Question

Enea, a promising entrepreneur, was excited for being recognized as having the best business plan at her college's annual entrepreneurial competition. An angel investor was so impressed with her plan that he offered her a loan at 6% compounded semi-annually to start her designer clothing store.

After two years, her business had savings of $80,654 and she used the entire amount to completely pay off her outstanding debt with the investor.

a. What was the loan amount provided to her by the angel investor and what was the accumulated interest over the two- year period?

b. What rate, compounded monthly, would have resulted in the same accumulated debt?

c. If Enea did not pay any amount throughout the term and the interest rate had remained at 6% compounded semi- annually, how long (in years and days) will it take for the debt to reach $100,000?

d. If she had obtained the same loan amount from a local bank, it would have accumulated to $80,654 in 18 months instead of two years. What is the interest rate compounded semi-annually charged by the local bank?

e. Calculate the loan amount provided to her by the angel investor if the loan had been issued to her at an annually
compounding frequency instead of a semi-annually compounding frequency. Compare your answer to (a) and determine what her savings would be.

f. If her contract with the investor required that she settle all dues in two years, how much could she have borrowed initially if she was sure that she could repay $25,000 in one year and $40,000 at the end of two years?

g. What was the size of the loan provided by the investor if she was charged 6% compounded semi-annually for the first
year and 8% compounded quarterly for the second year, and it accumulated to $80,654 in two years?

Answer 1

Explanation:

Let's break down each part of the problem step by step:

a. To calculate the loan amount (present value) provided by the angel investor and the accumulated interest over the two years, you can use the formula for compound interest:

A = P ( 1 + ( r / n ) ) ^ nt

Where:

• A is the amount of money saved ($80,654).

• P is the principal amount (the loan amount the investor provides).

• r is the annual interest rate (6% or 0.06 as a decimal).

• n is the number of times interest is compounded annually (semi-annually, so n = 2).

• t is the number of years (2 years).

We must find P (the loan amount) and the accumulated interest.

Using the formula above Simplify and solve for P:

Calculate P:

P \approx \frac{80,654}{1.1255089} \approx $71,505

So, the loan amount provided by the angel investor was approximately $71,505.

To find the accumulated interest, subtract the loan amount from the total savings:

Accumulated Interest = $80,654 - $71,505 = $9,149

b. To determine the interest rate compounded monthly that would result in the same accumulated debt, we can use the formula for compound interest with monthly compounding:

(orignal formula)

We already know A is $80,654, and we can use P = $71,505, n = 12 (for monthly compounding), and t = 2 years.

Solve for r (the monthly interest rate):

Now, take the 24th root of both sides to find the monthly interest rate:

Now, solve for r:

Calculate r:

r≈0.0768 or 7.68%

So, the equivalent monthly interest rate is approximately 7.68%.

c. To find out how long it will take for the debt to reach $100,000 if Enea did not make any payments and the interest rate remained at 6% compounded semi-annually, we can use the formula for compound interest with an unknown time (t):

(use the same formula)

We know A is $100,000, P is the initial loan amount, r is 6% (0.06 as a decimal), and n is 2 (semi-annual compounding).

Solve for t:

Now, take the natural logarithm (ln) of both sides:

Use the property of logarithms to bring down the exponent:

Now, solve for t:

We have the values for ln(1.06) and ln(100,000/P):

Simplify:

Now, let's say t is the number of years, so we'll calculate t in years and then convert it to years and days:

Now, plug in the values and calculate t:

Calculate t:

Use a calculator to find the natural logarithm and solve for t:

So, the debt will take approximately 2.819 years to reach $100,000.

To find the number of days, multiply the fractional part of the year (0.819) by the number of days in a year (365):

0.819

∗

365

≈

299.14

0.819∗365≈299.14

So, the debt will take approximately 2 years and 299 days to reach $100,000.

d. To find the interest rate compounded semi-annually charged by the local bank if the loan accumulated to $80,654 in 18 months (1.5 years), we can use the formula for compound interest:

(repeat) the same instructions to g.