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Question:

Ms.Smith has two grand children, Adam and Evelyn. Adam will begin college on 9/1/03 and Evelyn will start college on 9/1/05. Ms.Smith wants both Adam and Evelyn to receive $1,000 at the beginning of each of their 4 years in college.

Ms.Smith will fund these payments by making five level annual deposits of P into an account earning an annual effective interest rate of 7%, with the first deposit on 9/1/1998.

Determine the value of P.

Okay so if she begins payment on 9/1/1998 and makes 5 annual payments, then the last payment is on 9/1/02, which is one year before Adam starts college. Thus, the account would accumulate by (1+i) after the last payment is made since it is earning interest for a whole year from 2002-2003.

And this value should be equal to the present value of all the payments that Adam and Evelyn would receive which is basically deferred annuities of 1000A-angle[6] + 1000A-angle[4] - 1000*A-angle[2]

Note: A-angle refers to annuity-immediate and the number that follows in brackets is the n-term.

So PV = 4766.54 + 3387.21 - 1808.02 = 6,345.73

Since this present value must match the accumulated value of Ms.Smith's account, we have s angle 5 (1+i) where i=7% which gives:

6.15329074 * P = 6,345.73

P = 1031.27

However, my solution evaluates the accumulated value without the earned interest (1+i)

So s angle 5 = 5.75 thus,

5.75P = 6,345.73

P = 1,103.46

Can someone please explain to me why the textbook does not accumulate the fund by (1+i) even though it is sitting in the account for 1 year?

swordlordswamplord
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    I made a spreadsheet to compute the value in the account. Setting the final value to $0$ I agree with a deposit of $1103.46$ – Ross Millikan Dec 16 '20 at 06:13
  • can u send me a screen shot or tell me how I can do it? – swordlordswamplord Dec 16 '20 at 06:14
  • The textbook solution is 1103.46 but my answer was 1031-ish.. because I don't see how the account does not accumulate to (1+i) after the last deposit since last deposit is made in 2002 and its just sitting there until 2003 – swordlordswamplord Dec 16 '20 at 06:15
  • Just make a column with dates of $9/1$ at the year intervals. I made a special cell for the deposit amount. Each year you add $1.07$ times the previous balance plus new deposits minus withdrawals. Use Tools->Goal Seek to set the final value to $0$ by varying the deposit cell. – Ross Millikan Dec 16 '20 at 06:16
  • Using 1103.46 I have it at 6345.71 in 2002, then 5789.91 in 2003 after the interest and the withdrawal of 1000. That shows interest accumulates in that year. – Ross Millikan Dec 16 '20 at 06:19
  • Your 1031.27 produces 6345.71 in 2003 just before the 1000 is deducted. The book answer has that amount in 2002, then does collect interest before the first withdrawal is made. When I make the withdrawals from your amount I wind up -623.06 after the last one. – Ross Millikan Dec 16 '20 at 06:27
  • ah... crap. okay makes a lot more sense now.. I didn't think to compute the depreciating value of the account – swordlordswamplord Dec 16 '20 at 06:28

1 Answers1

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I'm just going to do it the way I think it should be done. I'm going to use the time of the last payment into the fund as the valuation time point; i.e., 01 Sep 2002. Therefore, the payments are an annuity-immediate with accumulated value $$\require{enclose} P s_{\enclose{actuarial}{5} i} = P(1+i)^4 + P(1+i)^3 + \cdots + P.$$ The fund is then withdrawn according to the following cash flow: $$\begin{align} 1000v + 1000v^2 + 2000v^3 + 2000v^4 + 1000v^5 + 1000v^6 &= 1000(a_{\enclose{actuarial}{4}i} + v^2a_{\enclose{actuarial}{4}i}) \\ &= 1000(1+v^2)a_{\enclose{actuarial}{4}i}.\end{align}$$ Assuming the interest rate is constant at $i = 0.07$ and $v = 1/(1+i)$, the resulting equation of value yields $$P = \frac{1000(1+v^2)a_{\enclose{actuarial}{4}i}}{s_{\enclose{actuarial}{5}i}} = 1103.46. $$ As it is unclear which answer is yours and which is the textbook's, I will let you sort out the issue.

Note that because there is only one year between the last payment and the first withdrawal, no matter which time point you choose for the equation of value, the annuities involved in the equation must either be both annuities-immediate, or annuities-due. Put another way, the equation should use $s$ and $a$, or $\ddot s$ and $\ddot a$. You cannot mix the two.

To illustrate an alternative method of solution, we can choose as the valuation time point 01 Sep 1998, the first payment. Then the equation of value looks like this: $$P\ddot a_{\enclose{actuarial}{5} i} = 1000 v^6 \ddot a_{\enclose{actuarial}{4} i} + 1000 v^8 \ddot a_{\enclose{actuarial}{4} i}.$$ The LHS is the present value of the fund, and the RHS is the present value of the withdrawals, appropriately deferred.

heropup
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    It isn't clear in the original post, but in the comments it becomes clear that the book has 1103.46 and the poster has 1031.27, so the book is correct. – Ross Millikan Dec 16 '20 at 06:29