Equation of Value

Study guide: Exam FM-2

An equation of value sets the value of all payments in equal to the value of all payments out, with every cash flow accumulated or discounted to one common comparison date $t_0$ . Each side is a Current Value at $t_0$ — the Present Value when every cash flow is discounted to $t_0$ (as with $t_0 = 0$ ), the Future Value when every one is accumulated to it. It is the fundamental tool for solving time-value-of-money problems:

$\text{Value}_{t_0}(\text{inflows}) = \text{Value}_{t_0}(\text{outflows})$

Cash flows before $t_0$ are accumulated to it; cash flows on or after $t_0$ are discounted to it.
Every cash flow must be moved to the same comparison date before comparing, each at the Interest Rate that applies to it — one equation can equate the balances of two funds earning different rates.
When every cash flow is valued at one compound interest rate, a different choice of comparison date gives a different equation but the same solution for the unknown: moving the date multiplies every term by the same power of $(1+i)$ .
Under Simple Interest this fails. SOA starts each cash flow's simple interest on its own date, so the solution can depend on the comparison date (second example).

Replacing Two Payments Debts of $1{,}000$ due in 2 years and $2{,}000$ due in 5 years are to be replaced by a single payment at the end of 3 years. Find the payment using $i = 6\%$ .

Answer Set time 3 as the comparison date: $X = 1000(1.06)^1 + 2000(1.06)^{-2} = 1060 + 1779.99 = 2839.99$

The Comparison Date Under Simple Interest Two payments of $100$ — one due now and one due at the end of 5 years — are to be replaced by a single payment $P$ at the end of 10 years. Money earns $5\%$ simple interest from the date each payment is made. Find $P$ using a comparison date of (a) 10 years and (b) 15 years.
Answer (a) Comparison date 10:
 $\begin{align*} P &= 100\,(1 + 10 \times 0.05) + 100\,(1 + 5 \times 0.05) \\ &= 150 + 125 \\ &= 275 \end{align*}$ 
(b) Comparison date 15: $P$ , paid at 10, earns simple interest for 5 years on its own:
 $\begin{align*} P\,(1 + 5 \times 0.05) &= 100\,(1 + 15 \times 0.05) + 100\,(1 + 10 \times 0.05) \\ 1.25\,P &= 175 + 150 \\ P &= 260 \end{align*}$ 
The two comparison dates give different answers; under compound interest they would agree.