FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 101
(d) Repeat part (c) using equation (3.14). You should get the same answer
as in part (c).
(e) Show that the future value of the annuity-immediate at the end of year
10, assuming future payments earn the spot rates of interest as at time
0, is
100 × (s 10⌉ 0.05 − s 6⌉0.05 ) + 100 × s 6⌉0.04 .
(f) Solve the problem in (e) numerically using the formula in Exercise
3.11. Do you get the same answer as using the formula in (e)?
3.13 Consider the accumulation function a(t) =1+0.05t.
(a) Compute the spot rates of interest for investments of 1, 2 and 2.5 years.
(b) Derive the accumulation function for payments due at time 2, assuming
the payments earn the forward rates of interest.
(c) Calculate the forward rates of interest for a payment due at time 2 with
time to maturity of 1, 2 and 2.5 years.
3.14 Suppose δ(t) = 10 + t
200 .
(a) Derive the accumulation function for payments due at time 5, assuming
the payments earn the forward rates of interest.
(b) Calculate the forward rates of interest for a payment due at time 5 with
time to maturity of 1 and 2 years.
3.15 Under the compound-interest accumulation function, i.e.,
show that
a(t) =(1+i) t ,
(a) i S t = i, forallt>0.
(b) The accumulation function for payments due at time t, assuming the
payments earn the forward rates of interest, is
a t (τ) =(1+i) τ .
(c) i F t,τ = i, fort>0 and τ>0.
3.16 Suppose v(t) = 20 . Assuming that future payments earn the forward
20 + t
rates of interest, calculate the value at time 2 of the following stream of payments: