5.1 Chapter Outline

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Slide 2

Consider the timeline below:

PV is the Present Value, that is, the value today.

FV is the Future Value, or the value at a future date.

The number of time periods between the Present Value andthe Future Value is represented by t.

The rate of interest is typically called r .

All time value questions involve the four values above: PV,FV, r, and t. Given three of them, it is always possible tocalculate the fourth.

Time Value Terminology

. . .PV

1 2 3 t0

FV


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Slide 3

By using a financial calculator, in general, you need to enterfour known values and calculate the unknown fifth value:

Where:

N = The number of periodsI = The interest rate during each periodPV = The present value (here: the initial investment)PMT = The value of cash flows that reoccur each period (e.g.

additional investments, withdrawals ordisbursements, see chapter 5, part 2)

FV = The future value

How to use a Financial Calculator

N I PV PMT FV

x x x 0 ?


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Slide 4

Some Notes on how to use a Financial Calculator

When to use Positive/Negative Cash Flows: On most calculators, the PV has to be entered as a negative cash flow

Why? Because the investment that you make today is a cash outflow for you.

Unfortunately, not all calculators use this rationale.

The best solution is to familiarize yourself with your calculator and understand how valueshave to be entered.

Ifyou getIncorrect Results:

Particularly in the calculations we perform in chapter 5, where we look at reoccurring cash flows(the PMT button), you may initially get incorrect results

Why? Because the factory settings for compounding periods per year (PMT/Y) and beginning- vs.end-of-period cash flows (BGN/END) may be set differently to what we will use in class

Solution: Set your compounding periods per year (P/Y) to 1 and change the settings tobeginning- vs. end-of-period cash flows carefully.

Operations Order:

There is another peculiarity of financial calculators that often leads to errors: Calculate thefollowing simple equation on a financial calculator and compare it to the result you get whenusing a scientific calculator: 2+2*3=?

Why are the results different? Because financial calculators dont adhere to the same order ofoperations that is normally used by scientific calculators.

Solution: Make sure that you enclose calculations you want the calculator to performseparately in brackets: Typing 2+(2*3) will give you the right result.


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Slide 5

5.1.1 Interest Rates: Simple Interest and Compound Interest

Simple interest: Is only applied to the principal (P).

Total interest earned = P*r*t

Ending balance or FV = P + (P*r*t)

Compound interest: Is earned on both the principal (P) and accrued interest.

Total interest earned = P(1 + r)t P

Ending balance or FV = P(1 + r)t


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Slide 6

Example I: suppose you invest $100 in a security earning a10% annual rate of return for a period of 3 years. How muchis simple interest? How much is compound interest?

Calculate simple interest:

Total interest earned = P*r*t = $100*0.1*3 = $30

Ending balance = P + (P*r*t) = $100 + ($100*0.1*3) = $130

Calculate compound interest:

Total interest earned = P(1 + r)t P = $100(1+0.1)3 $100 = $33.10

Ending balance = P(1 + r)t = $100(1+0.1)3 = $133.10

Notice that:

1. $110 = $100v (1 + .10)2. $121 = $110 v (1 + .10) = $100 v 1.10 v 1.10 = $100 v 1.102

3. $133.10 = $121 v (1 + .10) = $100 v 1.10 v 1.10 v 1.10 = $100 v 1.103

5.1.1 Interest Rates: Simple Interest and Compound Interest (contd)


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Slide 7

We can also use a financial calculator to compute the endingbalance or the FV of a compound interest account:


N I PV PMT FV

?


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Slide 8

5.1.1 Computing Simple and Compound Interests of $100 at 10 Percent:

Beginning Simple Interest on Compound Ending

Year Amount Interest Interest Interest Amount

1 $100.00 $10.00 $ 0.00 $10.00 $110.00

2 110.00 10.00 1.00 11.00 121.00

3 121.00 10.00 2.10 12.10 133.10

4 133.10 10.00 3.31 13.31 146.41

5 146.41 10.00 4.64 14.64 161.05

Totals $50.00 $ 11.05 $ 61.05


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Slide 9


Example II: you deposit $5,000 today in an account paying12%. How much will you have in 6 years? How much issimple interest? How much is compound interest?

The simple interest is:

0.12 v $5,000 = $600 per year.

After 6 years, this is 6 v $600 = $3,600

The compound interest is:

$5,000 v (1 + r)t - $,5000 = ($5,000 v (1.12)6) - $5,000

= ($5,000 v 1.9738227) - $5,000

= $4,869.11

The difference between compound and simple interest isthus $4869.11 - $3600 = $1269.11


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Slide 10


Example III: you opened a savings account with an initialinvestment of $500 three years ago. Today the accountbalance is $600. If the account paid interest compoundedannually, how much interest on interest was earned?

Step 1: calculate the annual compound interest (r):Ending balance = P v (1 + r)t

$600 = $500 v (1 + r)3

$600/$500 = (1 + r)3

1.2 1/3 = 1 + r

1.2 1/3 -1 = r

0.0627 = r

6.27% = r


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Slide 11

Alternative for step 1: We can also use a financialcalculator to compute the compound interest rate (r):


N I PV PMT FV

?


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Slide 12


Step 2: calculate the sum of all simple interest paymentsearned on the principal of $500:

= P*r*t = $500* 6.27%*3 = $94.05

Step 3: find the total interest earned in the account:

= $600 - $500 = $100

Step 4: calculate the interest on interest by subtracting thetotal simple interest from the total compound interest:

= $100 - $94.05 = $5.95


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Slide 13

Basis formula: FVt= PV(1 + r)t

The expression (1 + r)t is the compound factor.

Timeline:

5.1.2 Future Value of a Lump Sum

. . .

PV

1 2 3 t0

FV


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Slide 14

Suppose you invest $100 in a security earning a 10% annualrate. How much will you have after 3 years?

First define the variables:

PV = $100 r= 10 percent

t= 3 years FV = ?

Solve for the future value:

FV3= $100(1 + 0.1)3 = $133.10

5.1.2 Future Value of a Lump Sum (contd)


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Slide 15

We can also use a financial calculator to solve thisproblem.

FV = $133.10

5.1.2 Future Value of a Lump Sum (contd)

N I PV PMT FV

?


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Slide 16

Basis formula: PVt= FV*1/(1 + r)t

The expression 1/(1 + r)t is called the discount factor.

The discount factor is the reciprocal of the compound factor.

Timeline:

5.1.3 Present Value of a Lump Sum

. . .PV

1 2 3 t0

FV


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Slide 17

5.1.3 Present Value of a Lump Sum (contd)

Want to be a millionaire? No problem! Suppose you just turned 21years old, and can earn 10 percent on your money. How muchmust you invest todayin order to accumulate $1 million by thetime you reach age 65?

First define the variables:

FV = $1 million r= 10 percent

t= 65 - 21 = 44 years PV = ?

Set this up as a future value equation and solve for thepresent value:

$1 million = PVv

(1.10)44

PV = $1 million/(1.10) 44 = $15,091.


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Slide 18

We can also use a financial calculator to solve thisproblem.

PV = $15,091

5.1.3 Present Value of a Lump Sum (contd)

N I PV PMT FV

?


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Slide 19

5.1.4 Number of Periods

Number of Periods (t) : Is the life of the investment.

Can be easily computed using your financial calculator:

N I PV PMT FV

? x x 0 x


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Slide 20

5.1.4 Number of Periods (contd)

You need $15,000 to buy a new car. If you have $8,000 toinvest at a 5% interest rate compounded annually, howlong do you have to wait to buy the car?

First, define the variables:

FV = $15,000 r= 5 percent

PV = $8,000 t= ?

Set this up as a future value equation and solve for thenumber of periods (t):

FVt= PV(1 + r)t

$15,000 = $8,000(1 + 0.05) t

(1 + 0.05) t= $15,000/$8,000

t = 12.88


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Slide 21

Using the financial calculator to solve for t:

N = 12.88

5.1.4 Number of Periods (contd)

N I PV PMT FV

?


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Slide 22

Self-study problem

Suppose you deposit $5000 today in an account paying rpercent per year. If you will get $10,000 in 10 years, what rateof return are you being offered?

Set this up as present value calculation:

FV = $10,000 PV = $ 5,000 t= 10 years

PV = FVt/(1 + r)t

$5000 = $10,000/(1 + r)10

Now solve forr:

(1 + r)10 = $10,000/$5,000 = 2.00

r= (2.00)1/10 - 1 = .0718 = 7.18%


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5.1 Chapter Outline

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