FM for Actuaries

98 CHAPTER 3
5. Present values of future payments can be computed straightforwardly given
the term structure. To compute the future value of a future payment, we have
to make additional assumptions, such as (a) future payments earn forward
rates of interest, and (b) future payments earn current spot rates of interest.
In practice, under a varying interest rate environment, both assumptions may
not hold.
6. Spot and forward rates may be computed from accumulation functions. The
accumulation function of the forward rates of interest can be derived from
the current yield curve or the current accumulation function.
7. An interest rate swap is an agreement between two parties to exchange cash
flows based on interest rate movements in the future. It can be used to convert
a floating rate loan to a fixed rate debt or vice versa.
Exercises
3.1 Suppose the spot rates of interest for investment horizons of 1, 2 and 3 years
are, respectively, 4.0%, 4.2% and 4.3%, and the 1-year forward rates of interest
for payments due at time 1 and 2 are, respectively, 4.4% and 4.6%.
(a) Do the above rates violate the no-arbitrage relationship concerning spot
and forward rates of interest?
(b) If so, describe an arbitrageur’s strategy to make riskless profits.
3.2 Suppose the spot rates of interest for investment horizons of 1, 2, 3 and 4
years are, respectively, 3%, 3.5%, 3.5% and 3.75%.
(a) Calculate the forward rates of interest i F 1 ,iF 2 ,iF 3 and iF 4 .
(b) Calculate a 4⌉ and s 3⌉ .
3.3 Suppose the forward rates of interest for investments in year 1, 2, 3 and 4 are,
respectively, 6%, 6.2%, 6.5% and 7%.
(a) Calculate the spot rates of interest i S 1 ,iS 2 ,iS 3 and iS 4 .
(b) Calculate a 4⌉
and s 4⌉
.
3.4 You are given the following information: i S 1 =2.1%,iF 2 =2.6%,iS 3 =3.1%
and i F 4 =3.6%.
(a) Calculate the rates of interest i F 1 ,iS 2 ,iF 3 and iS 4 .
(b) Calculate ä 4⌉
and ¨s 4⌉
.