Calculate the present value needed to have $50,000 in three years with a 20% interest rate, compounded annually and continuously. Round the answer to the nearest integer.

Steps to Solve

1. Calculate the present value with annual compounding

The formula for present value (PV) with annual compounding is: $PV = \frac{FV}{(1 + r)^n}$ Where: $FV =$ Future Value = $50,000 $r =$ interest rate = 20% = 0.20 $n =$ number of years = 3

Plug in given values: $PV = \frac{50000}{(1 + 0.20)^3}$

2. Simplify the equation

Calculate $(1 + 0.20)^3$: $(1 + 0.20)^3 = (1.20)^3 = 1.728$

Now divide 50000 by 1.728: $PV = \frac{50000}{1.728} \approx 28935.185$

3. Calculate the present value with continuous compounding

The formula for present value with continuous compounding is: $PV = FV \cdot e^{-rt}$ Where: $FV =$ Future Value = $50,000 $r =$ interest rate = 20% = 0.20 $t =$ number of years = 3 $e \approx 2.71828$

Plug in given values: $PV = 50000 \cdot e^{-0.20 \cdot 3}$

4. Simplify the equation

Calculate the exponent: $-0.20 \cdot 3 = -0.6$

Calculate $e^{-0.6}$: $e^{-0.6} \approx 0.5488$

Now multiply 50000 by 0.5488: $PV = 50000 \cdot 0.5488 \approx 27440$

5. Provide both answers, rounded to integers

The present value with annual compounding is approximately $28,935$. The present value with continuous compounding is approximately $27,440$.