SOLUTION: You can afford a $200 per month car payment. You've found a 3 year loan at 4% interest. How big of a loan can you afford?

Use the standard formula for the monthly payment for a loan

    M = 


where L is the loan amount; r =  is the effective interest rate per month;
n is the number of payments (same as the number of months); M is the monthly payment.


From this formula, the expression for the maximum loan is

    L = .


In this problem  M = $200;  r = ,  n = 3*12 = 36 monthly payments.


Substitute these values into the formula and get for the maximum loan amount

    L =  = $6,774.15.


ANSWER.  The maximum loan amount is $6774.15.

Loan Affordability Calculation 

The maximum loan amount that can be afforded is determined by calculating the present value of an annuity, 
where the monthly payment is the annuity payment. 

Step-by-step Solution 

1. Identify the given values: 
	• Monthly payment, $PMT = \$200$. 
	• Annual interest rate, $r_{annual} = 4\%$. 
	• Loan term, $t = 3 \text{ years}$. 

2. Calculate the monthly interest rate: 
	• The monthly interest rate, $r_{monthly}$, is found by dividing the annual interest rate by $12$: 

$r_{monthly} = \frac{r_{annual}}{12} = \frac{0.04}{12} = 0.003333$. 

1. Calculate the total number of payments: 
	• The total number of payments, $n$, is found by multiplying the loan term in years by $12$: 

$n = t \times 12 = 3 \times 12 = 36 \text{ payments}$. 

1. Apply the present value of an annuity formula: 
	• The present value of an annuity, $PV$, which represents the maximum loan amount, is calculated using the formula: 

$PV = PMT \times \frac{1 - (1 + r_{monthly})^{-n}}{r_{monthly}}$. 

• Substitute the calculated values into the formula: 

$PV = 200 \times \frac{1 - (1 + 0.003333)^{-36}}{0.003333}$. 

• Calculate the term $(1 + 0.003333)^{-36}$: 

$(1.003333)^{-36} \approx 0.8874$. 

• Substitute this value back into the formula: 

$PV = 200 \times \frac{1 - 0.8874}{0.003333}$. 

• Perform the subtraction in the numerator: 

$PV = 200 \times \frac{0.1126}{0.003333}$. 

• Perform the division: 

$PV = 200 \times 33.78$. 

• Perform the final multiplication: 

$PV \approx \$6756$. 
Final Answer 
A loan of approximately $\$6756$ can be afforded. 

AI responses may include mistakes.