysis under Certainty The one investment certainty is that we are all frequently wrong

2. Up to now • Financial markets and instruments • Specifics of stocks, bonds, and derivatives • Trading process • Financial intermediaries FF 2005/06

3. Plan • Analysis under certainty • Term structure of interest rates • Fixed income instruments • Pricing • Risks • Capital budgeting FF 2005/06

4. Definitions of rates • Reinvestment: • Simple vs compound interest • Frequency of compounding: • Nominal (coupon) rate vs effective (annual) rate • Continuous compounding: • Log-return FF 2005/06

5. Definitions of rates (2) • Yield to maturity / internal yield / bond yield • Rate that equates cash flows on the bond with its market value • Return earned from holding a bond to maturity • Assuming reinvestment at same rate • Par yield • Coupon rate that causes the bond price to equal its face value • Current yield • Annual coupon payment divided by the bond’s price • Often quoted but useless FF 2005/06

6. Definitions of rates (3) • Zero rate • YTM of a zero-coupon bond • How to get zero rates from coupon bond prices? • Bootstrapping method: coupon bond as a ptf of zero-coupon bonds • Spot rate • One-period zero rate • Forward rate • Rate on a one-period credit from T to T+1 FF 2005/06

7. Term structure of interest rates • Relationship between yields and maturities • For bonds of a uniform quality (risks and taxes) • E.g., Treasury / Baa • Equivalent ways to describe TSIR: • Prices of zero-coupon bonds: P(t,T), with P(T,T)=1 • Zero rates: y(t, T) • Forward rates: f(t, T) • Upward sloping yield curve: • Fwd Rate > Zero Rate > Par Yield FF 2005/06

8. Theories of the term structure • Expectations theory: • Unbiased expectations hypothesis: f(t, T) = Et[r(T)] • Term structure is explained by expected spot rates • Upward sloping yield curve: signal that spot rate will increase • Liquidity preference theory: • Investors demand a premium for bonds with higher risk • Long-term bonds require a liquidity premium • Upward sloping yield curve: forward rates higher than expected future zero rates FF 2005/06

9. Theories of the term structure(2) • Preferred habitat: • Investors try to match the life of their assets with liabilities • There is a premium for maturities with insufficient demand • Market segmentation: • Different rates determined independently of each other • SR%: D – corporations financing sr obligations, S – banks • LR%: D – corporations financing lr inv projects, S – insurance co-s, pension funds • Investors don’t react to yield differentials between the maturities FF 2005/06

10. Empirical estimation of TSIR • Discrete rates: • Regression P = cD1 + cD2 + … + (c+F)DT where Dt = 1/P(0,t) = 1/y(0,t)t • Continuous rates: • Regression P = Σt=1:T ct (a0+a1t+a2t2+…) • P = a0[Σt=1:Tct]+a1[Σt=1:Ttct]+a2[Σt=1:Tt2ct]+… FF 2005/06

11. Modeling changes in bond prices • Due to passage of time: • E.g., flat yield curve: ΔP = r P0 • Unanticipated shift in the TSIR: • Need to approximate the function P = f(y) • Duration: sensitivity of a bond’s price to the change in the interest rates FF 2005/06

12. Macaulay’s duration • Wtd-avg maturity of bond payments • Generalized maturity for coupon bonds, D ≤ T • Elasticity of a bond’s price wrt ytm • The larger the duration, the riskier is the bond • For small changes in %: ΔP ≈ -D P Δy/y = -[D/y] P Δy • D* = D/y: modified duration FF 2005/06

13. Macaulay’s duration(2) • Properties: • C, coupon: – • Y, %: – • T, maturity: + • Limitations: • Assumes horizontal TSIR • Applies only to small changes in % FF 2005/06

14. Duration modifications • Convexity • Fisher-Weil duration • For parallel shifts of (non-horizontal) TSIR • Non-parallel shifts: • Two types: LR% usually more stable than SR% • Analytical approach: • E.g., assume d ln y(t,T) = KT-t+1 d ln r(t) • Empirical approach: • Separate estimation of duration for sr and lr % FF 2005/06

15. Conclusions