MM - Investing As You Go Lesson
Investing as You Go
Adapted from Course materials (VI.B Student Activity Sheet 4-7) for AMDM developed under the leadership of the Charles A. Dana Center, in collaboration with the Texas Association of Supervisors of Mathematics and with funding from Greater Texas Foundation.
In the previous lesson, you analyzed the future value of an investment over time. You began with $2,600 invested in a savings account for 30 years. After 30 years, your initial investment would be worth $9.062.70. In this lesson, you will look at the same investment in a different way. The question relates to the time value of money (TVM). What is that $9,062.70 future value worth at various times in the 30-year investment?
The following table lists the principal required to obtain the same future value of $9,062.70 for various investment lengths. So, in the table, the 30-year investment is the one you have already explored. The other values in the table show how much principal you would need to invest and the length of time of the investment for the same yield. This can be thought of as the present value of the investment.
|
Years Until Maturity |
Principal Required |
|---|---|
| 0 | $9062.70 |
| 5 | $7359.95 |
| 10 | $5977.16 |
| 15 | $4854.16 |
| 20 | $3942.20 |
| 25 | $3101.50 |
| 30 | $2600.00 |
Calculate the regression equation for the given data. Graph the regression equation on the scatterplot. Explain why the function model you used makes sense in the problem situation.
- Exponential decay is used to model the situation because the data points are decreasing at a decreasing rate.
- The greater the years of investment, the less you have to invest initially.
- y = 9,062.68 • 0.9592x
- The data, graph, and regression equation all support the fact that more principal is required the longer you wait to invest if you want the future value of the investment to be a specified amount.
TVM Calculator
In this activity, you will use a new tool, the TVM calculator. The TVM calculator is cash-flow based and calculates money values in terms of how money flows in and out of a person's wallet. The present value is negative. Most graphing calculators have some tool that calculates various variables in the future-value formula. Check your calculator manual for specifics to your calculator. Several websites, such as those listed in the Resources section, provide online TVM calculation tools. Spreadsheets can also be used to calculate these values utilizing a similar formula. There are many financially related formulas built into spreadsheets that can be used to calculate the different values involved in TVM.
Josephine is 20 years old and wants to save $1 million for retirement in 50 years. Assume she invests in a savings account that earns at least the current rate of inflation. Determine how much Josephine must save today to reach her retirement goal.
Recall the future-value formula:
Suppose Josephine does not want to begin saving for her retirement immediately. Fill in the following table to show the amount of money that Josephine must invest to retire 50 years from now with $1,000,000 based on the number of years that she waits to start saving.
|
Year |
Balance |
|---|---|
| 0 | $2600.00 |
| 5 | $3101.50 |
| 10 | $3942.20 |
| 15 | $4854.16 |
| 20 | $5977.16 |
| 25 | $7359.95 |
| 30 | $9062.70 |
Suppose Josephine believes in spending now and saving later. How could you use the table to convince her otherwise? The longer she waits to save for retirement, the more money she needs for her initial investment (principal).
Present Value
Present value is the current value of a deposit that is made in the present time. You can determine the present value of a single deposit investment or calculate how much a one time deposit should earn at a specific interest rate in order to have a certain amount of money saved for a future goal.
Using algebra, the present value formulas are derived from the future value formulas that you studied in the previous lesson.
Blaine wants to have $1,000 in 10 years. The following are the choices in which he can invest:
- a savings account earning 3% compounded quarterly,
- a checking account earning 1% compounded monthly, or
- a money market account earning 4.5% compounded semiannually.
Blaine plans on making no withdrawals or deposits for 10 years.
Rewrite the present-value formulas for each account that Blaine is considering. Make sure that the formulas include compounding periods other than annual and incorporate the different rates.
Graph the present-value formula for each account. Label the axes, scales, and curves, and provide titles.
Which factor has the most significant effect on the curve: the interest rate or compounding periods? Why? Interest rates have the greatest effect, because the present values or principal investment for 10 years is controlled by interest rate. A difference in interest rates leads to an increasingly large difference in future values across many compound periods.
In which account should Blaine invest? Why? Blaine should invest in the money market account because it has the lowest principal investment required to have 1,000 in 10 years!
Below are additional resources to explore TVM Calculations:
Online TVM Calculation Tools
JuFinance Calculator Links to an external site.
Arachnoid Finance Calculator Tool Links to an external site.
TVM Tutorials
TI-84 Plus Tutorial Links to an external site.
Microsoft Excel Tutorial Links to an external site.
Annuities
An annuity is a financial product that accepts and grows funds and then, upon annuitization, pays out regular payments to the investor. Annuities are often used as retirement funds. Some annuities are funded with a lump-sum investment, while others are funded with an initial investment and additional regular deposits before retirement. What complicates the time value of money (TVM) of an annuity that you pay into is that the investment increases in value due to both compound interest and increasing principal.
The following graph shows the value of a lump-sum investment of $1,000 earning 10% compounded per year (•) versus an annuity with an initial investment of $200 earning 10% compounded per year with additional $200 deposits made each year (♦).
How is the process different for calculating the future value of each investment?
- The future-value formula uses one amount to compound the interest, whereas the annuity process uses different amounts invested for varying times. The lump-sum investment earns compounded interest off the $1,000 principal, whereas the annuity earns interest off $200 and then adds another $200 to the principal.
Answer the following questions:
- Each year, for five years, you pay $200 into the annuity. After the five years, how long has the first $200 payment been invested?
- What is the future value of the first $200 payment at the end of the five years?
- After five years, how long has the second $200 payment been invested?
- What is the future value of the second $200 payment at the end of the five years?
An annuity can be thought of as a series of values connected by a common ratio. What common ratio connects the values of the annuity over time shown in the graph at the beginning of this activity sheet? How is the ratio related to the problem situation?
The following formula can be used to calculate the sum of a series connected by a common ratio, such as the previous annuity example.
Use the formula to calculate the value of the annuity described in the graph, and compare the results after five years.
The first term in the series is the balance at the end of the first year: 200 • 1.1 = 220
You learned to use a TVM calculator to determine different variables related to TVM. In your prior work with the TVM calculator, you only considered lump-sum investments (and the payment variable was always 0).
Explore using the TVM calculator to determine the future value of the $200 annuity over five years, and compare your answer with the known future value of $1,343.12. (Note: Interest is typically paid at the end of the compounding period. In this case, you make payments at the beginning of each period. Therefore, you must change appropriate variable from END to BEGIN.)
Amy is 25 years old and has attended some retirement planning seminars at work. Knowing she should start thinking about retirement savings early, Amy plans to invest in an annuity earning 5% interest compounded annually. She plans to save $100 from her monthly paychecks so that she can make annual payments of $1,200 into the annuity.
Use the TVM calculator to determine the future value of the investment after 35 years.
- The future value of the annuity is $113, 803.59
Amy seeks the advice of a financial planner, who recommends $850,000 for retirement. Will Amy's annuity plan provide the necessary funds for her retirement? If not, what could she do to increase the value of the investment at retirement? Of those actions, which does she have relative control over?
- No, Amy will not have enough for retirement.
- Amy could work more than 40 years. If she works until age 65, she has 40 years to make payments into the annuity. However, this still only gives her $152,207.72. Amy must increase the amount she saves from each paycheck to increase her annual payment to the annuity. Another option is to find an annuity that pays better interest.
Amy finds another annuity that accounts for monthly compounding and monthly payments. The annuity pays 6% annual interest, compounded monthly. Use the TVM calculator to determine the monthly payments Amy needs to make over 40 years to have $850,000 at the time of her retirement.
- Amy needs to make monthly payments of about $427 to reach her retirement goal.
Interest rates are a measure, among many other factors, of risk. The more risky an investment is in actuality and perception, the higher the rate of return. In general, stocks (an investment security that gives you ownership in a company) are riskier than bonds (a security in which you actually lend money to a company). Thus, the rate of return is much higher for stocks than bonds; on average, stocks have a rate of return of 10% annually and bonds 5% annually.
Use the following information when working through these activities:
- All investments have a rate of return (which sometimes can be negative).
- The rate of return on stocks is a percentage called a return on investment (ROI) that compounds not from interest payments but from an overall annual increase based on a price per share that changes daily.
- The rate of return on bonds is an actual interest rate percentage that is assumed to compound (much like a certificate of deposit), but may not if you decide not to reinvest the interest.
- Financial analysts use the time value of money (TVM) based on risk, rate of return, and the relationship it has with other investments to determine the market value or price of a share of stock or bond.
- Although interest rates are used in bonds, financial experts use interest as the lending rate that the Federal Reserve sets for banks. This may not seem related to stock prices or bonds, but the interest rate set by the Federal Reserve affects the value of all investments.
Stocks
Stock Texas is worth $14.92 per share on Monday. The interest rate drops on Tuesday, and Stock Texas is worth $15.04 per share. What type of relationship can you assume that Stock Texas has with interest rates? Why?
- Stocks vary inversely with respect to interest rate. As the interest rate goes down, the stock price goes up.
What does this relationship imply about the risk of stocks compared to bonds? Explain your reasoning.
- As the stock market goes down in value, bonds become a more attractive investment option because of their security and their higher rate of return. However, when the stock market recovers, bonds are less attractive to investors.
On Wednesday, Bond Austin has the best risk rating, Aaa, at a price of $72. On Thursday, the risk rating drops to a lower rating of Aa, and the price drops to $64. What type of relationship can you assume that the price of Bond Austin has with its risk ratings? Why?
- The price of the bond varies directly with its risk rating. As the rating goes down, the price goes down.
Assume losing a letter is considered one unit of risk and you assign the highest (meaning better) rating a 9. What does the price of Bond Austin drop to if the risk rating suddenly becomes Bb (a risk rating of 5)?
- Using proportional reasoning:
Stock Texas has a price of $156 per share when Bond Austin has a price of $23 per bond. Use an equation modeling the inverse variation between the stock and bond prices to predict the price of Stock Texas when Bond Austin is worth $75.
x = bond price
y = stock price
k = constant of proportionality
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