(a) A capital of $100 at time t = 0 grows to $110 at t = 2, while a capital of $100 at t = 2 grows to $120 at t = 5. Assume that the principle of consistency holds. Compute the accumulated value at t = 5 of a payment of $50 at t = 0, showing clearly where the principle of consistency is used.
(b) Assume in addition to the data in question (a) that the force of interest varies according to ?(t) = a + bt. Show that a ? 0.0424076 and that b ? 0.0052475.
(c) Compute the present value at t = 0 of a payment of $500 due at t = 3, assuming that the force of interest varies as in question (b).
(d) An annuity pays $75 at t = 1, t = 2 and t = 3. Compute its present value at t = 0, assuming that the force of interest varies as in questio.