What rate of return results in a present value of $432 for $250 received one year from now and another $250 received two years from now?

Steps to Solve

1. Define Present Value Formula

To calculate the present value of future cash flows, we use the formula:

$$ PV = \frac{C_1}{(1 + r)} + \frac{C_2}{(1 + r)^2} $$

where ( PV ) is the present value, ( C_1 ) and ( C_2 ) are cash flows at different times, and ( r ) is the rate of return we want to find.

2. Substitute Known Values

Given ( PV = 432 ), ( C_1 = 250 ) (cash flow in one year), and ( C_2 = 250 ) (cash flow in two years), we substitute these values into the formula:

$$ 432 = \frac{250}{(1 + r)} + \frac{250}{(1 + r)^2} $$

3. Multiply Through by ( (1 + r)^2 )

To eliminate the fractions, we multiply both sides of the equation by ( (1 + r)^2 ):

$$ 432(1 + r)^2 = 250(1 + r) + 250 $$

4. Expand and Rearrange the Equation

Expanding the left side gives:

$$ 432(1 + 2r + r^2) = 250 + 250 + 250r $$

Which simplifies to:

$$ 432 + 864r + 432r^2 = 500 + 250r $$

Rearranging terms leads to:

$$ 432r^2 + 614r - 68 = 0 $$

5. Apply the Quadratic Formula

Now we will use the quadratic formula:

$$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Where ( a = 432 ), ( b = 614 ), and ( c = -68 ). Calculate ( b^2 - 4ac ):

$$ b^2 - 4ac = 614^2 - 4(432)(-68) $$

6. Calculate the Values

Plugging in the numbers gives:

$$ b^2 - 4ac = 376996 + 117696 = 494692 $$

Now use the quadratic formula:

$$ r = \frac{-614 \pm \sqrt{494692}}{2 \cdot 432} $$

Calculate the square root and find the two possible values for ( r ).

7. Determine Valid Rate

Only the positive rate will be valid for this context. Calculate and round to two decimal places if needed.