Fin3400 - Chapter 5

Front 1

Describe the two methods of calculating future values for multiple cash flows.

Back 1

1) Compound the accumulated balance forward one year at a time.

2) Calculate the future value of each cash flow first and then add these up.

Front 2

You open a savings account that earns 7% interest compounded annually. You deposit $100 at the end of year 1, $200 at the end of year 2, and $300 at the end of year 3. How much will you have at the end of year 3? How much will have after 5 years?

Back 2

You will have $628.49 at the end of three years.

FV= [100*(1.07^2)]+(200*1.07)+300= 628.49

You will have $719.56 at the end of five years.

FV= [100*(1.07^4)]+[200*(1.07^3]+[300*(1.07^2)]= 719.558201

Front 3

You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year, and $800 at the end of the next year. You can earn 12% on very similar investments. What is the most you should pay for this one?

Back 3

You should pay no more than $1,432.93.

APV = [200*(1/1.12^1)]+[400*(1/1.12^2)]+[600*(1/1.12^3)]+[800*(1/12^4)]=1432.931591

Front 4

A stream of cash flows for a fixed period of time is called an _________.

Back 4

annuity

Front 5

ANNUITY PRESENT VALUE (APV) FACTOR EQUATION

Back 5

APVF = C*{1-[1/(1+r)^t]}/r

APV = C*(1-PVIF)/r = C*PVIFA

Present value of C per period for t periods at r percent per period.

Front 6

PRESENT INTEREST VALUE FACTOR FOR ANNUITIES (PVIFA) EQUATION

Back 6

PVIFA = {1-[1/(1+r)^t]}/r

Front 7

After carefully going over your budget, you have determined you can afford to pay $632 per month toward a new sports car. You call up your local bank and find out that the going rate is 1% per month for 48 months. How much can you borrow?

Back 7

You can borrow up to $24,000.

APVF = [1 - (1/1.01^48)]/.01 = 37.9740

AFV = PMT*APVF = 632*37.9740 = 23999.5424

Front 8

Suppose you wish to start up a new business that specializes in the latest of health food trends, frozen yak milk. To produce and market your product, you need to borrow $100,000. You propose to pay off the loan quickly by making five equal annual payments. If the interest rate is 18$, what will the payments be?

Back 8

The payments on this loan will be $31,977.80.

100000 = [C*(1-(1/1.18^5))]/0.18

100000 = C*3.12717021

C = 100000/3.12717021=31977.79247

Front 9

You ran a little short on your vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5% per month. How long will you need to pay off the $1,000?

Back 9

It will take you approximately 93.11 months, or 7.76 years.

Use APV = [C*(1-PVF)]/r. First solve for PVF:

1000 = 20*(1-PVF)/0.015

(1000/20)*0.015 = 1 - PVF

PVF =1-[(1000/20)*0.015]=1-(5*0.015)= 0.25

Then solve for t:

1.015^t = 1/0.25 = 4

t = ln(4)/ln(1.015) = 93.11105126

Front 10

ANNUNITY FUTURE VALUE FACTOR (AFVF) EQUATION

Back 10

AFVF = (FVF - 1)/r = [(1+r)^t - 1]/r

Future value of C invested per period for t periods at r per period.

Front 11

Suppose you plan to contribute $2,000 every year into a retirement account paying 8%. If you retire in 30 years, how much will you have?

Back 11

You will have $226,566.42 in your retirement account.

AFVF =(1.08^30-1)/0.08=9.062656...= 113.2832

AFV = 2000*113.2832111 = 226566.4222

Since we know t = 30 years and r = 8%, we can use the equation AFV = C*AFVF. First solve for AFVF, then solve for AFV.

Front 12

An annuity for which the cash flows occur at the beginning of the period, such as a lease, is called a(n) _____________.

Back 12

annuity due

ADV = ordinary annuity value*(1+r)

Front 13

ANNUITY DUE CALCULATOR TIP:

Back 13

To calculate an annuity due, you must put the calculator in "due mode." On the hp 12c, this is done by entering g BEG. To reset it, press g END.

Front 14

ANNUITY DUE VALUE (ADV) EQUATION

Back 14

ADV = ordinary annuity value x (1+r)

Front 15

An annuity in which the cash flows continue forever is called a(n) ___________.

Back 15

perpetuity

Front 16

Another name for a perpetuity, particularly in Canada and the UK, is a(n) _________.

Back 16

consol

Front 17

PERPETUITY PRESENT VALUE (PPV) EQUATION

Back 17

PPV = C/r

Present value of a perpetuity of C per period.

Front 18

When a corporation sells _________, the buyer is promised a fixed cash dividend every period forever. This dividend must be paid before any dividend can be paid to regular stockholders.

Back 18

preferred stock

Front 19

Suppose the Fellini Co., wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell?

Back 19

Fellini will have to offer a dividend of $2.50 per quarter.

40 = 1*(1/r)

1/r = 40/1 = 40

r = 1/40 = 0.025 = 2.5%

100 = C*(1/0.025)

C = 100/40 = 2.5 = $2.50

Front 20

Is 5% every six months the same as 10% per year?

Back 20

No. If you invest $100 at 10% per year, you will have $110.00 at the end of the year. If you invest that dollar at 5% every six months, then you'll have $110.25 at the end of the year.

Front 21

The interest rate expressed in terms of the interest payment made each period is called the __________ interest rate.

Back 21

quoted (or stated)

Front 22

The interest rate expressed as if it were compounded once per year is called the _________________.

Back 22

effective annual rate (EAR)

Front 23

EFFECTIVE ANNUAL RATE (EAR) EQUATION

Back 23

EAR = (1 + quoted rate/m)^m - 1

Front 24

Suppose you've shopped around and come up with the following three rates:

• Bank A: 15% compounded daily
  
• Bank B: 15.5% compounded quarterly
  
• Bank C: 16% compounded annually

Which of these is the best if you are thinking of opening a savings account? Which is the best if they represent loan rates?

Back 24

If you are going to open a savings account, Bank B is the best choice; however, if you are looking for a loan, Bank C is best.

Bank A =[(1+0.15/365)^365]-1=0.1618= 16.18%

Bank B =[(1+0.155/4)^4]-1=0.164244..= 16.42%

Bank C =[(1+0.16/1)^1]-1=0.16= 16%

Front 25

Anytime we do a present or future value calculation, the rate we use must be an _________ or _________ rate.

Back 25

actual, effective

Front 26

As a lender, you know you want to actually earn 18% on a particular loan. You want to quote a rate that features monthly compounding. What rate do you quote?

Back 26

You would quote a rate of 16.67% compounded monthly.

EAR = 18%, m = 12

0.18 = (1+q/12)^12 = - 1

1.18^(1/12) = 1 + q/12

q = [1.18^(1/12) - 1]*12=0.1666611642= 16.67%

Front 27

The interest rate charged per period multiplied by the number of periods per year is called the __________________.

Back 27

annual percentage rate (APR)

APR = PV*(1+r)^t

Front 28

True OR False:

By law, the APR is required to be calculated by multiplying the interest rate per period by the number of periods in a year.

Back 28

True. The APR is, in fact, a quoted (or stated) rate. The actual rate can be found by calculating the EAR.

Front 29

A typical credit card agreement quotes an interest rate of 18% APR. Monthly payments are required. What is the actual interest rate you pay on such a credit card?

Back 29

The interest rate you actually pay is 19.56%.

EAR =(1+0.18/12)^12-1=0.1956181715= 19.56%

Front 30

True OR False:

There can be a huge difference between the APR and EAR when interest rates are large.

Back 30

True. For example, consider a payday loan for $100 with a payment of $115 promised 14 days in the future. The interest rate on the loan is 15%; however, because the loan is so short, the APR is 391.07% and the EAR is 3,723.66%.

r= [(115/100)^(1/1)]-1 = 0.15 = 15%

APR = 0.15*365/14 = 3.9107 = 391.07%

EAR = (1+0.15)^(365/14)-1=37.2366...= 37.24%

Front 31

EAR/APR SPREADSHEET CALCULATION TIPS:

Back 31

1) To convert a quoted rate (or an APR) to an effective rate, use the formula EFFECT(nominal_rate, npery).

2) To convert an EAR to a quoted rate, use NOMINAL(effect_rate, npery).

Front 32

The three (3) basic types of loans are:

Back 32

pure discount loans
interest-only loans
amortized loans

Front 33

The ____________ is the simplest type of loan. The borrower repays the loan in a single lump sum at some time in the future. Treasury bills are one example.

Back 33

pure discount loan

PV = FV/(1+r)^t

Front 34

When the U.S. government borrows money on a short-term basis (a year or less), it does so by selling what are called ____________.

Back 34

Treasury bills (T-bills)

Front 35

If a T-bill promises to repay $10,000 in 12 months, and the market interest rate 7%, how much will the bill sell for in the market?

Back 35

The market price of this T-bill is $9,345.79.

PV = 10000/1.07 = 9345.794393 = $9,345.79

Front 36

A(n) ___________ loan has a repayment plan that calls for the borrower to pay interest each period and to repay the entire principal (the original loan amount) at some point in the future. Most corporate bonds have the general form of this type of loan.

Back 36

interest-only

Front 37

With a(n) ____________ loan, the lender may require the borrower to repay parts of the loan over time. Most mortgage loans fall into this category.

Back 37

amortized

Front 38

The process of paying off a loan by making regular principal reductions is called _____________ the loan.

Back 38

amortizing

Front 39

A series of constant cash flows that arrive or are paid at the end of the each period is called a(n)

_______________.

Back 39

ordinary annuity

Front 40

True OR False:

Interest rates can be quoted in a variety of ways. For financial decisions, it is important that any rates being compared by first converted to effective rates.

Back 40

True. In order to make the most informed decisions possible, all quoted (or stated) rates should be converted to actual/effective rates.

Front 41

The relationship between a quoted rate, such as an APR, and an effective rate, such as an EAR, is given by the equation:

Back 41

EAR = (1+quoted rate/m)^m - 1

Front 42

A quarterback has been signed to a three-year, $10 million dollar contract. The details provide for an immediate cash bonus of $1 million. The player is to receive $2 million in salary at the end of the first year, $3 million the next, and $4 million at the end of the last year. Assume a 10% discount rate. Is this package worth $10 million? How much is it worth?

Back 42

The package is not worth $10 million because the payments are spread out over 3 years. The bonus is paid today, so it's worth 1$ million. The package is worth a total of $8.3028 million.

1+2/1.1+3/1.1^2+4/1.1^3 = 8.302779865

Front 43

You plan to make a series of deposits in an interest-bearing account. You will deposit $1000 today, $2000 in two years, and $8000 in five years. If you withdraw $3,000 in three years and $5000 in seven years, how much will you have after eight years if the interest is 9%? What is the present value of these cash flows?

Back 43

The present value of these cash flows is $2,831.09.

First calculate the individual future values and add them together to get the total future value.

FV1 =1000*1.09^8=1992.6526...= $1,992.56

FV2 =2000*1.09^6=3354.200221...= $3,354.20

FV3 =-3000*1.09^5=-4615.8718...= -$4,615.87

FV4 =8000*1.09^3=10360.232= $10,360.23

FV5 =-5000*1.09^1=-5450.00= $-5,450.00

FV1+FV2+FV3+FV4+FV5 = $5,641.12

Then, use the PV equation to find the present value.

PV =5641.12/1.09^8=2831.087908= $2,831.09

Front 44

You are looking into an investment that will pay you $12,000 per year for the next 10 years. If you require a 15% return, what is the most you would pay for the investment?

Back 44

The most you would be willing to pay is the present value, which is $60,225.22.

This is an ordinary annuity, so first calculate the PVF and then use the PV equation.

[1 - (1/1.15^10)]/0.15 = 5.018768626

PV = 12000*5.018768626 = 60225.22351

Front 45

The going rate on student loans is quoted as 9% APR. The terms of the loan call for monthly payments. What is the effective annual rate, or EAR, on such a student loan?

Back 45

A rate of 9% with monthly payments is actually 0.75% a month. The EAR is thus 9.38%.

EAR = (1 + 0.9/12^12) - 1 = 0.093806... = 9.38%

Front 46

Suppose you borrow $10,000. You are going to repay the loan by making equal annual payments for five years. The interest rate on the loan is 14% per year. Prepare an amortization schedule for the loan. How much interest will you pay over the life of the loan?

Back 46

You will make a total of 5 payments of $2,912.83. The interest over the life of the loan will be $4,564.14.

PV = $10,000, r = 14%, t = 5 years

10000 = C*(1-1/1.14^5)/0.14 = C*3.43308969

C = 10000/3.43308969 = 2912.828065

5*2912.828065-10000 = 4564.140327

Front 47

You've recently finished your MBA at the Darnit School. Naturally, you must purchase a new BMW immediately. The car costs $42,000. The bank quotes an interest rate of 15% APR for a 72-month loan with a 10% down payment. What will your monthly payment be? What is the effective interest rate on the loan?

Back 47

Your monthly payment will be $799.28. The EAR on the loan is 16.08%.

PVF = 15/12 = 0.0125

APVF =[1-(1/1.0125^72)]/0.0125= 47.29247431

PV =42000-(42000*0.1)=37800= $37,800

C =37800/47.2924743=799.2815041= $799.28

EAR =1.0125^12-1=0.1607545177= 16.08%

Front 48

As you increase the length of time involved, what happens to the present value of an annuity? What happens to the future value?

Back 48

The present value decreases as the length of time increases. The future value increases as the length of time increases.

Front 49

What happens to the future value of an annuity if the interest rate increases? What happens to the present value?

Back 49

The present value decreases and the future value increases if you increase the interest rate.

Front 50

Rooster Co. has identified an investment project with the following cash flows. If the discount rate is 10%, what is the present value of these cash flows? What is the present value at 18%? At 24%?

Back 50

Y1 = $830, Y2 = $610, Y3 = $1,140, Y4 = $1,390

FV1 =830*1.10^4=1215.203= $1,215.20

FV2 =610*1.10^3=811.91= $811.91

FV3 =1140*1.10^2=1379.4= $1,379.40

FV4 =1390*1.10^1=1529= $1,529.00

FV =1215.203+811.91+1379.4+1529= 4935.513

PV =4935.513/1.1^4=3371.021788= $3,371.02

PV =5838.71061/1.18^4=3011.267..= $3,011.27

PV = 6880.93.../1.24^4=2910.452...= $2,910.45