Question

1. Derive FV =PV (1+R) N where symbols have their usual meanings.

Answer 1

Suppose you deposit $100 in a savings account earning 4% compounded annually. At the end of the first
year you will earn 4% (or $4), ending up with a balance of $104. Stated another way, your ending balance will
be 1.04 times as great as what you started with ($100 × 1.04 = $104). At the end of the second year you will
earn 4% on the $104 beginning balance (or $4.16), ending up with a balance of $108.16 This is compound
interest because you are earning interest on interest. Each year the balance increases by a multiple of 1.04:

(Yr 1) $100 (1.04)^1

(Yr 2) $100 (1.04) (1.04) = $100 (1.04)^2

(Yr 3) $100 (1.04) (1.04) (1.04) = $100 (1.04)^3

(Yr 4) $100 (1.04) (1.04) (1.04) (1.04) = $100 (1.04)
^4

Let’s use symbols to represent the process. The ending balance is a future value (FV), the initial deposit is
a present value (PV), the 1.04 is “1 + i” (1 + .04 = 1.04), and the exponent is the n-value. We end up with
Formula 1
FV = PV(1 + i) ^n