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Chapter 7

The Time Value of Money

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Time ValueA. Process of expressing

1. The present value of $1 invested now in future terms. (Compounding)

Compounding – Process by which interest is paid on interest that was previously earned.

2. The future value of $1 invested in terms of the present

Future Value of a dollar – Amount to which a single payment will grow at some rate of interest.

B. Payments are either: 1. a single payment 2. A series of equal payments (an annuity)

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Time Value

C. Time value of money problems may be solved by using:1. Interest tables2. Financial calculators3. Software

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Variables for Time Value of Money Problems

PV = present valueFV = future valuePMT = annual paymentN = number of time periodsI = interest rate per period

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Future Value

Future value of $1 takes a single payment in the present into the future.

General equation for the future value of $1:

P0(1 + i)n = Pn

FV = PV (1 + i)n

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Future Value Illustrated

PV = -100I = 5N = 20PMT = 0FV = ?

= 265.30

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Greater Terminal Values

Higher interest rates

Longer time periods

Result in greater terminal values

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Greater Terminal Values

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Present Value

Present value of $1 brings a single payment in the future back to the present.Present Value – Current value of a dollar to be

received in the future.Discounting – Process of determining present

value.

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Present Value

General equation for the present value of $1:P0 = Pn

(1+i)n

PV = FV [ 1 / (1 + i)n ]

1. PV = Present Value

2. FV = Future Value

3. i = Interest Rate per Period

4. n = Number of Compounding Periods

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Present Value Illustrated

FV = 100I = 6N = 5PMT = 0PV = ?

= -74.73

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Lower Present Values

Higher interest rates

Longer time periods

Result in lower present values

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Lower Present Values

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Financial Calculators and Excel

Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows

At least one of the cash variables must be –an inflow (+)–an outflow (-)

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Simple Interest

Simple Interest – No Compounding– Good to have if you withdraw interest each

period.

– SI = Principal (PV) x rate (i) x time (n)

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Future Value of a Single Amount Example

You buy a stock for $10 and expect the price to increase 9 percent annually. After 10 years, what is the anticipated price of the stock?

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Future Value of a Single Amount Example

The unknown: FVThe givens:

–PV = 10

–PMT = 0

–N = 10

–I = 9

The answer: $23.67

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FV Interpretation

A $10 stock will be worth $23.67 after 10 years if its price grows 9% annually.

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Present Value of a Single Amount Example

What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?

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Present Value of a Single Amount Example

The unknown: PVThe givens:

–FV = 23.67

–PMT = 0

–N = 10

–I = 9

The answer:$10

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Interpretation

$23.67 received after ten years is worth $10 today if the rate of return is 9 percent.

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Interpretation of Future and Present Values

These two problems are mirror images: In the first case, the $10 is compounded

into its future value ($23.67).In the second case, the future value

($23.67) is discounted back to its present value ($10).

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Rate of Return Example

A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?

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Future Determination of the Interest Rate(can use Present too)

The unknown: IThe givens:

–PV = 10

–PMT = 0

–N = 0

–FV = 23.67

The answer: 9%

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Interpretation

The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.

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Non-annual Compounding

More than one interest payment a year

State Interest rates are always annual interest rates.

More frequent compounding

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Non-annual Compounding

Multiply number of years by frequency of compounding

Divide interest rate by frequency of compounding

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Periods less than One Year

Same variables as in all time value problems except N < 1.

Calculate by dividing the number of days by 365.

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Illustration for Return on Investment

What is the return on an investment that costs $98,543 and pays $100,000 after 45 days?

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Determination of Return

The unknown: IThe givens:

–PV = -98,543

–N = 0.1233 (45/365)

–FV = 100,000

–PMT = 0

The answer: 12.64%

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Interpretation

$98,543 invested for 45 days grows to $100,000 at 12.64 percent.