Investments Chapter 13: Interest Rates and Bond Valuation
Definition: Bonds and Yields
Represent a claim on future cash flows (coupon payments and par value).
Yield to Maturity
The annualized discount rate that makes the present value of the future cash flows equal to the current price of the bond.
Other Measures of Bond Yields
Coupon yield / nominal yield.
Yield to call.
Yield Curve: I
Bond prices are related inversely to market interest rates.
This relation is NOT tautological, however:
bond prices and market interest rates are determined simultaneously by the underlying economic forces that drive the supply and demand for money.
Yield Curve: II
The market generally has various interest rates for various maturities.
The relationship between the market interest rate and the time to maturity is known as the yield curve.
Yield Curve: III Exhibit 13.1 Examples of actual yield curves
Spot and Forward Rates
The Spot Rate
The yield to maturity of a zero-coupon bond that has a stated maturity.
The Forward Rate
The yield to maturity of a zero-coupon bond that an investor agrees to purchase at some future specified date.
Spot- & forward rates for pure discount bonds Spot rate is the yield (return) of a pure discount bond, which is sold att discount, since discount bonds, P < F. If a one-year pure discount bond, just issued, is sold at P = €90.9 and has an F = €100, the spot rate is :
Spot & forward rates for pure discount bonds Forward rate is the interest rate an investor will pay to buy a bond in the future , no matter its true interest rate (or its bond price), that date. If I sign a forward contract to buy next year a two-years bond at a P = SEK892.9 (with F = SEK1000, to be paid in two years from now), the forward rate is : If the interest rate next year becomes 10%, the price of the bond will be SEK909. In that case, I gain since I buy the bond at SEK892.9 .
Spot- & forward rates for pure discount bonds Forward rates are derived from spot rates and provide a good information on the expected interest rates in the future. Maturity (n) Spot (R m ) Forward (f n ) 1 5 - 2 5.8 6.606 3 6.3 7.307 4 6.4 6.701 5 6.45 6.65
Spot- & forward rates for pure discount bonds If we graph the spot yield curve, it is 5 % for 1-year and 6.45 % for 5-years. But why is it f 2 = 6.606 % ? Strategy 1 : Save 1 $ directly in 2-years and get: 1(1 + R 2 ) 2 = 1(1.058) 2 = 1.11936. Strategy 2 : (a) Save first 1 $ for 1-year, and (b) sign a contract to invest your $ and its return in an implied rate , (i.e. f 2 ), in order to get the same as in strategy 1.
Spot- & forward rates for pure discount bonds [1(1.05)]*[1 + r impl ] = 1.11936 , i.e. 1 + r impl = 1.11936 / 1.05, => r impl = f 2 = 0.06606. Alternatively, R 2 = (R 1 + f 2 ) / 2. The Formula to estimate implied forward rates (f n ) from one periods’ spot rates (R n ) is: