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Chapter 4: Problem 76

Carlos invested $$\$ 5000$$ in a money market mutual fund that pays interest on a daily basis. The balance in his account at the end of 8 mo ( 245 days) was \(\$ 5170.42\). Find the effective rate at which Carlos's account earned interest over this period (assume a 365-day year).

Short Answer

Expert verified
The effective interest rate for Carlos's account over the 8-month period is approximately \(4.318\% \).

Step by step solution

01

Identify the compound interest formula

The formula for compound interest is: \[ A = P(1 + \frac{r}{n})^{nt} \] Where: - \(A\) is the final amount in the account - \(P\) is the initial principal investment - \(r\) is the annual interest rate (expressed as a decimal) - \(n\) is the number of times the interest is compounded per year - \(t\) is the number of years the money is invested for In our case, the interest is compounded daily, so \(n = 365\), and we need to find the interest rate \(r\).
02

Plug in the known values

\[ A = 5170.42\] \[ P = 5000 \] \[ n = 365 \] \[ t = \frac{245}{365} \]
03

Rewrite the formula based on the given values

\[ 5170.42 = 5000(1 + \frac{r}{365})^{365 \cdot \frac{245}{365}} \]
04

Simplify the equation

First, simplify the exponent by canceling out the 365: \[ 5170.42 = 5000(1 + \frac{r}{365})^{245} \] Now, we need to solve for the interest rate, \(r\).
05

Solve for the interest rate

First, isolate the term with the interest rate by dividing both sides by 5000: \[ \frac{5170.42}{5000} = (1 + \frac{r}{365})^{245} \] Then, find the 245th root of both sides to eliminate the exponent: \[ \sqrt[245]{\frac{5170.42}{5000}} = 1 + \frac{r}{365} \] Next, subtract 1 from both sides: \[ \sqrt[245]{\frac{5170.42}{5000}} - 1 = \frac{r}{365} \] Finally, multiply both sides by 365 to find the interest rate: \[ r = 365 \cdot \left( \sqrt[245]{\frac{5170.42}{5000}} - 1 \right) \] Use a calculator to find the value of \(r\): \[ r \approx 0.04318 \]
06

Convert the interest rate to a percentage

To find the effective interest rate as a percentage, multiply the decimal value by 100: \[ 0.04318 \cdot 100 = 4.318 \% \] The effective interest rate for Carlos's account over the 8-month period is approximately 4.318%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Effective Interest Rate
The effective interest rate refers to the actual annual rate that an investment earns or a loan accrues after accounting for compounding. It gives a more accurate picture than the nominal interest rate, especially when interest is compounded more than once a year. In Carlos's case, the account balance increased from $5000 to $5170.42 due to daily compounding. To find the effective interest rate for his investment for this 8-month period, we applied the compound interest formula. The effective interest rate provides a means for investors to assess the real value and performance of their investments over time. This rate, especially with frequent compounding, can significantly differ from the nominal annual rate.

In financial contexts, the effective interest rate helps compare investment options with different compounding frequencies. It provides transparency and clarity, aiding investors in making educated decisions based on how interest accumulates over time.
Interest Compounding
Interest compounding is the process by which interest earned on an investment or loan is reinvested or added to the principal, so that in future periods, interest is earned on the total amount. In simple terms, it's when you're earning 'interest on interest.'

For Carlos, the daily compounding implies that the interest was added to the account balance every day, meaning each day's interest was calculated on the balance of the previous day. Despite appearing trivial during short periods, this effect becomes substantial amid longer durations or higher frequencies of compounding.

Carlos's investment demonstrates daily compounding, implying that interest calculations occur 365 times within a year. This contrasts with annual or quarterly compounding, where interest accumulates less frequently. The compounding frequency matters because more frequent compounding typically yields greater total returns, leading investors to consider options with more frequent compounding intervals.
Financial Mathematics
Financial mathematics involves using mathematical formulas and calculations to solve problems related to finance, banking, investments, and other money-related matters. Understanding how money grows over time through methods like compound interest falls under this domain.

In the exercise, financial mathematics provided the tools to determine the interest rate and effective yield from Carlos's investment. Using the well-known compound interest formula, we navigated through known variables such as initial investment (P), time period (t), and frequency of compounding (n) to solve for the effective interest rate (r).

Financial mathematics is crucial in decision making, helping to analyze investment opportunities, evaluate financial products, and comprehend risk levels. Whether it's through calculating returns on investments or determining the financial impact of variable interest rates, financial mathematics underpins myriad decisions individuals and businesses make daily.

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Most popular questions from this chapter

The management of Gibraltar Brokerage Services anticipates a capital expenditure of $$\$ 20,000$$ in 3 yr for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of \(10 \% /\) year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.

Suppose payments will be made for \(6 \frac{1}{2}\) yr at the end of each semiannual period into an ordinary annuity earning interest at the rate of \(7.5 \% /\) year compounded semiannually. If the present value of the annuity is $$\$ 35,000$$, what should be the size of each payment?

Suppose an initial investment of $$\$ P$$ grows to an accumulated amount of $$\$ A$$ in \(t\) yr. Show that the effective rate (annual effective yield) is $$ r_{\text {eff }}=(A / P)^{1 / t}-1 $$ Use the formula given in Exercise 71 to solve Exercises \(72-76 .\)

Josh purchased a condominium 5 yr ago for $$\$ 180,000$$. He made a down payment of \(20 \%\) and financed the balance with a 30 -yr conventional mortgage to be amortized through monthly payments with an interest rate of \(7 \% /\) year compounded monthly on the unpaid balance. The condominium is now appraised at $$\$ 250,000$$. Josh plans to start his own business and wishes to tap into the equity that he has in the condominium. If Josh can secure a new mortgage to refinance his condominium based on a loan of \(80 \%\) of the appraised value, how much cash can Josh muster for his business? (Disregard taxes.)

Joe secured a loan of $$\$ 12,0003$$ yr ago from a bank for use toward his college expenses. The bank charged interest at the rate of \(4 \% /\) year compounded monthly on his loan. Now that he has graduated from college, Joe wishes to repay the loan by amortizing it through monthly payments over \(10 \mathrm{yr}\) at the same interest rate. Find the size of the monthly payments he will be required to make.
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