Time Value of Money Equations

Study guide: Exam FM-2

Time value of money equations express the equivalence of cash flows at different points in time under a given Interest Rate or Accumulation Function. The core principle: a dollar today is worth more than a dollar in the future because it can be invested to earn interest.

• Standard equation linking Present Value, Future Value, rate, and time under Compound Interest:

\text{FV} = \text{PV} \cdot (1+i)^n

\text{PV} = \text{FV} \cdot v^n

• More generally, the Equation of Value at a comparison date t_0 equates the value of all inflows C_k^+ with the value of all outflows C_k^-. Under compound interest, a force of interest, or any accumulation function a question gives other than simple interest, each cash flow is moved to t_0 by the factor a(t_0)/a(t_k):

\sum_k C_k^+ \cdot \frac{a(t_0)}{a(t_k)} = \sum_k C_k^- \cdot \frac{a(t_0)}{a(t_k)}

• Under Simple Interest, SOA starts each cash flow's simple interest on its own date instead: a cash flow before t_0 is multiplied by 1 + i(t_0 - t_k) and one after t_0 is divided by 1 + i(t_k - t_0). The answer can then depend on the choice of t_0.
• Solving these equations under Variable Force of Interest requires evaluating integrals.

Solve for Unknown Time How long does it take 5{,}000 to double at an effective annual rate of 7\%?

Answer 10000 = 5000(1.07)^n \implies (1.07)^n = 2 \implies n = \frac{\ln 2}{\ln 1.07} = \frac{0.6931}{0.0677} \approx 10.24 \text{ years}