Question

a) If the interest rate decreases, how will it impact the market price of a bond?

b) Do long-term bonds have higher price risk than short-term bonds?

Please illustrate your point by comparing the price change of two bonds of 1yr and 10yr maturity respectively due to interest rate changes. Assume both bonds have $1000 par and $100 annual coupon payment. show work

Answer 1

a)if the interest rate of the bond decreases,then the market price of bond will inceases.

b)yes,usually longterm bonds will have higher price risk than shorterm bonds.

we can illustrate this conclusion by comparing price of following two bond.

market price of the bond is the present value of future cash flows,discounted at market interest rate.

assume market interest rate is 8%.

1.then, market price of the first bond having 1 year maturity and 1000 par value and 100 coupen payment.

so total cash flow at end of the year =1100(sum of both interest and principal)

present value factor $1 recieved at end of the year=1/(1+interst rate)

=1/1+.08)=.926

so present value of future cash flow of the first bond having maturity 1 year =.926*1100=1018.6.

2.market price of the 2nd bond having having ten year maturity and 1000 par value and 100 coupen payment.=sum of present value par value and present value of annuity

presnt value facot for $1 at end of the 10th year =1/(1+.08)^10=.463

so present value of par value recieved at end of the year=.463*1000=463

present value of $1 annuity =1-(1/1+r)^n)/r ,where

r=discount rate

  n=number of periods.

=1-(1/1+.08)^10)/.08=6.71

so presnt value of 100$annuity for period 10 at interst rate 8%=6.71*100=67.1

so market value of 2nd bond at interst rate of 8%=671+463=1134

assume market interest rate changed to 10%

1.market value of the first bond having maturity one year and 1000 par value and 100 interst payment=1100*(1/(1+.1))=1000.

2.market value of second bond having maturity 10 years=sum of presnt value of 1000 recieve at end and 100$ annuity recieve at every year.

presenet value of 1000 will recieve at end of 10th year=1000*(1/(1+.1)^n), where, n=number of period=10.

=.386*1000=386

present value of 100$ annuity=100((1/(1+r)^n)/r) , where r= interest rate and n=number of periods

= 100*6.14=614

so market value of second bond=614+386=1000.

from the above calculations we can see that both bond having greater market value at lower interest rate. so we can conclude the interest rate decreases market value of bond will increases.

and also we can find greater variability in price for second bond(1134 to 1000) than the first bond(1018 to 1000).so we can conclude that longterm bond has higher price risk than shorterm bonds.