Formally show the relationship between a bond yield and bond price for a zero co

Question

Formally show the relationship between a bond yield and bond price for a zero coupon bond (using the notation from lectures). Find the derivative of the yield with respect to the price and comment on the relationship. a) [10] b) Assume that investors require a risk premium on top of the return given by the pure Expectations Theory to hold long-term assets. Today's one-year bond rate is 2%, the expected one-year rate in year 2 is 3%, in year 3 it is 4%, in year 4 it is 5% and in year 5 it is 6%. If today's 5-year bond rate is 3.5%, estimate the risk premium that investors require to hold a 5-year bond c) Set out two reasons why a risk premium might be required in this case. 17)

Explanation / Answer

a) Price of the bond = Face Value/(1+r)n

first order derivative

dP/dr = Face Value*-1/(1+r)n+1 *n

dP/dr = - n*Face Value/(1+r)n+1

b) five year rate = (1.02*1.03*1.04*1.05*1.06)1/5 - 1 = 3.99%

risk premium required = 3.99% - 3.50% = 0.49%

c) A risk premium may be required because:

There is a maturity risk premium associated

There is a possibility that the credit rating could change within the five year tenure

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