$1.(24$ points$)$ One source of risk, or uncertainty in horizon analysis lies with the rate at which we are able to reinvest interim cash flows $($coupon payments$).$ Given the notorious difficulty in predicting future interest rates $($those at which future cash flows will be invested$),$ there may be some sense in utilizing the market$$s consensus view on the path of future short$-$term interest rates.$$ Imagine we are analyzing the same bond from question $1,$ and the purchase price of the bond remains the same.$$ a$.$ Assume the same Treasury spot rate curve reproduced below.$$ Begin by calculating the implied $6-$month forward rates that would be applicable for the reinvestment of all interim cash flows from the bond.$$ That is$,$ the rates at which the $$3$ coupon payment in $6$ months will be reinvested, the rates at which the $$3$ coupon payment in $12$ months will be reinvested, and so on for the $3-$year holding period.$$ Period Years Annual Par YTM Price Spot Rate Semiannual $10\mathrm{.}53\mathrm{.}00\%3\mathrm{.}0000\%1\mathrm{.}5000\%21\mathrm{.}03\mathrm{.}30\%3\mathrm{.}3000\%1\mathrm{.}6500\%31\mathrm{.}53\mathrm{.}50\%1003\mathrm{.}5053\%1\mathrm{.}7527\%42\mathrm{.}03\mathrm{.}90\%1003\mathrm{.}9164\%1\mathrm{.}9582\%52\mathrm{.}54\mathrm{.}40\%1004\mathrm{.}4376\%2\mathrm{.}2188\%63\mathrm{.}04\mathrm{.}70\%1004\mathrm{.}7520\%2\mathrm{.}3760\%73\mathrm{.}54\mathrm{.}90\%1004\mathrm{.}9622\%2\mathrm{.}4811\%84\mathrm{.}05\mathrm{.}00\%1005\mathrm{.}0650\%2\mathrm{.}5325\%94\mathrm{.}55\mathrm{.}10\%1005\mathrm{.}1701\%2\mathrm{.}5851\%105\mathrm{.}05\mathrm{.}20\%1005\mathrm{.}2772\%2\mathrm{.}6386\%115\mathrm{.}55\mathrm{.}30\%1005\mathrm{.}3864\%2\mathrm{.}6932\%126\mathrm{.}05\mathrm{.}40\%1005\mathrm{.}4976\%2\mathrm{.}7488\%136\mathrm{.}55\mathrm{.}50\%1005\mathrm{.}6108\%2\mathrm{.}8054\%147\mathrm{.}05\mathrm{.}55\%1005\mathrm{.}6643\%2\mathrm{.}8322\%157\mathrm{.}55\mathrm{.}60\%1005\mathrm{.}7193\%2\mathrm{.}8597\%168\mathrm{.}05\mathrm{.}65\%1005\mathrm{.}7755\%2\mathrm{.}8878\%178\mathrm{.}55\mathrm{.}70\%1005\mathrm{.}8331\%2\mathrm{.}9166\%189\mathrm{.}05\mathrm{.}80\%1005\mathrm{.}9584\%2\mathrm{.}9792\%199\mathrm{.}55\mathrm{.}90\%1006\mathrm{.}0863\%3\mathrm{.}0432\%2010\mathrm{.}06\mathrm{.}00\%1006\mathrm{.}2169\%3\mathrm{.}1085\%$ Semiannual Annual $6-$month rate in $6$ months $6-$month rate in $12$ months $6-$month rate in $18$ months $6-$month rate in $24$ months $6-$month rate in $30$ months $$ b$.$ Next, determine the $4-$year forward rate in $3$ years, given the spot rates above. $$ c$.$ Assuming that the rate calculated in part $($b$)$ holds, at what price would we expect to sell the bond in $3$ years?$($For simplicity, use the forward rate calculated in part $($b$)$ as the single discount rate for valuation.$)$ d$.$ Now, compete the following table laying out the expected cash flows from the bond over the $3-$year holding period, along with their future values assuming that all cash flows are reinvested at the calculated forward rates appropriate to the time of receipt of each cash flow.$$ Coupon and Principal Payment Amount Future Value $1$ $$3\mathrm{.}002$ $$3\mathrm{.}003$ $$3\mathrm{.}004$ $$3\mathrm{.}005$ $$3\mathrm{.}006$ e$.$ What compounded annual rate of return is implied by the total future accumulation at the end of the $3-$year holding period?$$ f$.$ Now, assume that you find the market$$s expectations around future interest rates to be inaccurate.$$ Specifically, you think the $4-$year rate in $3$ years will be $3\mathrm{.}60\%$ on a semiannual basis.$($Again$,$ for simplicity, use this single rate as the discount rate in computing the selling price of the bond in $3$ years.$)$ Furthermore, you expect the following reinvestment rates to be available over your planned holding period.$$ Assumed reinvestment rates Semiannual $6-$month rate in $6$ months $2\mathrm{.}60\%6-$month rate in $12$ months $2\mathrm{.}80\%6-$month rate in $18$ months $2\mathrm{.}90\%6-$month rate in $24$ months $2\mathrm{.}95\%6-$month rate in $30$ months $3\mathrm{.}00\%$ i$.$ At what price would you expect to sell the bond after $3$ years assuming your rate forecast is correct? $$ ii$.$ Next, determine the future value of each coupon payment and principal repayment at our horizon date $3$ years from today.$$ Be careful to use the appropriate reinvestment rates for each semiannual compounding period for each cash flow. $$ Coupon and Principal Payment Amount Future Value $1$ $$3\mathrm{.}002$ $$3\mathrm{.}003$ $$3\mathrm{.}004$ $$3\mathrm{.}005$ $$3\mathrm{.}006$ iii.$$ What total future dollar amount have you accumulated?$$ What compounded annual rate of return does this imply, based again upon the $$72\mathrm{.}1150$ purchase price? $$