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. Present Value Formula

The present value (PV) formula is given by:

$$ PV = \frac{C}{(1 + r)^n} $$

where ( C ) is the cash flow received in the future (in this case, $250), ( r ) is the rate of return, and ( n ) is the number of years until the cash flow is received.

2. Calculate Present Value for Each Cash Flow

For the first cash flow of $250 received in one year:

$$ PV_1 = \frac{250}{(1 + r)^1} $$

For the second cash flow of $250 received in two years:

$$ PV_2 = \frac{250}{(1 + r)^2} $$

3. Set Up the Equation

We know that the sum of the present values of both cash flows must equal $432:

$$ PV_1 + PV_2 = 432 $$

Substituting our equations for ( PV_1 ) and ( PV_2 ):

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

4. Find a Common Denominator

To solve the equation, we can multiply everything by ( (1 + r)^2 ) to eliminate the denominators:

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

5. Expand and Rearrange the Equation

Expanding and rearranging gives us:

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

Combining like terms:

$$ 250r + 500 = 432(1 + 2r + r^2) $$

6. Simplifying Further

Now distribute the ( 432 ):

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

Rearranging gives us a standard quadratic form:

$$ 432r^2 + (864 - 250)r + (432 - 500) = 0 $$

This simplifies to:

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

7. Use the Quadratic Formula

To find ( r ), we will use the quadratic formula:

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

Where ( a = 432 ), ( b = 614 ), and ( c = -68 ).

8. Calculate the Discriminant and Roots

First, calculate the discriminant:

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

Then substitute into the quadratic formula to find ( r ).