chapter 2

Chapter 2. The Time Value of Money

The "Chapter" worksheet performs the calculations required for Chapter 2, and was used to create many of the chapter exhibits (Tables and Figures). We pasted in a few dialog boxes for specific Excel functions and features and show then off to the right of where they apply, but in general we encourage students who want to know more about Excel to use the Excel Tutorial and refer to it as necessary. We also like to let students know that Excel models can be used to create tables and graphs that can then be copied into Word documents, which is the way we prepared the text manuscript for submission to the publisher. That procedure is used often in business to prepare reports.

Although answers to the Self-Test questions within the chapter are generally quite easy and were found with a calculator, we also solved them with Excel as a check and also to provide some information on the solutions in case students have questions. The tabs at the lower part of this screen take you to these solutions. Even if students are not familiar with Excel, they should still be able to see the solution setup and then work out the answer with a calculator. Although we did not create the model specifically for use in lectures, it could be used as such in a classroom where a projector is attached to a computer. The instructor could scroll through the model and lecture on points as they come up. This would be more useful if the students have some familiarity with Excel, but that is not really necessary because everything the model does can also be done with a financial calculator.

FUTURE VALUES (Section 2.2)

A dollar in hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest it, earn interest, and end up with more than a dollar in the future. The process of going to future values (FVs) from present values (PVs) is called compounding.

To illustrate, refer to our 3-year time line and assume that you plan to deposit $100 in a bank that pays a guaranteed 5% interest each year. How much would you have at the end of Year 3?

Figure 2-1. Summary of Future Value Calculations

-$100.005.00%

3

Periods: 0 1 2 3| | | |

Cash Flow Time Line: -$100 FV = ?

Step-by-Step Approach: $100 $105.00 $110.25 $115.76

= $115.76

3 5 -$100.00 $0

Calculator Approach: N I/YR PV PMT FV$115.76

Excel Approach: FV Function:

Fixed inputs: $115.76

Cell references: $115.76

The Compounding Process: A Graphic View

Investment = CF0 = PV =Interest rate = I =No. of periods = N =

Formula Approach: FVN = PV(1+I)N FV3 = $100(1.05)3

FVN = =FV(I,N,0,PV)

FVN = =FV(0.05,3,0,-100) =

FVN = =FV(C15,C16,0,C14) =

In the Excel formula, the terms are entered in this sequence: interest, periods, 0 to indicate no intermediate cash flows, and then the PV. The data can be entered as fixed numbers or as cell references.

Figure 2-2 (shown below) shows how a $1 investment grows over time at different interest rates. The curves were created by solving for FV at different values for N and I. This allows you to simultaneously see the effects of varying time and the interest rate.

The data table used to create this figure is shown to the right. For instruction on data tables, refer to the Excel Tutorial.

0 1 2 3 4 5 6 7 8 9 100.00

1.00

2.00

3.00

4.00

5.00

6.00

Periods

Future Value of $1

I = 20%

I = 0%

I = 5%

I = 10%

Figure 2-3. Summary of Present Value Calculations

$115.76Interest rate = I = 5.00%No. of periods = N = 3

Periods: 0 1 2 3| | | |

Cash Flow Time Line: PV = ? $115.76

Step-by-Step Approach: $100.00 $105.00 $110.25 $115.76

= $100.00

3 5 $0 $115.76Calculator Approach: N I/YR PV PMT FV

-$100.00

Excel Approach: PV Function:

Fixed inputs: -$100.00Cell references: -$100.00

PRESENT VALUES (Section 2.3)

In many ways the present value is just the opposite of the future value. Instead of compounding a value forward, you discount it back. If you know the PV, you can compound to find the FV, while if you know the FV, you can discount to find the PV.

To illustrate, refer to the time line and assume that $115.76 is due in 3 years. If a bank pays a guaranteed 5% interest rate each year, how much would you need to deposit now to have $115.76 in 3 years?

Future payment = CFN = FV =

Formula Approach: PV = FVN / (1 + I)N PV = $115.76/(1.05)3

PV = =PV(I,N,0,FV)

PV = =PV(0.05,3,0,115.76) =PV = =PV(C65,C66,0,C64) =

In the Excel function, 0 indicates that there are no intermediate cash flows.

The Discounting Process: A Graphic View

Present value (PV) -$100.00Future value (FV) $150.00No. of years (N) 10Interest rate (I) = RATE(N,0,PV,FV)Interest rate (I) 4.14%

Present value (PV) -$500,000Future value (FV) $1,000,000Interest rate (I) 4.50%No. of years (N) =NPER(I,0,PV,FV)No. of years (N) 15.75

Figure 2-4 shows how a $1 payment in the future has a lower and lower present value as the interest rate and time until receipt increase. The data table to the right provides the data used to draw the figure.

FINDING THE INTEREST RATE (Section 2.4)

Previously, we solved equations to find FV and PV. However, we could just as easily solve for I or N. For example, suppose we know that a given bond has a cost of $100 and that it will return $150 after 10 years. Thus, we know PV, FV, and N, and we want to find the rate of return we would earn if we bought the bond.

FINDING THE NUMBER OF YEARS (Section 2.5)

Sometimes we need to know how long it will take to accumulate a given sum of money, given our beginning funds and the rate we will earn on those funds. For example, suppose we believe that we could retire comfortably if we had $1 million, and we want to find how long it will take us to reach that goal, assuming that we now have $500,000 invested at 4.5%.

0 10 20 30 40 500.00

0.20

0.40

0.60

0.80

1.00

Periods

Present Value of $1

I = 0%

I = 20%

I = 10%

I = 5%

Figure 2-5. Summary: Future Value of an Ordinary Annuity

Payment amount = PMT = $100.00Interest rate = I = 5.00%

3

Periods: 0 1 2 3| | | |

Cash Flow Time Line: -$100 -$100 -$100

-$100.00Multiply each payment by -$105.00

-$110.25-$315.25

Formula Approach:

= $315.25

3 5 $0 -$100.00Calculator Approach: N I/YR PV PMT FV

$315.25

Excel Function Approach: FV Function:

Fixed inputs: $315.25 Cell references: $315.25

FUTURE VALUE OF AN ORDINARY ANNUITY (Section 2.7)

An ordinary annuity has regular, periodic payments that occur at the end of each period. Methods for solving the future value of an ordinary annuity are shown below.

Number of periods = N =

Step-By-Step Approach.

(1+I)N-t and sum these FVs tofind FVAN:

FVAN =

FVAN = =FV(I,N,PMT,PV)

FVAN = =FV(0.05,3,-100,0) =FVAN = =FV(C132,C133,-C131,0) =

PMT×((1+ I )N

I− 1

I )

Summary: Future Value of an Annuity Due (Not in Text)

Payment amount = PMT = $100.00Interest rate = I = 5.00%Number of periods = N = 3

Periods: 0 1 2 3| | | |


-$105.00Multiply each payment by -$110.25

-$115.76-$331.01

Formula Approach:

= $331.01

BEG MODE 3 5 0 -100Calculator Approach: N I PV PMT FV

331.01

Excel Function Approach: FV Function: Fixed inputs: 331.01 Cell references: 331.01

FUTURE VALUE OF AN ANNUITY DUE (Section 2.8)

An annuity due also has regular, periodic payments, but unlike an ordinary annuity, the payments occur at the beginning of each period.


(1+I)N-t and sum these FVs tofind FVAN:

FVAN(due) =

FVAN = =FV(I,N,PMT,PV,Type)FVAN = =FV(0.05,3,-100,0,1) =FVAN = =FV(C163,C164,-C162,0,1) =

In the Excel formula, the 1 at the end of the formula indicates that cash flows occur at the beginning of each period. A 0 or nothing would indicate end of period payments.

PMT×((1+ I )N

I− 1

I ) (1+ I )

Figure 2-6. Summary: Present Value of an Ordinary Annuity


Periods: 0 1 2 3| | | |


$95.24Divide each payment by $90.70

$86.38$272.32

Formula Approach:

= $272.32

3 5 -100 0Calculator Approach: N I PV PMT FV

272.32

Excel Function Approach: PV Function: Fixed inputs: 272.32 Cell references: 272.32

PRESENT VALUE OF AN ORDINARY ANNUITY (Section 2.9)

The present value of an ordinary annuity is the sum of the PVs of the individual cash flows. Methods for solving the present value of an ordinary annuity are shown below.


(1+I)t and sum these PVs tofind PVAN:

PVAN =

PVAN = =PV(I,N,PMT,FV)PVAN = =PV(0.05,3,-100,0) =PVAN = =PV(C197,C198,-C196,0) =

PMT×(1I

- 1

I (1+I )N )

Summary: Present Value of an Annuity Due (Not in text)


Periods: 0 1 2 3| | | |


-$100.00Divide each payment by -$95.24

-$90.70-$285.94

Formula Approach:

= $285.94

BEG MODE 3 5 -100 0Calculator Approach: N I PV PMT FV

285.94

Excel Function Approach: PV Function: Fixed inputs: 285.94 Cell references: 285.94

FINDING PMT

No. of years (N) 5Interest rate (I) 6%Present value (PV) $0Future value (FV) $10,000

END MODE BEGIN MODEPayment (PMT) -$1,773.96 Payment (PMT) -$1,673.55

=PMT(I,N,PV,FV) =PMT(I,N,PV,FV,Type)

PRESENT VALUE OF AN ANNUITY DUE (not in text)

The difference between the present value of an ordinary annuity and an annuity due is that payments are received earlier in an annuity due.


(1+I)t and sum these PVs tofind PVAN:

PVAN =

PVAN = =PV(I,N,PMT,FV,Type)PVAN = =PV(0.05,3,-100,0,1) =PVAN = =PV(C229,C230,-C228,0,1) =

FINDING ANNUITY PAYMENTS, PERIODS, AND INTEREST RATES (Section 2.10)

Fundamentally, this section is no different than previous TVM exercises. When solving for PMT, N, or I, you must be given values for the other variables, and then you solve the problem.

Suppose we need to accumulate $10,000 and have it available 5 years from now. Suppose further that we can earn a return of 6% on our savings, which are currently zero.

PMT×(1I

- 1

I (1+ I )N ) (1+ I )

FINDING N

Interest rate (I) 6%Present value (PV) $0Payment (PMT) -$1,200Future value (FV) $10,000

No. of years (N) 6.96=NPER(I,PMT,PV,FV)

FINDING I

No. of years (N) 5Present value (PV) $0Payment (PMT) -$1,200Future value (FV) $10,000

Interest rate (I) 25.78%=RATE(N,PMT,PV,FV)

Payment (PMT) $25 Interest rate (I) 5.2%

Interest rate (I) $480.77

Suppose you decide to make end-of-year deposits, but you can only save $1,200 per year. Again assuming that you would earn 6%, how long would it take you to reach your $10,000 goal?

Now suppose you can only save $1,200 annually, but you still want to have the $10,000 in 5 years. What rate of return would enable you to achieve your goal?

PERPETUITIES (Section 2.11)

Perpetuities are securities that promise to make payments forever. The present value of a perpetuity can be found with a simple formula: Value = I / r . Note that we cannot calculate the future value of a perpetuity because, since payments go on forever, this value would be infinitely large and thus meaningless.

Consider a British consol that pays a $25 annual payment. If interest rates are currently 5.2%, what is the value of the consol?

If an annuity makes constant payments, then adding more payments to the security adds less value for each additional payment. This helps explain why perpetuities' values are finite, while payments are infinite. To see this better, consider the figure below (not in the text). The data used to construct the graph is shown to the right in columns I through L. One hundred payments are analyzed and their present values, the total value of an annuity of N number of years, and the contribution of the Nth payment are all shown in the table.

Summary of Uneven Cash Flow Present Value Calculations (Annuity plus Lump Sum)

Interest rate = I = 12%

Periods: 0 1 2 3 4 5| | | | | |

Annuity CFs: $0 $100 $100 $100 $100 $100Lump sum CFs: $1,000

Total CFs: $0 $100 $100 $100 $100 $1,100PV of CFs

$89.2979.7271.1863.55

624.17$927.90 = PV of cash flow stream = value of the asset

UNEVEN CASH FLOWS (Section 2.12)

An annuity has constant payments. Although many financial decisions do involve annuities, many others involve uneven, or nonconstant, cash flows. With a spreadsheet, the present value of a series of uneven cash flows (called the net present value) can be calculated easily .

First, consider a security that pays $100 for 5 years and a lump sum of $1,000 at the end of 5 years. We can find the PV using either the PV function or the NPV function.

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 73 77 81 85 89 93 97

$0

$50

$100

Years (N)

PV of Additional Payments in an

Annuity

Value of 25-Year Annuity: $907.70Value of 50-Year Annuity: $991.48Value of 100-Year Annuity: $999.93Value of Perpetuity: $1,000.00

Amt. Added: Years 1-25: $907.70 26-50: $83.78 51-100: $8.45 From 101 to infinity: $0.07

Excel Function Approach: Fixed inputs: =PV(0.12,5,-100,-1000) 927.90

Cell references =PV(B335,G337,-C339,-G340) 927.90

Fixed inputs: =NPV(0.12,100,100,100,100,1100) 927.90 Cell references: NPV = =NPV(B335,C341:G341) 927.90

Now consider an irregular cash flow stream (where CFs can take on any value).

Figure 2-7. PV of an Uneven Cash Flow Stream


Periods: 0 1 2 3 4 5| | | | | |

CF Time Line: $0 $100 $300 $300 $300 $500PV of CFs

$89.29239.16213.53190.66283.71

$1,016.35 = PV of cash flow stream = value of the asset

Excel Function Approach: Fixed inputs: =NPV(0.12,100,300,300,300,500) 1,016.35 Cell references: NPV = =NPV(B358,C362:G362) 1,016.35

Figure 2-8. FV of an Uneven Cash Flow Stream


Periods: 0 1 2 3 4 5| | | | | |

CF Time Line: $0 $100 $300 $300 $300 $500$500.00336.00376.32421.48

PV =

PV =

NPV =

NPV =

Our Excel formula ignores the initial cash flow (in Year 0). When entering a cash flow range, Excel assumes that the first value entered occurs at the end of the first year. If there is an initial cash flow, as we will see later, that cash flow must be separately added to the NPV formula result. Notice too that you can enter cash flows one-by-one, or if the cash flows appear in consecutive cells, you can enter the cell range.

FUTURE VALUE OF AN UNEVEN CASH FLOW STREAM (Section 2.13)

We find the future value of uneven cash flow streams by compounding rather than discounting. The step-by-step approach works the same, but unfortunately, Excel does not have a net future value (NFV) function. One way around this is to solve for the NPV and find the FV of this amount to the end of the cash flow stream.

157.350.00

FV of cash flow stream = $1,791.15

Excel Function Approach: First find NPV: =NPV(B383,C387:G387) 1,016.35Then compound NPV for 5 years: NFV = =FV(B383,G385,0,-G397) 1,791.15

NPV =

Finding the Interest Rate, Annuity Plus Lump Sum

Annuity pmts $100 Future lump sum $1,000

Periods: 0 1 2 3 4 5| | | | | |

CF Time Line: -$927.90 $100 $100 $100 $100 $1,100

Excel Function Approach: Cell references: RATE = =RATE(G411,B408,B413,B409) 12.00%Excel Function Approach: Cell references: =IRR(B413:G413) 12.00%

Figure 2-9. IRR of an Uneven Cash Flow Stream

Periods: 0 1 2 3 4 5| | | | | |

CF Time Line: -$1,000 $100 $300 $300 $300 $500

Excel Function Approach: Cell references: =IRR(B423:G423) 12.55%

Nominal annual rate = 10%Amount invested = $100 Number of years = 5

Table 2-1. The Impact of Frequent Compounding

Future ValueAnnual 10% 1 10.00% 10.000% $161.05 Semiannual 10% 2 5.00% 10.250% $162.89 Quarterly 10% 4 2.50% 10.381% $163.86 Monthly 10% 12 0.83% 10.471% $164.53

The NFV result using the Excel formulas is a negative number. This is because we used Excel's FV function and entered the NPV as a positive value as the PV.

SOLVING FOR I WITH UNEVEN CASH FLOWS (Section 2.14)

Assume that an investment with the following positive cash flows has a cost of $927.90. Find the rate of return on this investment.

IRR =

IRR =

SEMIANNUAL AND OTHER COMPOUNDING PERIODS (Section 2.15)

If $100 is invested in an account at an annual nominal interest rate of 10% for 5 years, what are the future values and effective interest rates for annual, semiannual, quarterly, monthly and daily compounding?

Frequency of Compounding

Nominal Annual Rate

Number of periods

per year (M)Periodic

Interest RateEffective

Annual Rate

D438

The periodic rate is the nominal rate divided by the number of periods.

E438

The effective annual rate is the rate such that $1 invested for 1 year with annual compounding at the effective rate grows to the same value as $1 invested for 1 year with multiple compounding. It is defined as: EAR (or EFF%) = (1 + INOM/M)M -1, where M is the number of periods of compounding per year.

F438

The future value is found by compounding the periodic rate for M*N periods, where M is the number of periods per year and N is the number of years.

Daily 10% 365 0.03% 10.516% $164.86

AMORTIZED LOANS (Section 2.17)

If a loan is to be repaid in equal amounts on a monthly, quarterly, or annual basis it is said to be an amortized loan.

Table 2-4 (replicated below) illustrates the amortization process. A homeowner borrows $100,000 on a mortgage loan, and the loan is to be repaid in 5 equal payments at the end of each of the next 5 years. The lender charges 6% on the balance at the beginning of each year.

First, we solve for the required payment, then we construct an amortization table.

N 5I 6%

PV $100,000FV $0

PMT -$23,739.64

Table 2-2. Loan Amortization Schedule, $100,000 at 6% for 5 YearsAmount borrowed: $100,000

Years: 5Rate: 6%PMT: $23,739.64

Year1 $100,000.00 $23,739.64 $6,000.00 $17,739.64 $82,260.362 $82,260.36 $23,739.64 $4,935.62 $18,804.02 $63,456.343 $63,456.34 $23,739.64 $3,807.38 $19,932.26 $43,524.084 $43,524.08 $23,739.64 $2,611.44 $21,128.20 $22,395.895 $22,395.89 $23,739.64 $1,343.75 $22,395.89 $0.00

Beginning Amount

(1)Payment

(2)Interesta

(3)

Repayment of Principalb

(2) - (3) = (4)

Ending Balance

(1) - (4) = (5)

a Interest in each period is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefore, interest in Year 1 is $100,000(0.06) = $6,000; in Year 2 it is $82,260.36(0.06) = $4,935.62; and so on.

b Repayment of principal is equal to the payment of $23,739.64 minus the interest charge for the year.

Growing Annuities (Section 2.18)

SECTION 2.2SOLUTIONS TO SELF-TEST

2a What would the future value of $100 be after 5 years at 10% compound interest?

N 5I 10%

PV $100PMT $0 FV $161.05

N 3I 4%

PV $2,000PMT $0 FV $2,249.73

3b How would your answer change if the interest rate were 5%, or 6%, or 20%?

Interest rate $2,249.73 5% $2,315.25 6% $2,382.03

20% $3,456.00

N 10I 8%

PV ($M) $100PMT $0 FV ($M) $215.89

5a How much would $1, growing at 5% per year, be worth after 100 years?

N 100I 5%

PV $1PMT $0 FV $131.50

5b What would FV be if the growth rate were 10%?

N 100I 10%

PV $1PMT $0 FV $13,780.61

3a Suppose you currently have $2,000 and plan to purchase a 3-year certificate of deposit (CD) that pays 4 percent interest compounded annually. How much will you have when the CD matures?

4 A company’s sales in 2006 were $100 million. If sales grow at 8 percent, what will they be 10 years later, in 2016?


N 3I 4%

PMT $0FV $2,250 PV $2,000.00

3b How would your answer change if the bond matured in 5 rather than 3 years?

N 5I 4%

PMT $0FV $2,250 PV $1,849.11

3c What if the interest rate on the 5-year bond were 6% rather than 4%?

N 5I 6%

PMT $0FV $2,250 PV $1,681.13

4a How much would $1,000,000 due in 100 years be worth today if the discount rate were 5%?

N 100I 5%

PMT $0FV $1,000,000 PV $7,604.49

4b If the discount rate were 20%?

N 100I 20%

PMT $0FV $1,000,000 PV $0.0121

3a Suppose a U.S. government bond promises to pay $2,249.73 three years from now. If the going interest rate on 3-year government bonds is 4%, how much is the bond worth today?


N 10PMT $0PV $585.43FV $1,000 I 5.50%

1b What rate would you earn if you could buy the bond for $550?

N 10PMT $0PV $550.00FV $1,000 I 6.16%

1c For $600?

N 10PMT $0PV $600.00FV $1,000 I 5.24%

N 10PMT $0PV $0.12FV $1.04 I 24.10%

2b If EPS in 2004 had been $0.65 rather than $1.04, what would the growth rate have been?

N 10PMT $0PV $0.12FV $1 I 18.41%

1a The U.S. Treasury offers to sell you a bond for $585.43. No payments will be made until the bond matures 10 years from now, at which time it will be redeemed for $1,000. What interest rate would you earn if you bought this bond for $585.43?

2a Microsoft earned $0.12 per share in 1994. Ten years later, in 2004, it earned $1.04. What was the growth rate in Microsoft’s earnings per share (EPS) over the 10-year period?


1a How long would it take $1,000 to double if it were invested in a bank that pays 6% per year?

I 6%PMT $0PV $1,000FV $2,000 N 11.90

1b How long would it take if the rate were 10%?

I 10%PMT $0PV $1,000FV $2,000 N 7.27

I 24.1%PMT $0PV $1.04FV $2.08 N 3.21

2a Microsoft’s 2004 earnings per share were $1.04, and its growth rate during the prior 10 years was 24.1% per year. If that growth rate were maintained, how long would it take for Microsoft’s EPS to double?


N 5I 10% Years of int 4

PMT -$100PV $0 Payments FV $146.41

1b Answer this same question for the 5th payment.

N 5I 10% Years of int 0

PMT -$100PV $0 Payments FV $100.00

N 5I 4%

PMT -$2,500PV $0 FV $13,540.81

2b How would your answer change if the interest rate were increased to 6%, or lowered to 3%?

N 5I 6%

PMT -$2,500PV $0 FV $14,092.73

N 5I 3%

PMT -$2,500PV $0 FV $13,272.84

1a For an ordinary annuity with 5 annual payments of $100 and a 10% interest rate, how many years will the 1st payment earn interest, and what will this payment’s value be at the end?

2a Assume that you plan to buy a condo 5 years from now, and you estimate that you can save $2,500 per year. You plan to deposit the money in a bank that pays 4% interest, and you will make the first deposit at the end of the year. How much will you have after 5 years?


N 5I 4%

PV $0PMT -$2,500 FV $14,082.44

2b How much would you have if you made the deposits at the end of each year?

N 5I 4%

PV $0PMT -$2,500 FV $13,540.81

3a Assume that you plan to buy a condo 5 years from now, and you need to save for a down payment. You plan to save $2,500 per year, with the first payment made immediately, and you will deposit the funds in a bank account that pays 4%. How much will you have after 5 years?


N 10I 10%

PMT -$100FV $0 PV $614.46

3b What would PVA be if the interest rate were 4%?

N 10I 4%

PMT -$100FV $0 PV $811.09

3c What if the interest rate were 0%?

N 10I 0%

PMT -$100FV $0 PV $1,000.00

3d How would the PVA values differ if we were dealing with annuities due?

Part a Part b Part cN 10 N 10 N 10I 10% I 4% I 0%

PMT -$100 PMT -$100 PMT -$100FV $0 FV $0 FV $0PV $675.90 PV $843.53 PV $1,000.00

N 10I 8%

PMT -$100FV $0 PV $671.01

4b If the payments began immediately, how much would the annuity be worth?

N 10I 8%

PMT -$100FV $0 PV $724.69

3a What is the PVA of an ordinary annuity with 10 payments of $100 if the appropriate interest rate is 10%?

4a Assume that you are offered an annuity that pays $100 at the end of each year for 10 years. You could earn 8% on your money in other investments with equal risk. What is the most you should pay for the annuity?


N 10I 7%

PV $100,000FV $0 PMT -$14,237.75

1b How would your answer change if you made withdrawals at the beginning of each year?

N 10I 7%

PV $100,000FV $0 PMT -$13,306.31

I 7.0%PV $100,000

PMT -$10,000FV $0 N 17.8

2b How long would they last if you earned 0%?

I 0.0%PV $100,000

PMT -$10,000FV $0 N 10.0

2c How long would they last if you earned the 7% but limited your withdrawal to $7,000 per year?

I 7.0%PV $100,000

PMT -$7,000FV $0 N #NUM!

N 12PMT $12,000PV $100,000FV $0 I 6.11%

1a Suppose you inherited $100,000 and invested it at 7% per year. How much could you withdraw at the end of each of the next 10 years?

2a If you had $100,000 that was invested at 7% and you wanted to withdraw $10,000 at the end of each year, how long would your funds last?

* This result means that with $7,000 withdrawals, you would never exhaust the funds.

3 Your rich uncle named you as the beneficiary of his life insurance policy. The insurance company gives you a choice of $100,000 today or a 12-year annuity of $12,000 at the end of each year. What rate of return is the insurance company offering?

N 10PMT -$10,000PV $60,000FV $0 I 13.70%

4b If you think a “fair” return would be 6%, how much should you ask for the annuity?

N 10I 6%

PMT -$10,000FV $0 PV $78,016.92

4a Assume that you just inherited an annuity that will pay you $10,000 per year for 10 years, with the first payment being made today. A friend of your mother offers to give you $60,000 for the annuity. If you sell it, what rate of return would your mother’s friend earn on his investment?


PMT $1,000I 5% PV $20,000

1b What would the value be if the annuity began its payments immediately?

PMT $1,000I 5% PV $21,000

1a What’s the present value of a perpetuity that pays $1,000 per year, beginning one year from now, if the appropriate interest rate is 5%?

**The perpetuity value formula values payments 1 through infinity. If a payment is received immediately, it must be added to the formula result.


Interest rate 6%

Year 0 1 2 3 4 5Ann Pmt $0 $100 $100 $100 $100 $100

Lump Sum $500Total CFs $0 $100 $100 $100 $100 $600

NPV $794.87

Interest rate 6%

Year 0 1 2 3 4 5 6 7 8 9 10Ann Pmt $0 $100 $100 $100 $100 $100 $100 $100 $100 $100 $100

Lump Sum $500Total CFs $0 $100 $100 $100 $100 $100 $100 $100 $100 $100 $600

NPV $1,015.21

Interest rate 8%

Year 0 1 2 3 4CFs $0 $100 $200 $0 $400

2a What’s the present value of a 5-year ordinary annuity of $100 plus an additional $500 at the end of Year 5 if the interest rate is 6%?

2b How would the PV change if the $100 payments occurred in Years 1 through 10 and the $500 came at the end of Year 10?

3a What’s the present value of the following uneven cash flow stream: $0 at Time 0, $100 in Year 1 (or at Time 1), $200 in Year 2, $0 in Year 3, and $400 in Year 4 if the interest rate is 8%?

NPV $558.07


Interest rate 15%

Year 0 1 2 3CFs $0 $100 $150 $300

FV of CFs $0.00 $132.25 $172.50 $300.00

NFV $604.75

3a What is the future value of this cash flow stream: $100 at the end of 1 year, $150 due after 2 years, and $300 due after 3 years if the appropriate interest rate is 15%?


Interest rate 6%

Year 0 1 2 3 4Ann Pmt -$465 $100 $100 $100 $100

Lump Sum $200Total CFs -$465 $100 $100 $100 $300

IRR 9.05%

Year 0 1 2 3CFs -$465 $100 $200 $300

IRR 11.71%

1 An investment costs $465 and is expected to produce cash flows of $100 at the end of each of the next 4 years, then an extra lump sum payment of $200 at the end of the 4th year. What is the expected rate of return on this investment?

2 An investment costs $465 and is expected to produce cash flows of $100 at the end Year 1, $200 at the end or Year 2, and $300 at the end of Year 3. What is the expected rate of return on this investment?


N 3I 8%

PV -$100PMT $0 FV $125.97

2b Compounded monthly?

N 36I 0.67%

PV -$100PMT $0 FV $127.02

N 3I 8%

PMT $0FV $100 PV $79.38

3b Compounded monthly?

N 36I 1%

PMT $0FV $100 PV $78.73

Nominal rate 18%Comp/year 12

Effective rate 19.56%

2a What’s the future value of $100 after 3 years if the appropriate interest rate is 8%, compounded annually?

3a What’s the present value of $100 due in 3 years if the appropriate interest rate is 8%, compounded annually?

6 Credit card issuers must by law print their annual percentage rate (APR) on their monthly statements. A common APR is 18%, with interest paid monthly. What is the EFF% on such a loan?


Loan $1,000,000 Interest rate 9%Days/year 360Interest pd (days) 30

Interest paid $7,500

1b What would the interest be if the bank used a 365-day year?

Loan $1,000,000 Interest rate 9%Days/year 365Interest pd (days) 30

Interest paid $7,397.26

Loan $1,000 Interest rate 7%Comp/year 365 Time period (months) 7

Effective rate 7.250098% Account value $1,041.67

1a Suppose a company borrowed $1 million at a rate of 9%, simple interest, with interest paid at the end of each month. The bank uses a 360-day year. How much interest would the firm have to pay in a 30-day month?

2a Suppose you deposited $1,000 in a credit union that pays 7% with daily compounding and a 365-day year. What is the EFF%, and how much could you withdraw after 7/12 of a year?


N 3I 8%

PV $30,000FV $0

PMT -$11,641.01

Loan Amortization Schedule, $30,000 at 8% for 3 YearsAmount borrowed: $30,000 Years: 3Rate: 8%PMT: -$11,641.01

Year1 $30,000.00 $11,641.01 $2,400.00 $9,241.01 $20,758.992 $20,758.99 $11,641.01 $1,660.72 $9,980.29 $10,778.713 $10,778.71 $11,641.01 $862.30 $10,778.71 $0.00

Rather than focus on Year 1 data, it was easier to just construct a full amortization schedule.

1 Suppose you borrowed $30,000 on a student loan at a rate of 8% and now must repay it in 3 equal installments at the end of each of the next 3 years. How large would your payments be, how much of the first payment would represent interest, how much would be principal, and what would your ending balance be after the first year?

Beginning Amount (1)

Payment (2)

Interest (3)

Repayment of Principal

(4)Ending

Balance (5)


10%Inflation 5%

4.7619%

3 If the nominal interest rate is 10% and the expected inflation rate is 5%, what is the expected real rate of return?

rNOM

rr

3 If the nominal interest rate is 10% and the expected inflation rate is 5%, what is the expected real rate of return?

chapter 2

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