Suppose the price of the 3-year zero coupon bond is $800 and the 6-year zero coupon bond is $600.
1) How could you lock the 3-year borrowing rate starting at the end of year 3?
2) What is the implied forward rate on a 3-year bond which will start at the end of year 3?
3) According to the liquidity premium theory of the term structure, what is the expected rate on a 3-year bond at the end of year 3?
Are the attached answers correct? How do I decide whether to sell/buy the 3-year bond vs sell/buy the 6-year bond? Thanks!
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If you buy a 3-year zero-coupon bond at discount of 200 for 800 and sell a 6-year zero-coupon bond at a discount 400 for 600, you are initially out of pocket by a net 200, not 0.
At the end of three years, let's say rates are the same as they are today. At that instant, you get 1000 from the maturing 3-year bond and can buy back the 6-year bond initially sold short for 800, giving you back your initial outlay of net 200. You can also borrow 800 for the same discount as today, namely 200. The out of pocket cost of borrowing 800 dollars for three years will be 200. of course there is also an opportunity cost, namely the 50 you could have earned on your net outlay of 200, for a total economic cost of 250. (All this assumes no transaction costs, equal rates for lending and borrowing, etc.)
At the end of three years, let's say rates have gone down from where they are today so that the 6-year bond (now maturing in three years) has a market value of 850. You get 1000 from the maturing three year bond, but must close out the short sale at 850, giving you a net cash gain of 150. But you were initially out of pocket net 200, so net net you are 50 in the hole. Therefore you now need to borrow 850, not 800. The out of pocket cost of borrowing 850 for three years will be 150 (1000 - 850) so your net cash outlay will be 50 + 150 = 200. But again you will have an opportunity cost of 50 for a total economic cost of 200 + 50 = 250. (Again, all this assumes no transaction costs, equal rates for lending and borrowing, etc.)
At the end of three years, let's say rates have gone up from where they are today so that the 6-year bond (now maturing in three years) has a market value of 750. You get 1000 from the maturing three year bond, but can close out the short sale at 750, giving you a net cash gain of 250. But you were initially out of pocket net 200, so net net you are 50 to the good. 50 to the good is a - 50 out-of-pocket expenditure. Furthermore, you now need to borrow only 750, not 800. The out of pocket cost of borrowing 750 for three years will be 250 (1000 - 750) so your net cash outlay will be -50 + 250 = 200. But again you will have an opportunity cost of 50 for a total economic cost of 200 + 50 = 250. (Again, all this assumes no transaction costs, equal rates for lending and borrowing, etc.)
Turning all this into rates is simply mechanics. And because I am not sure of the exact definitions used in your book for liquidity premia etc, I'll let you figure out the answer to those questions. Alternatively you can tell us exactly what definitions you are supposed to use and what you think the answer is, and we can confirm or point out an error.
