Annuities-Immediate: Present and Accumulated Value Formulas
Master annuity-immediate formulas for present and accumulated values on Exam FM.
Annuity-Immediate Definition
An annuity-immediate (or ordinary annuity) makes payments at the end of each period. For n payments of 1 at the end of each period, the present value (at time 0) is a-angle-n = (1 - v^n) / i, where v = 1/(1+i). The accumulated value (at time n) is s-angle-n = ((1+i)^n - 1) / i.
These are the most important formulas on Exam FM. For a payment of R per period, multiply by R: PV = R * a-angle-n and FV = R * s-angle-n.
Derivation and Understanding
The present value is the sum of a geometric series: a-angle-n = v + v^2 + ... + v^n = v(1 - v^n)/(1 - v) = (1 - v^n)/i. Similarly, s-angle-n = 1 + (1+i) + ... + (1+i)^(n-1) = ((1+i)^n - 1)/i. Note that s-angle-n = a-angle-n * (1+i)^n; this relationship connects present and accumulated values.
Understanding the derivation helps when you encounter non-standard annuities on Exam FM. Any sequence of level payments can be valued using these building blocks.
Exam FM Problem Types
Exam FM problems with annuities-immediate typically give three of {payment amount, interest rate, number of periods, present value, accumulated value} and ask for the fourth. Set up the equation and solve. For example: find the quarterly payment needed to accumulate $50,000 in 10 years at 8% compounded quarterly. Here i = 0.02 per quarter, n = 40 quarters, and R * s-angle-40 at 2% = 50,000. Since s-angle-40 = ((1.02)^40 - 1)/0.02 = 60.402, R = 50,000/60.402 = $827.85.