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?
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value $$\require{enclose} P s_{\enclose{actuarial}{5} i} = P(1+i)^4 + P(1+i)^3 + \cdots + P.$$ The fund is then withdrawn according to the following cash flow: $$\begin{align} 1000v + 1000v^2 + 2000v^3 + 2000v^4 + 1000v^5 + 1000v^6 &= 1000(a_{\enclose{actuarial}{4}i} +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.