Question

At the beginning of each of her four years in college, Miranda took out a new Stafford loan. Each loan had a principal of $5,500, an interest rate of 7.5% compounded monthly, and a duration of ten years. Miranda paid off each loan by making constant monthly payments, starting with when she graduated. All of the loans were subsidized. What is the total lifetime cost for Miranda to pay off her 4 loans?

Answer 1

At the beginning of each of her four years in college, Miranda took out a new Stafford loan. Each loan had a principal of $5,500, an interest rate of 7.5% compounded monthly, and a duration of ten years. Miranda paid off each loan by making constant monthly payments, starting with when she graduated. All of the loans were subsidized. The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

Answer 2

The total lifetime cost of all the loans is $31336.8

Further Explanation:

Present Value: It is the current value of a future cash flow or streams of cash flow calculated at given a specific rate of return. The present value the  as:

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

Calculate the total lifetime cost of the loan:

$\begin{aligned}\text{Total lifetime cost of loans}&=\text{Monthly cost of loans}\times\text{Total years}\times12\\&=\ 261.14\times10\times12\\&=\ 31,336.8\end{aligned}$

Thus, the total lifetime cost of all the loans is $31336.8

Working note 1:

Calculate the monthly payment:

$\begin{aligned}A&=\ 22,000\\r&=7.5\%\\n&=10\,{\text{years}}\\\\\end{aligned}$

Then,

$\begin{aligned}PV&=A\times\dfrac{\left[1-\left(\dfrac{1}{(1+r)^{n}}\right)\right]}{r}\\&=\ 22,000\times\dfrac{\left[1-\left(\dfrac{1}{\left(1+\dfrac{7.5}{12}\right)^{120}}\right)\right]}{\dfrac{7.5}{12}}\\&=\ 261.14\end{aligned}$

Thus, the total amount of monthly payment is $261.14

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Answer details:

Grade: High School

Subject: Financial Management

Chapter: Time Value of Money

Keywords: At the, beginning, of each, of, her, four, years, in college, Miranda, took, out, a new, Stafford, loan, Each, loan, had, a principal, of $5,500, an interest, rate, of, 7.5%, compounded, monthly, and, a duration, often, year,. Miranda, paid, off, each loan, by making, constant, monthly payments, starting, with, when, she, graduated, All, of the loans, were subsidized, What is the total, lifetime, cost, for Miranda, to pay, off her, 4 loans.