MM - Planning for the Future Lesson

Planning for the Future Lesson

 

 

Regression equations can be used to find a curve of best fit to data. Any prediction can be inaccurate due to rounding or outliers in the data. The regression equation is used as a model and does not always contain each data point.

 

Amanda is considering three job offers in educational publishing.

1. One is a full-time position as an editor that pays a salary of $37,500 per year.
2. Another is a full-time position as an e-Learning designer that pays an hourly wage of $26.50. The job assumes five 8-hour days per week.
3. The final offer is for a sales representative that pays a 5% commission. Sales representatives typically sell an average of $100,000 per month in textbooks.

There are many things to consider and the exercise below will walk you through it.

Amanda is analyzing how to invest $500. She is considering the two investments described below.

1. Savings accounts are insured and vary in the way in which interest is calculated. Some accounts pay simple interest, but other accounts compound interest at varying frequencies. Amanda is considering a savings account that pays 3.75% interest compounded annually.
2. A certificate of deposit (CD) is an interest-bearing instrument that is similar to a savings account—it is insured and pays interest. Unlike savings accounts, CDs have a fixed time period and usually a fixed interest rate. CDs also vary in the way in which interest is calculated. Sometimes the interest is compounded, but simple-interest CDs also exist. Simple interest is calculated only on the original deposit. The CD must be held until the date of maturity, at which time the original money deposited may be withdrawn with the accrued interest. Amanda is considering a CD that pays 4% simple annual interest for five years.

Amanda wants to evaluate each investment for the first five years. She used the spreadsheet below to record her calculations.

Amanda is using this investment as an emergency fund, in which should she invest?

Write a function rule to model each investment. Let y represent the value of the investment at the end of any year x.

• In the CD account calculations, you used repeated addition of $20. What operation represents repeated addition? multiplication
• In the savings account calculations, you used repeated multiplication of 0.0375 (or 1.0375). How can you represent repeated multiplication? with exponents
• • Savings Account = 500 + 500 x 0.0375^x
  • Can you use a single multiplication to calculate the ending balance in the savings account instead of both a multiplication to calculate the interest and a sum to calculate the ending balance?
  • Savings Account = 500 (1+1 x 0.0375^x)=500 x 1.0375^x
  • What types of functions did you use to model each investment option? How are the functions related to the type of interest earned in each option?
  • The savings account earns compound interest and is modeled by an exponential relationship.
  • The CD earns simple interest and is modeled by a linear function.

  The future value of an investment is the amount it will be worth after so many months or years of earning interest. The following table lists a savings account's future values in selected years.

  

 

• Create a scatterplot of the given data. Label the axes and scales and be sure to provide a title.

 

Data Modeling

What type of function would best model the data? Explain your reasoning. Exponential growth best models the data because the data are increasing at an increasing rate. 

Calculate the regression equation for the given data. Graph the regression equation on the scatterplot.

According to the model, what is the interest rate of the savings account?

• (1.0425 - 1) • 100 = 4.25%

Is the interest simple or compound? How do you know? The interest is compound because there is exponential growth resulting from the previous total being multiplied by the constant multiplier, rather than just the initial amount.  

Using the model, how much will be in the account in 50 years? If you round the regression equation, they get y= 2,600 * 1.0425  =$ 20,834.19 if you use the graphing calculator tp evaluate the regression equation without rounding, they get y = $ 20,834.27

• The balance your account grows to at some point in the future is called the future value of the deposit investment.

Use the regression equation from the previous problems to write a general formula for future value of an investment compounded annually. Use the following variables:

• FV for future value
• t for time (in years)
• i for interest rate (in decimal form)
• PV for the principal or present value
• FV =PV(1+i)^t

All of the investments so far have compounded and paid interest annually. However, some investments compute interest in compounding periods that are semi-annually quarterly or monthly. If the annual interest rate is divided evenly, how would the interest rate be calculated for each compounding period?

• For interest accrued monthly, the interest rate is the annual interest rate divided by 12.
• For interest accrued quarterly, the interest rate is the annual interest rate divided by 4.
• For interest accrued semi-annually, the interest rate is the annual interest rate divided by 2.

Using the variables from above, in addition to n for number of compound periods in one year you can write a general formula for future value that takes into account any compounding period.

Suppose you invest $2,600 into a savings account with a 4.25% annual interest rate that compounds interest quarterly. Use the formula to determine the balance in the account after five years. $3,211.99

How much would the same savings account be worth in 50 years if the interest is compounded quarterly?

REFLECTION:

Is there a difference between the account balance that was compounded quarterly after 5 years and the account balance that was only compounded annually shown in the table above? If so, is the difference large or small? How might this difference influence your decision about investments?

• By compounding quarterly, the investment is worth $10.49 more. This is a small increase. This small amount of difference might not matter too much since investments are generally considered over an extended amount of time.

Is there a difference between the account balance in 50 years using the formula and the account balance using the model? If so, is the difference large or small? How might this difference influence your decision about investments?

• By compounding quarterly, the investment is worth about $692.68 more after 50 years. This difference emphasizes the importance of the factor of time in investing. An investor would be wise to take this into account when making decisions about where to invest money.

 

 

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