COMM298-CLASS02.pdf

8/9/2019 COMM298-CLASS02.pdf

1/13

Class 2: The Time Value of Money

COMM 298 – Introduction to Finance

Cornelia Kullmann


2/13

2

Objectives

In this class we learn how to compare sums of money received atdifferent points in time. – Simple interest – Compound interest

Visual aid: Time lines To be able to compare, add, or subtract sums of money received atdifferent points in time we use the concepts of compounding anddiscounting. – Three rules of time travel.

Understanding this part of the course is crucial. So please takeadvantage of office hours or ask questions in class if you are confusedabout something (or everything)!


3/13

The Time Value of Money

Financial decisions involve cash outflows and inflows that arespread out over time.

Financial decision-makers therefore have to compare the values ofsums of money at different dates.

The time value of money (TVM) refers to the fact that a dollar youhave today is worth more than the expectation of that dollar to bereceived at a future date.

– To make cash flows received at different times comparable, we use‘Three Rules of Time Travel ’ introduced later today.

3


4/13

Why?

– “ A bird in the hand is worth two in the bush ”

There are three main reasons why a dollar today is worth more thanthe expectation of one dollar in the future – Inflation changes the purchasing power of one dollar over time

You know what you can get for $1 today, but you don’t know what you can buywith $1 in the future.

One dollar 101 years ago bought you more goods than one dollar can buy youtoday.

In 1912, you could buy with 4 cents what you can buy with $1 now. Saiddifferently, with $1 you could buy the equivalent of $25 worth of ‘stuff’ today.

– Risk: Future money is oftentimes uncertain. Depending on how you invest your money or who promises it to you, there is

almost always counterparty risk. Increases the longer you are away from thepromised payment.

– Opp ortunity Cost: If you have a dollar today, you can invest it to earn areturn on that dollar and end up with more money in the future.

The promised return on your investment has to compensate you for expectedinflation and the risk of the investment (see above). 4


5/13

Visual Aid: The Timeline

A timeline is a linear representation of the timing of potential cash flows.

– Drawing a timeline of the cash flows lets you visualize a financial problem thatinvolves cash outflows and inflows at different points in time.

-1 0 1 2 Periods

----|--------------------|--------------------------|-------------------------|---------$X $Y Cash Flows

One period ago Today End of Period 1=

Beginning of Period 2

Example: The beginning of period 29 is at t = 28. 5

Period 1 Period 2


6/13

Time Line Examples

0 1 2 3 4 Periods|----------|----------|----------|---------|----

Cash Flows (CFs)

-1 0 1 2 3 4 Periods-|---------|----------|----------|----------|---------|---

CFs

T S R Q K P Periods-|---------|----------|----------|----------|---------|--

CFs

Note: ‘Periods’ can be of any length, e.g. days, weeks, months, years, etc.

Fireworks clip art is from sweetcliparts.com

6


7/13

Cash Flows and The Timeline

In most cases (in this course), cash flows are either all inflows or alloutflows. In this case, you don’t need to sign them in calculations or thetime line – as long as you get the end result right.

One can differentiate between two types of cash flows:

– Inflows are positive cash flows.

– Outflows are negative cash flows, which are indicated with a –(minus) sign.

– Example: You borrow $10,000 from a friend today. You repay himin two installments of $5,500 at the end of each of the next twoyears.

– Your cash flows look like this: -1 0 1 2 Periods

----|--------------------|--------------------------|-------------------------|---------

$10,000 - $5,500 - $5,500

Your friend’s cash flows are the reverse of yours.

7


8/13


9/13

Simple Interest

Interest that is paid only on the amount originally invested but not onany interest that accrues subsequently. – >Not very common in finance<

Interest payments are given by I =r*P each period, where r is the

simple rate of interest Over n periods, the original amount invested grows to FV n = P + r*P+ r*P + …r*P = P + n*r*P = P(1 + n*r)

Example: – Firm borrows $100 at 5% simple interest due at the end of 4 years.

What amount must firm repay after 4 years? Timeline: 0 1 2 3 4 5 Years

--------|----------|----------|----------|----------|----------|--100 -120 Cash Flows

I = 4*0.05*100 = 20FV4 = $100+$100*4*0.05 = $100*(1+0.05*4) = $120 9


10/13

iClicker : Simple Interest Example

A Firm borrows $100 at 5% simple interest due at the end of 4 years. Oneyear later, the firm borrows another $100 at 4% simple interest for threeyears. What amount must firm repay after 4 years?

a) $209b) $264

c) $232d) $241e) None of the above

Draw a timeline!

0 1 2 3 4 5

__|____|____|____|____|____|_100 100 ?

Correct answer is c)FV(4) = $100*(1+0.05*4) + $100*(1+0.04*3) = $232

10


11/13


12/13

How Compound Interest Works

An Example: Assume you invest P = $100 at an annual rate of interest r = 5% for n = 4 years.You take no money out of your account before the end of year 5.

Draw a timeline first! Then :

– After one year, at t=1, the future value of P, denoted by FV 1 is given by

– From the end of year 1 to the end of year two, the $105 will grow at 5%, i.e.

– After 3 years, you will have:

– By repeated multiplication, you will find the value after 4 years to be:

12

FV 2

= FV 1 *(1 + r ) = P *(1 + r )*(1 + r ) = P *(1 + r )

2

= $105*(1.05) = $100*1.05*1.05 = $100 *1.05 2 = $110.25

FV 1

= P + rP = P *(1 + r ) = $100 *(1.05) = $105

FV 3

= FV 2 !(1.05) = $100 !(1.05) 2 !(1.05) = $100 !(1.05) 3 = $115.76

FV 4

= FV 3 !(1.05) = $100 !(1.05) 3 !(1.05) = $100 !(1.05) 4 = 121.55


13/13

Documents

COMM298-CLASS02.pdf