You are currently looking at two bonds:

Bond $1$: $5$ year, $6\%$ coupon, face value $$1$ million, yield to maturity $6\%.$

Bond $2$: $5$ years $4\%$ coupon, Face value $$1$ million, yield to maturity $4\%.$

a$.$ If interest rates drop by $0\mathrm{.}5\%,$ how would the prices of these two bonds change? $(10\%)$

You have to start by finding the prices of bond $1$ and bond $2$

Price of bond $1=$ PV $=$ $$717,257($assuming coupons are paid

semiannually and using your calculator to discount future cash flows$)$

Price of bond $2=$ PV $=$ $$821,827$

With the drop in interest rates by $1\mathrm{.}5\%,$ PV of bond $1$ will rise to $802,451($ a gain of $$85,154),$ while PV of bond $2$ will rise to

$883,853($a gain of $$61,926).$

b$.$ Explain what accounts for the difference in price changes of these two bonds? $(5\%)$

Bond $1$ with the higher coupon and higher YTM is more sensitive to the drop in interest rate than bond $2.$ The bond with the higher coupon becomes more valuable when interest rates decline, its price responds higher.

Also $1\mathrm{.}5\%$ drop of $4\%$ YTM is $0\mathrm{.}375\%$ While $1\mathrm{.}5\%$ drop of $6\%$ YTM is $0\mathrm{.}25\%$

because $0\mathrm{.}375\%>0\mathrm{.}25\%,$ then the bond $2$ will be affected more by the $1\mathrm{.}5\%$ drop in interest rate than bond $1$

c$.$ If you want to add one of these two bonds to your existing portfolio,

and at the same time you want to reduce your systematic risk, which bond would you pick and why? $(10\%)$

Duration$=$X$1(1)+$X$2(2)+$X$3(3)+$X$4(4)+$X$5(5)$

Where X$1=$PV$1/$Price of the bond, etc.

Duration of bond $1=4\mathrm{.}47$