Part $2.$ Determining the price of a bond.

The market interest rate is $4$ percent.

You are thinking about purchasing a zero$-$coupon bond with a face value of $$5,000$ that matures in $t=3$ years.

Q$1.$ First, determine the present value of this bond. Use the formula and show your work.

Q$2.$ Of course, you would like to buy this bond at the lowest possible price.

As a buyer in a competitive market, though, you don't get to choose the price. You see the price and then decide whether to buy the bond. What we want to do now is determine the highest price you'd be willing to pay.

a$.$ If the price of the bond is $$4200,$ should you buy the bond? If you do$,$ you'll get $$5000$ in three years. If you don't buy the bond, you can put the $$4200$ in an account that pays the market interest rate. How much will that account be worth in three years? Answer this question and show your work. Then tell me whether you're better off buying the bond or putting the money into the account.

b$.$ If the price of the bond is $$4400,$ should you buy it$?$ Determine how much money you'd have in three years if$,$ instead of buying the bond, you put the $$4400$ into an account that pays $4$ percent interest. Then tell me whether you should buy the bond if its price is $$4400.$

c$.$ If the price of the bond is $$4440,$ should you buy the bond? Say yes or no$,$ and back up your answer with the relevant calculation.

d$.$ If the price of the bond is $$4450,$ should you buy the bond? Say yes or no$,$ and back up your answer with the relevant calculation.

Q$3.$ What is the highest price at which you should be willing to purchase this bond? Why would you not buy the bond if its price is any higher?

Next, let's approach the same question from the seller's perspective. You $,$