BONDS AND FORWARD RATES

8.2 BONDS AND FORWARD RATES

A forward rate is denoted by F(t, t 1 , t 2 ) and is agreed on at time t, but for payments starting to take effect at a future time t 1 and for a certain amount of time t 2 −t 1 . In Figure 8.2, these times are specified.

A relationship between forward rates and spot rates hinges on an arbitrage argument. Roughly, this argument states (as we saw earlier), that two equivalent

BONDS AND FORWARD RATES

Figure 8.2

investments (from all points of view) have necessarily the same returns. Say that at time t we invest $1 for a given amount of time t 2 − t at the available spot rate (its yield). The price of such an investment using a bond is then: B(t, t 2 ). Alternatively, we could invest $1 for a certain amount of time, say t 1 − t, t 1 ≤t 2 at which time the moneys available will be reinvested at a forward rate for the remaining time interval: t 2 −t 1 . The price of such an investment will then be

B (t, t 1 )B f (t 1 , t 2 ) where B f (t 1 , t 2 ) = [1 + F(t, t 1 , t 2 )] −(t 2 −t 1 ) is the value of the bond at time t 1 paying $1 at time t 2 using the agreed-on (at time t) forward rate F(t, t 1 , t 2 ). Since both payments result in $1 both received at time t 2 they have the same value, for otherwise there will be an opportunity for arbitrage. For this reason, assuming no arbitrage, the following relationship must hold (and see Figure 8.3):

B (t, t 2 )

B (t, t 2 ) = B(t, t 1 )B f (t 1 , t 2 ) implying B f (t 1 , t 2 )=

B (t, t 1 ) In discrete and continuous time, assuming no arbitrage, this leads to the following

forward rates:

[1 + y(t, t 2 )]

t 2 −t

[1 + F(t, t

1 , t 2 )] t 2 −t 1

(discrete time)

[1 + y(t, t 1 )] t 1 −t

F (t, t − t) 1 , t 2 )= (continuous time)

y (t, t 2 )(t 2 − t) − y(t, t 1 )(t 1

(t 2 −t 1 )

In practice, arbitrageurs can make money by using inconsistent valuations by bond and forward rate prices. For complete markets (where no arbitrage is possible), the spot rate (yield) contains all the information regarding the forward market rate and, vice versa, the forward market contains all the information regarding the spot market rate, and thus it will not be possible to derive arbitrage profits. In practice, however, some pricing differences may be observed, as stated above, opening up arbitrage opportunities.

Btt (,) 2

Btt (,) 1 Btt f (,) 1 2

Figure 8.3

FIXED INCOME , BONDS AND INTEREST RATES

Problem

An annuity pays the holder a scheduled payment over a given amount of time (finite or infinite). Determine the value of such an annuity using bond values at the current time. What would this value be in two years using the current observed rates?

Problem

What will be the value of an annuity that starts in T years and will be paid for M years afterwards? How would you write this annuity it is terminated at the time the annuity holder passes away (assuming that all payments are then stopped)?

Problem

Say that we have an obligation whose nominal value is $1000 at the fixed rate of 10 % with a maturity of 3 years, reimbursed in fine. In other words, the firm obtains a capital of $1000 whose cost is 10 %. What is the financial value of the obligation? Now, assume that just after the obligation is issued the interest rate falls from 10 to 8 %. The firm’s cost of finance could have been smaller. What is the value of the obligation (after the change in interest rates) and what is the ‘loss’ to the firm.