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

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

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

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

NES FF 2005/06 5 Definitions of rates (2) Yield to maturity / internal yield / bond yield 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 Assuming reinvestment at same rate Par yield Par yield –Coupon rate that causes the bond price to equal its face value Current yield Current yield –Annual coupon payment divided by the bond’s price –Often quoted but useless

NES FF 2005/06 6 Definitions of rates (3) Zero rate 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 Spot rate –One-period zero rate Forward rate Forward rate –Rate on a one-period credit from T to T+1

NES FF 2005/06 7 Term structure of interest rates Relationship between yields and maturities Relationship between yields and maturities –For bonds of a uniform quality (risks and taxes) –E.g., Treasury / Baa Equivalent ways to describe TSIR: 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: Upward sloping yield curve: –Fwd Rate > Zero Rate > Par Yield

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

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

NES FF 2005/06 10 Empirical estimation of TSIR Discrete rates: Discrete rates: –Regression P = cD 1 + cD 2 + … + (c+F)D T where D t = 1/P(0,t) = 1/y(0,t) t where D t = 1/P(0,t) = 1/y(0,t) t Continuous rates: Continuous rates: –Regression P = Σ t=1:T c t (a 0 +a 1 t+a 2 t 2 +…) –P = a 0 [Σ t=1:T c t ]+a 1 [Σ t=1:T tc t ]+a 2 [Σ t=1:T t 2 c t ]+…

NES FF 2005/06 11 Modeling changes in bond prices Due to passage of time: Due to passage of time: –E.g., flat yield curve: ΔP = r P 0 Unanticipated shift in the TSIR: 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

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

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

NES FF 2005/06 14 Duration modifications Convexity Convexity Fisher-Weil duration Fisher-Weil duration –For parallel shifts of (non-horizontal) TSIR Non-parallel shifts: 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) 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 % Separate estimation of duration for sr and lr %

Conclusions