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The force of interest at any time t (measured in years) is given by: 0.04 0<t< 1 0.05t- 0.01 1<t<= 5 0.24 t>5 (i) What is the total accumulated value at any time t ( > 0 ) of investments of 1 at times 0, 4 and 6? (ii) What is the present value at time 0 of a payment stream paid at a rate of rho(t) = 5t - 1 received between t = 1 and t = 5 ?
Its because there is a payment of 1 at time 0, a second payment of 1 at time 4 and a third payment of 1 at time 6. So if you need to know the total accumulated value between times 4 and 6 you need to accumulate a payment of 1 from time 0 to time (4 to 6) and a payment of 1 from time 4 to (4 to 6). This is a good example of a typical exam question because the timings of the payments don't coincide with the times that the force of interest changes, so you have to break the calc up into many components. Tim
Well, no, because you are asked to calculate the total accumulated value at any time t (>0) of the investments In your original question you asked why the calculation for the accumulated value within times 4<=t<=5 includes the second term. In between times 4 and 5 (inclusive) the total accumulated value includes two elements: the accumulation of 1 at time 0 and the accumulation of 1 at time 4. Part (i) of this question is similar to the example on page 14 of the notes but with the added complication that there is more than one investment involved. It might be a good idea for you to write the elements along a timeline as follows: - at time 0 an investment of 1 is made - force of interest is 0.04 0<t<=1 - force of interest is 0.05t - 0.01 1<t<=4 - at time 4 an investment of 1 is made - force of interest is 0.05t - 0.01 4<t<=5 - force of interest is 0.24 5<t<=6 - at time 6 a investment of 1 is made - force of interest is 0.24 t>6 You need to break down your accumulation over all these elements. Tim
