Foundations of Finance 10th Keown - Chapter 5 Notes PDF

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Lecture notes from text. Chapter 5, Foundations of Finance Keown, 10th edition....

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CHAPTER 5

The Time Value of Money CHAPTER ORIENTATION In this chapter, the concept of the time value of money is introduced; that is, a dollar received today is worth more than a dollar received a year from now. If we are to compare projects and financial strategies logically, we must either move all dollar flows back to the present or out to some common future date.

CHAPTER OUTLINE I. Compound Interest, Future Value, and Present Value Compound interest results when the interest paid on the investment during the first period is added to the principal so that during the second period the interest is earned on the original principal plus the interest earned during the first period. A. Using Timelines to Visualize Cash Flows 1. To visualize cash flows, it is helpful to construct a timeline, a linear representation of the timing of cash flows. For example, a $100 investment today that pays $30 at the end of year 1, $20 at the end of year 2, costs $10 at the end of year 3, and pays $50 at the end of year 4 would be represented as follows: Years Cash flow

0 –100

1 30

2 20

3 –10

4 50

2. The future value of an investment is the amount to which an investment will grow. Mathematically, the future value of an investment if compounded annually at a rate of r for n years will be FVn where n

=

PV (l + r)n

=

r PV

= =

the number of years during which the compounding occurs the annual interest (or discount) rate the present value or original amount invested at the beginning of the first period 5-1

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(5-1)

5-2  Keown/Martin/Petty

Instructor’s Solutions Manual

FVn

=

the future value of the investment at the end of n years

3. (1 + r)n is referred to as the future value factor. This is the value used as a multiplier to calculate an amount’s future value. 4. Simple interest is the interest earned only on the initial investment. Interest is not earned on any accumulated interest. 5. The future value of an investment can be increased either by increasing the number of years we let the investment compound or by compounding the investment at a higher rate. B. Techniques for Moving Money Through Time A time value of money problem can be solved using three different approaches. 1. Mathematical calculations—Be sure to use the correct formula. 2. Financial calculators a. For calculations with a financial calculator, cash outflows (money disbursed) are usually entered as a negative number. Cash inflows (money received) are usually entered as a positive number. b. Any variable with a value of zero must be specifically entered with a value of zero. Otherwise, be sure to clear the calculator before beginning a new problem. c. Enter the interest rate as a percent, not as a decimal. 3. Spreadsheets a. Many common financial calculations are preprogrammed. Follow the correct format for these formulas. Refer to the “Help” function when needed. b. Cash outflows are entered as a negative number. Cash inflows are entered as a positive number. c. The interest rate is entered as a decimal rather than as a percent in whole numbers, in contrast to a financial calculator. C. Two Additional Types of Time Value of Money Problems The future value and present value formula can also be used to determine the interest rate, r, or the number of periods, n, needed to accumulate a certain amount. To solve for these items, three of the four variables must be known, then the basic future value/present value formula can be used to solve for the fourth unknown variable. D. Applying Compounding to Things Other Than Money The basic future value/present value formula can be applied to any problem that involves compound growth, for example, predicting the total number of turtles in a nature preserve if the population grows at a certain rate per year.

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Foundations of Finance, Tenth Edition  5-3

E. Present Value 1. Present value is the value in today’s dollars of a sum of money to be received in the future. It involves nothing other than inverse compounding. The differences in these techniques are simply different points of view. 2. Mathematically, the present value of a sum of money to be received in the future can be determined by the following equation:

PV = FVn

 1   1  rn 

   

(5-2)

where n= the number of years until payment will be received, r = the annual interest rate or discount rate PV = the present value of the future sum of money FVn = the future value of the investment at the end of n years

  1    1  r n   is referred to as the present value factor. It is the value used as a 3.  multiplier to calculate an amount’s present value. 4. The present value of a future sum of money is inversely related to both the number of years until the payment will be received and the opportunity rate. II. Annuities An annuity is a series of equal dollar payments for a specified number of years. Because annuities occur frequently in finance—for example, bond interest payments—we treat them specially. A. Compound Annuities A compound annuity involves depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow. 1. This can be calculated by using our compounding equation and compounding each one of the individual deposits to the future, or by using the following compound annuity equation:

FVn

 (1 r )n  1 n  1 t 1 r         r   = PMT  PMT  t 0

=

where PMT

=

the annuity value deposited at the end of each year

r

=

the annual interest or discount rate

n

=

the number of years for which the annuity will last

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(5-3)

5-4  Keown/Martin/Petty

Instructor’s Solutions Manual =

FVn

the future value of the annuity at the end of the nth year

 (1 r )n  1    r  is referred to as the annuity value factor. It is used as a multiplier to 2.  calculate the future value of an annuity.

B. The Present Value of an Annuity Pension funds, insurance obligations, and interest received from bonds all involve annuities. To compare these financial instruments, we would like to know the present value of each of these annuities. 1. This can be done by using our present value equation and discounting each one of the individual cash flows back to the present or by using the following present value of an annuity equation:

= (5-4)

PV

  t PMT 

1    1  (1  r) n   n   1  r    1  1  r  t    = PMT

where PMT

=

the annuity withdrawn at the end of each year

r

=

the annual interest or discount rate

PV

=

the present value of the future annuity

n

=

the number of years for which the annuity will last

  1   1 (1  r) n    r     2. is referred to as the annuity present value factor. It is used as a multiplier to calculate the present value of an annuity. C. Annuities Due 1. An annuity due is an annuity in which the payments occur at the beginning of each period. Annuities due are just ordinary annuities in which all of the annuity payments have been shifted forward by one year. 2. The formula for calculating the future value of an annuity due is FVn (annuity due) = where

 (1  r)n  1    (1  r) r   PMT

(5-5)

PMT

=

the annuity deposited at the beginning of each year

r

=

the annual interest or discount rate

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Foundations of Finance, Tenth Edition  5-5 FVn

=

the future value of the annuity

n

=

the number of years for which the annuity will last

3. The formula for calculating the present value of an annuity due is

1    1  (1  r) n    r     (1 + r) = PMT

PV (annuity due) where

(5-6)

PMT

=

the annuity withdrawn at the beginning of each year

r

=

the annual interest or discount rate

PV

=

the present value of the annuity

n

=

the number of years for which the annuity will last

D. Amortized Loans This procedure of solving for PMT, the annuity value when r, n, and PV are known, is also the procedure used to determine what payments are associated with paying off a loan in equal installments. Loans paid off in this way, in periodic payments, are called amortized loans. Here again, we know three of the four values in the annuity equation and are solving for a value of PMT, the annual annuity. III. Making Interest Rates Comparable A. Interest rates on investments or loans may be compounded at different intervals. To assist borrowers when comparing interest rates, the U.S. Truth-in-Lending Act requires the calculation of the annual percentage rate (APR). The annual percentage rate (APR) is the interest rate that indicates the amount of interest earned in one year without compounding. 1. The formula for calculating the annual percentage rate is APR = (interest rate per period) × (compounding periods per year, m)

(5-7)

2. The APR is often referred to as the nominal or stated interest rate. 3. If the interest rates being compared are not compounded for the same number of periods per year, the APR may not be very helpful. To correctly compare interest rates with different compounding periods, we calculate the effective annual rate (EAR), the annual compound rate that produces the same return as the nominal (or quoted) rate when the interest rate is compounded on a nonannual basis. The EAR provides the true rate of return.

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Instructor’s Solutions Manual

4. The formula for calculating the effective annual rate is m

EAR

=

 quoted rate   1   1 m  

(5-8)

B. Calculating the Interest Rate and Converting It to an EAR 1. An annual percentage rate can be converted to a periodic rate with the following formula: Periodic rate = APR or quoted annual rate Compounding periods per year (m) 2. The APR can be calculated by multiplying the periodic rate by the number of compounded periods per year. C. Finding Present and Future Values with Nonannual Periods 1. The logic that applies to calculating the EAR also applies to calculating present and future values with nonannual periods. 2. The relevant equation is then

FVn = where FVn =

r  PV  l   m 

m n

(5-9)

the future value of the investment at the end of n years

n

=

the number of years during which the compounding occurs

r

=

APR

PV

=

the present value or original amount invested at the beginning of the first year

m

=

the number of times compounding occurs during the year

D. Amortized Loans with Monthly Compounding The present value/future value formula with nonannual periods can be used to calculate monthly mortgage payments. IV. The Present Value of an Uneven Stream and Perpetuities A. The present value of projects with uneven cash flows can be calculated by discounting or accumulating the individual cash flows by the appropriate interest rate and then summing these cash flows. 1. If some sequential payments are the same, the formula for the present value/future value of an annuity may be used. 2. Any payments that are not the same must be discounted individually and then summed with the other payments.

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Foundations of Finance, Tenth Edition  5-7 B. A perpetuity is an annuity that continues forever; that is, every year from now on, this investment pays the same dollar amount. An example of a perpetuity is preferred stock that yields a constant dollar dividend infinitely. C. The following equation can be used to determine the present value of a perpetuity: PV

=

PP r

where PV

=

the present value of the perpetuity

PP

=

the constant dollar amount provided by the perpetuity

r

=

the annual interest or discount rate

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Instructor’s Solutions Manual

ANSWERS TO END-OF-CHAPTER REVIEW QUESTIONS 5-1.

The concept of time value of money is a recognition that a dollar received today is worth more than a dollar received a year from now, or at any future date. It exists because there are investment opportunities on money; that is, we can place our dollar received today in a savings account, earn interest, and one year from now have more than a dollar.

5-2.

Compounding and discounting are inverse processes of each other. In compounding, money is moved forward in time, while in discounting, money is moved back in time. This can be shown mathematically in the compounding equation: FVn

=

PV (1 + r)n

We can derive the discounting equation by multiplying each side of this equation by 1

1

 r

n

and we get: 1

PV 5-3.

We know that FVn

=

=

FVn  1  r 

n

PV(1 + r)n

Thus, an increase in r will increase FVn, and a decrease in n will decrease FVn. 5-4.

Bank C, which compounds daily, pays the highest interest. This occurs because, while all banks pay the same interest, 5 percent, bank C compounds the 5 percent daily. Daily compounding allows interest to be earned more frequently than semiannual or annual compounding. Continuous compounding (not included with this question) allows interest to be earned more frequently than any other compounding period.

5-5.

An annuity is a series of equal dollar payments for a specified number of years. Examples of annuities include mortgage payments, interest payments on bonds, fixed lease payments, car loan payments, and any fixed contractual payment. A perpetuity is an annuity that continues forever; that is, every year from now on this investment pays the same dollar amount. The difference between an annuity and a perpetuity is that a perpetuity has no termination date, whereas an annuity does.

5-6.

This problem involves a comparison of three websites, all of which are very good. Answers will vary.

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Foundations of Finance, Tenth Edition  5-9

SOLUTIONS TO END-OF-CHAPTER STUDY PROBLEMS 5-1.

5-2.

a.

N I/Y CPT PV PMT FV

= = = = =

45 10 –68,596.06 dollars 0 $5,000,000

FVn

=

PV (1 + r)n

FV10

=

$5,000(1 + 0.10)10

FV

10 FV10

=

$5,000 (2.5937)

=

$12,969

N I/Y PV PMT CPT FV

= = = = =

10 10 –5,000 0 12,969 dollars

FVn

=

PV (1 + r)n

FV7

=

$8,000 (1 + 0.08)7

FV7

=

$8,000 (1.7138)

FV7

=

$13,711

N I/Y PV PMT CPT FV

= = = = =

7 8 –8,000 0 13,711 dollars

FVn FV 12 FV 12

=

PV (1 + r)n

=

$775 (1 + 0.12)12

=

$775 (3.896)

OR

b.

OR

c.

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5-10  Keown/Martin/Petty FV12

=

Instructor’s Solutions Manual $3,019

OR N = I/Y PV PMT CPT FV d.

= = = =

12 12 –775 0 3,019 dollars

FVn = PV (1 + r)n FV5

= $21,000 (1 + 0.05)5

FV5

= $21,000 (1.2763)

FV5

= $26,802

OR

5-3.

N I/Y PV PMT CPT FV

= = = = =

5 5 –21,000 0 26,802 dollars

a.

I/Y PV PMT FV CPT N

= = = = =

5 –500 0 1,039.50 15 years

b.

I/Y PV PMT FV CPT N

= = = = =

9 –35.0 0 53.87 5 years

c.

I/Y PV PMT FV CPT N

= = = = =

20 –100 0 298.60 6 years

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Foundations of Finance, Tenth Edition  5-11

5-4.

5-5.

d.

I/Y PV PMT FV CPT N

= = = = =

2 –53 0 78.76 20 years

a.

N = CPT I/Y = PV = PMT = FV =

12 12 percent –500 0 1,948

b.

N CPT I/Y PV PMT FV

= = = = =

7 5 percent –300 0 422.10

c.

N CPT I/Y PV PMT FV

= = = = =

20 9 percent –50 0 280.20

d.

N CPT I/Y PV PMT FV

= = = = =

5 20 percent –200 0 497.60

a.

N = I/Y CPT PV PMT FV

= = = =

10 10 –308 dollars 0 800

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5-6.

Instructor’s Solutions Manual

b.

N I/Y CPT PV PMT FV

= = = = =

5 5 –235 dollars 0 300

c.

N I/Y CPT PV PMT FV

= = = = =

8 3 –789 dollars 0 1,000

d.

N I/Y CPT PV PMT FV

= = = = =

8 20 –233 dollars 0 1,000

a.

FVn = PV (1 + r)n compounded for 1 year FV1

=

$10,000 (1 + 0.06)1

FV1

=

$10,000 (1.06)

FV1

=

$10,600

OR N I/Y PV PMT CPT FV

= = = = =

1 6 –10,000 0 10,600 dollars

compounded for 5 years FV5

=

$10,000 (1 + 0.06)5

FV5

=

$10,000 (1.3382)

FV5

=

$13,382

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Foundations of Finance, Tenth Edition  5-13 OR N I/Y PV PMT CPT FV

= = = = =

5 6 –10,000 0 13,382 dollars

compounded for 15 years FV15

=

$10,000 (1 + 0.06)15

FV15

=

$10,000 (2.3966)

FV15

=

$23,966

OR N I/Y PV PMT CPT FV b.

= = = = =

15 6 –10,000 0 23,966 dollars

FVn = PV (1 + r)n compounded for 1 year at 8% FV1

=

$10,000 (1 + 0.08)1

FV

=

$10,000 (1.080)

=

$10,800

1 FV1 OR N I/Y PV PMT CPT FV

= = = = =

1 ...