A First Course in Probability, 5th Ed scanned + Solutions by Sheldon Ross

By Sheldon Ross

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T, fixed coupons Cl , . . ,CT, face values Fl , . . ,FT, and (observed) market prices M P l , . . , MPT. For the bond j, the present value is expressed as For the first bond, the yield to maturity tR1 is, according to (6), calculated from the relation For the second bond we use or bootstrap the information from the first bond (tR1 already ascertained) using the relation and by recursion, having known Rl, . . ,t Rj-1, we calculate the yield t R j from the relation Care should be taken if the maturities are not equally spaced.

12. We will now make use of It6 formula to characterize the developnient of logarithiliic prices. 9) Put f ( P ) := In P . The first and second derivatives off with respect to P are 1 / P and -1/P2, respectively. After some algebra we obtain the solution to (11) for the logarithmic prices: din P = ( p - f 02)dt+ odW. (13) The discrete version of the last equation is (recall that InP = p) (14) lnPt+at-hPt=ln(Pt+at/Pt)=pt+at-pt=(p-$02)At+or& with E distributed as N(0, 1) again. 02)t,02t), (16) where by the symbol L N ( m , s2) we mean the distribution of the random variable e x p { N ( m , s 2 ) ) ,the log-normal distribution with paranieters m and s2 which are not its mean and variance, respectively.

Accrued interest is a reward to the seller of the bond compensating the loss of the next forthcoming coupon. The difference between the dirty price and the accrued interest is calledpu~eprice, pure value, net value of the bond which therefore takes the form which is also quoted in the financial press. A very important measure of a bond is the so called yield to maturity. Let us suppose that the market price of the bond is MP. Consider the value of the bond expressed in terms of interest rate i, either from (1) or (3), P V ( i ), ceteris paribus.

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