Daniel received a loan of $51,000, 7 years ago.

Question

Daniel received a loan of $51,000, 7 years ago. The interest rate charged on the loan was 4.68% compounded quarterly for the first 6 months, 5.55% compounded semi-annually for the next 2 years, and 5.67% compounded monthly thereafter.

a. Calculate the accumulated value of the loan at the end of the first 6 months.

Round to the nearest cent

b. Calculate the accumulated value of the loan at the end of the next 2 year period.

Round to the nearest cent

c. Calculate the accumulated value of the loan today.

Round to the nearest cent

d. Calculate the amount of interest charged on this loan over the past 7 years.

Round to the nearest cent

Answer 1

a.) We have a $51,000 loan with an interest rate of .0468 compounded quarterly for the first 6 months. Ok, 6 months is half a year. Quarterly is 1/4 of a year or every 3 months, so for the first 6 months we're going to compound (add up the interest to the balance and take that new bigger balance to charge interest on) our loan twice.

The first 6 months are compounded Quarterly, or every 3 months at a 4.68% interest rate.

The first 3 months we'd take (51,000 x .0468) + 51,000 = $53,386.80

The next 3 months we'd use our new balance of (53,386.8 x .0468) + 53,386.8 = $55,885.30

b.) The next two years (24 months) are compounded semi annually (twice a year or every 6 months) at a 5.55% rate.

So we know during that two year period we're going to compound the interest on our loan twice a year or 4 times over that period. We take the balance we got from A and use it to start our calculations on B.,

The first 6 months would be ($55,885.3 x 5.5%) + $55,885.3 or $58,958.99

The next 6 months would be ($58,958.99 x .055) + $58,958.99 or $62,201.73

We would then do that two more times to find the balance at the end of the two year period of 5.5% interest compounded semiannually.

c.) The loan was take out 7 years ago. We've calculated 6 months plus another 24 months, or 30 months total so far. 7 years is 84 months (7x12), so we have 54 months (84-30) of interest left to calculate at a monthly compounded rate of 5.67%

At this point we could take our balance of $62,201.73, multiply it by .0567, add it to the previous balance and do it again 53 more times or we could use a formula. I vote formula.

Total = Principal(1 + interest rate/ number of times compounding yearly)^number of years.

In our case: T = $62,201.73[(1 + .0567)/12]^4.42 (4.42 comes from 53 months remaining/ 12 months in a year to get 4.4166666 years left)

PEMDAS says do parenthesis first, then exponents, so lets simplify it a bit:

T= $62,201.73(1.0567)^4.42

T= $62,201.73 times 1.276

T= $79,369.41

d.) Now we just go back through and add up all the interest over the past 7 year