Suppose at time t = 0, we are given four zero-coupon bond prices {B1, B2, B3, B4} that mature at times t = 1, 2, 3, 4. This forms the term structure of interest rates. We also have one-period forward rates {f0, f1, f2, f3}, where each fi is the rate contracted at time t = 0 on a loan that begins at time t = i and ends at time t = i + 1. In other words, if a borrower borrows $N at time t = i, he or she will pay back N(1 + fi) at time t = i + 1. The spot rate is denoted by ri. By definition we have
r0 = f0
The {Bi} and all forward loans are default free. At each time period there are two possible states of the world, denoted by {ui, di:= 1, 2, 3, 4, }.
(a) Looked at from time i = 03 how many possible states of the world are there at time i = 3?
(b) Suppose
{Bi = 0.9, B2 = 0.87, B3 = 0.82, B4 = 0.75}
and
{f0 = 8%, f1 = 9%, f2 = 10%, f3 = 18%
Form three arbitrage portfolios that will guarantee a net positive return at times i = 1, 2, 3 with no risk.
(c) Form three arbitrage portfolios that will guarantee a net return at time i = 0 with no risk.
(d) Given a default-free zero-coupon bond, Bn, that matures at time t = n, and all the forward rates {f0, . . ., fnˆ’1}, obtain a formula that expresses Bn as a function of fi.
(e) Now consider the Fundamental Theorem of Finance as applied to the system:

Can all Bi be determined independently?
(f) In the system above can all the {fi} be determined independently?
(g) Can we claim that all fi are normally distributed? Prove your answer.

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