SEVERAL EXAMPLES ON ANNUITY

1. [2010 paper question 6] The deterministic force of interest at time t isgiven by

δ(t) =

{0.01 t ≤ 50.03 t > 5

.

Find the present value of a 10 year continuous annuity that pays 1 perannum during the first 3 years and 2 per annum during the last 7 years.Calculate also its accumulated value after 10 years.

2. [2012 paper question 3] The force of interest at time t is given by

δ(t) =1

10 + t.

An increasing annuity pays an amount of 11 at the end of the year 1, 12at the end of year 2 and so on till final payment which is 20 at the endof year 10. Calculate the present value of the annuity at time 0 and theaccumulation of the annuity at time 10.

3. [2007 paper question 2] The force of interest is given by

δ(t) =

{0.08 0 ≤ t < 11

1+t 1 ≤ t < 3.

(a) Find the present value for a 3-year increasing annuity paid contin-uously with a stream of payments as follows:

ρ(t) =

1 0 ≤ t < 12 1 ≤ t < 23 2 ≤ t < 3

.

(b) Find the accumulated value after 3 years of the annuity describedin the previous part.

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