An immunized bond portfolio is a portfolio which, for a given fixed target date, gives the holder a fixed return no matter what happens to interest rates from

the initial date and the future target date. Essentially, an immunized bond portfolio has the cash flow structure of a zero$-$coupon bond with maturity equal to

the fixed target date.

Suppose you can only trade the following three bonds:

Given the three bonds, the spot rates in this economy are: $r_1=5\mathrm{.}5\%,r_2=7\%,$ and $r_3=8\%.$

$($a$)(**)$ Suppose that you have $$1000$ and you want to construct an immunized portfolio with a target date of three years. In other words, you want to create

a $($synthetic$)3-$year zero$-$coupon bond and invest in it your capital of $$1000.$ What annual return will you get?

$($Use at least three decimals for intermediate calculations. Write your final answers in percentage and round to one decimal: for instance, if the answer is

$3\mathrm{.}45\%$ or $3\mathrm{.}47\%,$ write $3\mathrm{.}5$ without the percentage sign; if the answer is $3\mathrm{.}44\%$ or $3\mathrm{.}42\%,$ write $3\mathrm{.}4.)$

The annual return on the immunized portfolio is:

What bonds would you buy or sell and in what amounts $($denoted by $b_A,b_B,$ and $b_C).?$

$($Use at least three decimals for intermediate calculations. Write your final answers as positive numbers if you want to buy, and as negative numbers if you

want to sell a given bond. Also, round your final answers to two decimals: for instance, if the number of bonds to buy is $3\mathrm{.}855$ or $3\mathrm{.}859,$ write $3\mathrm{.}86$ ; if the

number of bonds to sell is $3\mathrm{.}854$ or $3\mathrm{.}851,$ write $-3\mathrm{.}85.)$

The number of Bond $A{b}_A$ is:

The number of Bond $B{b}_B$ is:

The number of Bond $C{b}_C$ is: