Solving PV questions with varying force of interest

[dextar]

Hi I have a general question in solving questions where force of interest change. for example, q2 , apr, 2007 exam asks ot calculate.

Calculate the present value at time t = 0 of a payment stream, paid continuously from
time t = 9 to t = 12, under which the rate of payment at time t is 50e0.01t

Now the problem is it is asking us to calculate the value at t=0 . But when we integrate the expression using limits 9-12 in the question, aren't we calculating the value at t=9 instead of t=0. How would the solution change if we are told to calculate at
1) t=9 (would the expression of exp starts from taking the last limit (i.e 0.06)?
2) t=3 (should we then change our lower limit and make intervals for exp?)

I'm getting confused on the various permutations. can anyone explain the concept?

 

[John Lee]

Hi I have a general question in solving questions where force of interest change. for example, q2 , apr, 2007 exam asks ot calculate.

Calculate the present value at time t = 0 of a payment stream, paid continuously from
time t = 9 to t = 12, under which the rate of payment at time t is 50e0.01t

Now the problem is it is asking us to calculate the value at t=0 . But when we integrate the expression using limits 9-12 in the question, aren't we calculating the value at t=9 instead of t=0. How would the solution change if we are told to calculate at
1) t=9 (would the expression of exp starts from taking the last limit (i.e 0.06)?
2) t=3 (should we then change our lower limit and make intervals for exp?)

I'm getting confused on the various permutations. can anyone explain the concept?

Probably the easiest thing to do is to calculate the PV at the start of the payment stream, ie t=9.

You have then replaced the payment stream with an equivalent single payment at time 9.

You then discount this back to time 0 using the usual methods.