Question 5 You can afford a $$ \$ 200 $$ per month car payment. You've found a 4 year loan at $$ 2 \% $$ interest. How big of a loan can you afford? Question Help: $$ \square $$ Video 1 Video 2 Message instructor Submit Question

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We begin with the formula for the present value of an annuity (the loan amount), given by

$$
\text{PV} = P \times \frac{1 - \left(1+r\right)^{-n}}{r},
$$

where  
$$ P $$ is the monthly payment,  
$$ r $$ is the monthly interest rate, and  
$$ n $$ is the total number of monthly payments.

1. The monthly payment is  
   $$
   P = 200.
   $$

2. The annual interest rate is $$ 2\% $$; thus the monthly interest rate is

$$
   r = \frac{0.02}{12} \approx 0.00166667.
   $$

3. The loan term is 4 years, which gives

$$
   n = 4 \times 12 = 48 \text{ months}.
   $$

4. Substitute these values into the formula:

$$
   \text{PV} = 200 \times \frac{1 - \left(1 + 0.00166667\right)^{-48}}{0.00166667}.
   $$

5. First, compute the term $$\left(1+0.00166667\right)^{-48}$$. Calculate

$$
   \left(1+0.00166667\right)^{48} \approx e^{48 \times 0.00166667} \approx e^{0.08} \approx 1.0833.
   $$

Then

$$
   \left(1+0.00166667\right)^{-48} \approx \frac{1}{1.0833} \approx 0.9233.
   $$

6. Next, compute the numerator:

$$
   1 - 0.9233 = 0.0767.
   $$

7. Now, divide by the monthly interest rate:

$$
   \frac{0.0767}{0.00166667} \approx 46.02.
   $$

8. Finally, multiply by the monthly payment:

$$
   \text{PV} \approx 200 \times 46.02 \approx 9204.
   $$

Thus, the size of the loan you can afford is approximately $$ \$9,\!200 $$.
You can afford a loan of approximately \$9,200.

Question 5 You can afford a \( \$ 200 \) per month car payment. You've found a 4 year loan at \( 2 \% \) interest. How big of a loan can you afford? Question Help: \( \square \) Video 1 Video 2 Message instructor Submit Question

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