Present value (with changing interest rates). Marty has been offered an injury settlement of $$ \$ 10,000 $$ payable in 3 years. He wants to know what the present value of the injury settlement is if his opportunit) cost is $$ 4 \% $$. (The opportunity cost is the interest rate in this problem.) What if the opportunity cost is $$ 6.5 \% $$ ? What if it is $$ 10.5 \% $$ ? If Marty's opportunity cost is $$ 6.5 \% $$, what is the present value of the injury settlement? $$ \$ 8278.49 $$ (Round to the nearest cent.) If Marty's opportunity cost is $$ 10.5 \% $$, what is the present value of the injury settlement? $$ \$ \square $$ (Round to the nearest cent.)

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To find the present value of the injury settlement, we can use the formula for present value:

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

where:
- $$ PV $$ is the present value,
- $$ FV $$ is the future value (in this case, the injury settlement of $10,000),
- $$ r $$ is the opportunity cost (interest rate),
- $$ n $$ is the number of years until the settlement is paid.

Given that Marty's opportunity cost is 4%, 6.5%, and 10.5%, we can calculate the present value for each interest rate.

1. For an opportunity cost of 4%:
$$ PV = \frac{10000}{(1 + 0.04)^3} $$

2. For an opportunity cost of 6.5%:
$$ PV = \frac{10000}{(1 + 0.065)^3} $$

3. For an opportunity cost of 10.5%:
$$ PV = \frac{10000}{(1 + 0.105)^3} $$

Let's calculate the present value for each opportunity cost.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{10000}{\left(1+0.04\right)^{3}}$$
- step1: Add the numbers:
  $$\frac{10000}{1.04^{3}}$$
- step2: Convert the expressions:
  $$\frac{10000}{\left(\frac{26}{25}\right)^{3}}$$
- step3: Simplify the expression:
  $$\frac{10000}{\frac{26^{3}}{25^{3}}}$$
- step4: Rewrite the expression:
  $$\frac{156250000}{26^{3}}$$
- step5: Reduce the fraction:
  $$\frac{19531250}{2197}$$
Calculate or simplify the expression $$ 10000/(1+0.065)^3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{10000}{\left(1+0.065\right)^{3}}$$
- step1: Add the numbers:
  $$\frac{10000}{1.065^{3}}$$
- step2: Convert the expressions:
  $$\frac{10000}{\left(\frac{213}{200}\right)^{3}}$$
- step3: Simplify the expression:
  $$\frac{10000}{\frac{213^{3}}{200^{3}}}$$
- step4: Rewrite the expression:
  $$\frac{10000\times 200^{3}}{213^{3}}$$
Calculate or simplify the expression $$ 10000/(1+0.105)^3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{10000}{\left(1+0.105\right)^{3}}$$
- step1: Add the numbers:
  $$\frac{10000}{1.105^{3}}$$
- step2: Convert the expressions:
  $$\frac{10000}{\left(\frac{221}{200}\right)^{3}}$$
- step3: Simplify the expression:
  $$\frac{10000}{\frac{221^{3}}{200^{3}}}$$
- step4: Rewrite the expression:
  $$\frac{10000\times 200^{3}}{221^{3}}$$
The present value of the injury settlement for an opportunity cost of 4% is approximately $8,889.96.
The present value of the injury settlement for an opportunity cost of 6.5% is approximately $8,278.49.
The present value of the injury settlement for an opportunity cost of 10.5% is approximately $7,411.62.
The present value of the injury settlement is: - At a 4% opportunity cost: \$8,889.96 - At a 6.5% opportunity cost: \$8,278.49 - At a 10.5% opportunity cost: \$7,411.62

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