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Given:

i) X is the current value at the end of year two of a 20-year annuity-due of 1 per annum.

ii) The annual effective rate for year t is: $$i_t = \frac {1}{8+t}$$

Calculate X.

  1. $$ a(t) = (1+i_t) = \frac {9+t}{8+t}$$

From this point, I honestly have no idea how to evaluate the annuity...

swordlordswamplord
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  • From your prior question, I thought you were studying life contingencies, but now it appears to me that you're studying compound interest. In that case, the Jordan book I sent you that link to won't help, but keep it in mind for when you get to life contingencies. – saulspatz Nov 28 '20 at 20:27

1 Answers1

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At the end of year $2$, there are $18$ payments of $1$ still to come. The first one come immediately, so its present value is $1$. The second one comes at the end of year $3$, in which the interest rate is $1/11$ so the present value is $$\left(1+\frac1{11}\right)^{-1}=\frac{11}{12}$$ The third comes at the end of year $4$, so the present value is $$\left(1+\frac1{11}\right)^{-1}\left(1+\frac1{12}\right)^{-1}=\frac{11}{12}\frac{12}{13}=\frac{11}{13}$$

I'm sure you see how to continue. We have to discount each payment by the rate in every year from the end of yer $2$ to the payment date, so we get for the present value of the future payments $$1+11\left(\frac1{12}+\frac1{13}+\cdots+\frac1{28}\right)$$

We must add to this the accumulated value of the past payments, of which there have been $2$. The payment at the beginning of year $1$ has grown by $1+\frac1{10}=\frac{11}{10}$ and the payment at the beginning of year $1$ has grown by $\left(1+\frac19\right)\frac{11}{10}=\frac{11}9$ so the current value is $$11\left(\frac19+\frac1{10}+\cdots+\frac1{28}\right)=\sum_{t=9}^{28}\frac{11}t$$

saulspatz
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  • sorry but i do not follow..from what I see, payment of 1 is immediately made and after the 2nd year, a total of 3 payments have already been made. thus there are 17 remaining payments from t=3 to t=19.. so the present value is the accumulated value of the first 3 payments plus the present value of the future 17 payments – swordlordswamplord Nov 28 '20 at 20:28
  • You're right. I misinterpreted the question. I have to take the accumulated value of the past payments into account. I'll correct it. – saulspatz Nov 28 '20 at 20:30
  • basically, i say that the first 3 payments is 1 + 10/9 + 10/9 * 11/10 = 1 + 10/9 + 11/9 however, the textbook is telling me that it is incorrect – swordlordswamplord Nov 28 '20 at 20:33
  • the textbook evaluates it as 11/9 + 11/10 + 1 for the first 3 payments which i disagree with or at least dont know how that is the case – swordlordswamplord Nov 28 '20 at 20:35
  • I agree with your text. Look at my edited version. – saulspatz Nov 28 '20 at 20:38
  • "The first one come immediately, so its present value is 1. The second one comes at the end of year 3." Why do u say this? the second payment comes at the end of year 1 – swordlordswamplord Nov 28 '20 at 20:41
  • the payment schedule I get is 1 + 10/9 + 10/9 * 11/10 for the first 2 years – swordlordswamplord Nov 28 '20 at 20:42
  • @swordlordswamplord I'm talking about the second of the future payments. – saulspatz Nov 28 '20 at 20:42
  • @swordlordswamplord You are computing the current value as of the annuity starting date, instead of the current value at the end of the first two years. – saulspatz Nov 28 '20 at 20:44
  • okay the textbook says that the answer is Rieman Sum from t=9 to n=28 of 11/t .. can u explain to me how this relates to the question? – swordlordswamplord Nov 28 '20 at 20:52