Ct1 Q 5.15

Q 5.15
A rich woman pays £2m and in return expects to receive a continuous cashflow for the next six years with a constant rate of payment. Calculate the annual payment from this cashflow and the accumulated amount of the cashflow after six years if the interest rate is 9% per annum effective.

The solution says to solve the equation 2,000,000 = integral between 6 and 0 of Cv^t dt.

But if it's a constant rate of payment then where is delta(t) force of interest...?

The formula given was including force of interest like C = C times integral between T and 0 of force of interest(t) times v(t)dt + Cv(T).

So where did integral Cv^t come from?

So they say that = C[v^t/logv]between 6 and 0.

And then with the limits in there = C(1-v^6)/log(1+i)

And then plug numbers in to get 2,000,000/4.68489 = 0.427m.

So is that the annual payment right? Well I kind of get why it's integrate for Cv(t) but what happened to force of interest - why isn't that included in working this out??

2nd part of solution is Accumulated value = 2,000,000 x 1.09^6 = £3.354m yeah I get that bit.

So I'm sorry this post is so long but I needed to type it out to get my head around it. Only thing I don't get is what happened to force of interest? Can someone please explain - thanks.

Edit: I also don't get why when you plug limits 6 and 0 you get C(1-v^6)/ln(v)... like shouldn't it be v^6 - 1???

 

Can you explain where the equation to solve comes from?
According to the material, the present value of the income is defined by C times integral between T and 0 of force of interest(t) times v(t)dt (page 20). The equation indicates that the present value of the income is 2000000 but this is actually the capital? Furthermore the question asks for the annual payment but this equation seems to calculate the present value of the income received over 6 years ?