**SAS 6 and 7**

**How is the process different for calculating the future value of each investment?**

2. Refer to the future-value formula in Student Activity Sheet 3. How is the process

different in calculating the future value of an annuity when compared to using the

future-value formula?

4. The following formula can be used to calculate the sum of a series connected by a

common ratio, such as the previous annuity example.

Sn = a1

(1 − rn)

(1 − r) , where

a1 = the first term in the series, n = the number of terms in the series, and r = the

common ratio.

Use the formula to calculate the value of the annuity described in the graph, and

compare the results after five years.

220(1-1.1^5)

_______________

(1-1.1)

3. 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?

We add 10% and it is related to the problem situations interest and payment.

1. 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?

What does this relationship imply about the risk of stocks compared to bonds? Explain

your reasoning

The relationship that Stock Texas has with interest rates is an inverse relationship.

Inverse goes with stocks and bonds get interest rates.

2. 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?

Do you think that this is a reasonable assumption about the relationship between bonds

and risk ratings? Why or why not?

Direct variation because one of the variable is always constant to the other.

3. 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)?

AAA 9

AA 8

A 7

BBB 6

BB 5

72/9=x/5

9x=362

x=40

4. 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.

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 (♦).

(23)(156)=75(x)

3588= 71.76(y)

50=Y

3588= 75(X)

47.84=x

REFLECTION: How certain is this prediction? What other factors could affect the price of

either investment?

Not very certain, factors that could affect the price of either investment include: interest rates, inflation/deflation, etc.

EXTENSION: Emily, who is 25 years old, has $25,000 to invest. She wants to invest in stocks, bonds, and/or cash accounts (collectively called an investment portfolio). Currently interest rates (and inflation) are relatively low, but seem to be on the rise.

Decide the percentage and amount that Emily should invest in each category.

Stocks: 30% or $7500

Bonds: 20% or $5000

Cash Accounts: 50% or $12500

Stocks: $7500(1+0.12)^35

$395997.15

Bonds: $5000(1+0.06)^35

$38430.43

Cash Accounts: $12500(1+0.03)^35

$35173.28

FV=PV(1=r)^n

The future value (FV) measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return. The FV is calculated by multiplying the present value by the accumulation function. The value does not include corrections for inflation or other factors that affect the true value of money in the future. The process of finding the FV is often called capitalization.

Source: Boundless. “The Relationship Between Present and Future Value.” Boundless Finance. Boundless, 20 Jan. 2015. Retrieved 19 May. 2015 from https://www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values-59/the-relationship-between-present-and-future-value-271-1835/

If you know how much you can invest per period for a certain time period, the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate

5. In Student Activity Sheet 5, 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. List the

values you assigned to each variable and explain why.

(Note: Interest is typically paid at the end of the compounding period. In this case, you

make payments at the beginning

5

10

0

200

1

1

1221.1

6. 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.

5

3

5

0

1200

108384

1

1

7. 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?

-Bigger payment

-Work more than 35 years

-Better Investment

8. 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.

480

6

0

12

12

424

85000