Consider a $15-$year $10\%$ coupon bond with a par value of $$1,000$ and semi$-$annual coupon

payments. The bond price is $$900.$ The YTM of the bond is $11\mathrm{.}4\%.$

i$)$ Assuming an investor of the bond has an income tax rate of $20\%$ and a capital gain

tax rate of $15\%.$ What is the investor$$s after$-$tax YTM$?$

ii$)$ Another investor, David, bought the bond for $$900$ and plan to sell the bond in $5$

years. David put the bond in his retirement account, so he does not need to worry

about any tax consequence of this investment. He assumes that he can reinvest the

coupons at a rate of $9\%.$ What is his horizon yield $($i$.$e$.,$ total yield$)$ during his

investment horizon? Assuming the market discount rate at the end of his investment

horizon is $10\%.$

iii$)$ Right after David bought the bond, the market interest rate unexpectedly went up

to $13\%$ and is expected to stay at this level until the bond matures. At this new

reinvestment rate, what is his interest$-$on$-$interest income at the end of his

investment horizon $($i$.$e$.,$ after $5$ years$)?$

iv$)$ Right after David bought the bond, the market interest rate unexpectedly went up

to $13\%$ and is expected to stay at this level until the bond matures. After this rate

hike, what is the capital gain$/$loss for David at the end of his investment horizon?

Hint: Carrying value is the hypothetical price of the bond if the bond$$s yield

remains at the initial yield to maturity.