**CHAPTER 4 – TIME VALUE OF MONEY 1: ANALYZING SINGLE CASH FLOWS**

**Questions**

1. List and describe the purpose of each part of a time line with an initial cash inflow and a future cash outflow. Which cash flows should be negative and which positive? Why?

The cash flow timeline is a visual depiction of inflows and outflows relative to the period under consideration. Cash flows are illustrated above the cash flow line with the corresponding periods that apply appearing under the cash flow diagram. Inflows are represented by positive numbers and outflows by negative numbers.

2. How are the present value and future value related?

The measure that relates present values to future values is the interest rate i. A present value can be moved forward in time with interest to arrive at the future value

A future value can be discounted back to the present by rearranging the equation so that the FV is divided by the interest factor.

3. Would you prefer to have an investment earning 5 percent for 40 years or an investment earning 10 percent for 20 years? Explain.

Investments of $1 will grow to $7.04 in 40 years (= $1.05^{40}) and $1 will grow to $6.73 in 20 years (= $1.10^{20}). The 5 percent investment for 40 years is worth more. This example illustrates the importance of time in building wealth.

4. How are present values affected by changes in interest rates?

Interest rates have an inverse relationship to present values. Increases in expected interest rates result in lower present values because future values are discounted at a high rate to become smaller present values. Decreases in expected interest rates result in higher present values because future values are discounted at a lower rate.

5. What do you think about the following statement. “I am going to receive $100 two years from now and $200 three years from now, so I am getting a $300 future value.” How could the two cash flows be compared or combined?

Cash flows may only be combined when they are moved to the same point in time. The statement above is incorrect in that it compares the $100 cash flow after year 2 with the $200 cash flow after year 3. To make comparisons meaningful, the cash flows need to be considered at the same point in time. Either the $100, 2^{nd} year cash flow could be moved to year 3, or the $200, 3^{rd} year cash flow could be moved to year 2 for combining.

6. Show how the Rule of 72 can be used to approximate the number of years to quadruple an investment.

The Rule of 72 is a rule of thumb that approximates the amount of time necessary for an investment to double given a certain level of interest expressed in percentage form. Therefore, an investment of 8% interest will take approximately 9 years (= 72/8) to double according to the Rule of 72. It would then take another 9 years for this amount to double. Therefore, it would take 18 years for the original investment to quadruple at an 8 percent rate.

7. Without making any computations, indicate which of each pair has a higher interest rate?

a. $100 doubles to $200 in 5 years *or* 7 years.

b. $500 increases in 4 years to $750 *or* to $800.

c. $300 increases to $450 in 2 years *or* increases to $500 in 3 years.

a. $100 doubling to $200 in 5 years has the higher interest rate.

b. $500 increasing to $800 in 4 years has a higher interest rate.

c. $300 increasing to $450 in 2 years has the higher interest rate.

The Rule of 72 predicted that $1000 will double to $2000 in 8 years at 9 percent interest. Another doubling from $2000 to $4000 will occur in eight more years at 9 percent, again predicted by the Rule of 72.

Problems

4-1 **Time Line** Show the time line for a $300 cash inflow today, a $363 cash outflow in year two, and a 10 percent interest rate.

The time line for this problem is:

4-2 **Time Line** Show the time line for a $400 cash outflow today, a $518 cash inflow in year three, and a 9% interest rate.

The time line for this problem is:

Cash Flow -400 518

Period 0 9% 1 9% 2 9% 3 years

4-3 **One Year Future Value** What is the future value of $500 deposited for one year earning a 9% interest rate annually.

*FV _{N}* =

*PV*× (1 +

*i*)

^{N}*FV _{1}* = 500 × (1 + 0.09)

^{1}

= 500 × 1.09

= 545

4-4 **One Year Future Value** What is the future value of $400 deposited for one year earning an interest rate of 11 percent per year?

*FV _{N}* =

*PV*× (1 +

*i*)

^{N}*FV _{1}* = 400 × (1 + 0.11)

^{1}

= 400 × 1.11

= 444

4-5 **Multi-Year Future Value** How much would be in your savings account in 8 years after depositing $150 today if the bank pays 7 percent per year?

*FV _{N}* =

*PV*× (1 +

*i*)

^{N}*FV _{8}* = 150 × (1 + 0.07)

^{8}

= 150 × 1.71819

= 257.73

4-6 **Multi-Year Future Value** Compute the value in 25 years of a $1,000 deposit earning 10 percent per year.

*FV _{N}* =

*PV*× (1 +

*i*)

^{N}*FV _{25}* = 1000 × (1 + 0.10)

^{25}

= 1000 × 10.83471

= 10,834.71

4-7 **Compounding with Different Interest Rates **A deposit of $350 earns the following interest rates:

· 8 percent in the first year,

· 7 percent in the second year, and

· 5 percent in the third year.

What would be the third year future value?

The time line for this problem is:

*
*

*FV* = *PV* × (1 + *i*) (1 + *j*) (1 + *k*)

*FV* = 350 × (1 + 0.08) (1 + 0.07) (1 + 0.05)

= 350 × 1.08 × 1.07 × 1.05

= 424.68

4-8 **Compounding with Different Interest Rates** A deposit of $750 earns interest rates of 10 percent in the first year and 12 percent in the second year. What would be the second year future value?

The time line for this problem is:

*
*

*FV* = *PV* × (1 + *i*) (1 + *j*)

*FV* = 750 × (1 + 0.10) (1 + 0.12)

= 750 × 1.10 × 1.12

= 924.00

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