Fbf 2015 Lect 02 Tvm 1

7/25/2019 Fbf 2015 Lect 02 Tvm 1

1/34

TIME VALUE OF MONEY 1

Spring 2015

FBF Lecture 2 1

7/25/2019 Fbf 2015 Lect 02 Tvm 1

2/34

explain the time value of money (TVM) concept

identify the differences between simple interestand compound interest

be familiar with TVM terminology calculate the future value of a single amount

calculate the present value of a single amount

calculate effective rates and distinguish themfrom nominal rates

draw time lines to represent the DCF process

Spring 2015FBF Lecture 2

2

Lecture 2 Do List

7/25/2019 Fbf 2015 Lect 02 Tvm 1

3/34

Topic In Perspective


3

Last week Three major financial management decisions

Cash flow is important

Financial markets This week Valuing cash flows at different periods

TVM is critical to this valuation

Future costs and benefits TVM has applications throughout financeInvestment decision analysis

Financial markets

7/25/2019 Fbf 2015 Lect 02 Tvm 1

4/34

The financial manager makes decisions aboutproposals with cash flows over long periods oftime

An important consideration is the timing of thesecash flows The time value of money must be recognised

It is based on the fact that a dollar today is worth

more than a dollar tomorrow Would you prefer $2 million today, or

$2 million in five years time?


4

Time value of money (TVM)

7/25/2019 Fbf 2015 Lect 02 Tvm 1

5/34

You invest $1,000 in a bank today for a period ofone year

The bank will pay interest at a rate of 5% pa.

How much will you have in the bank next year?

Solution

Interest = $1000 x 5% (or 0.05) = 50

Add interest to the original investment $1000 + 50 = 1050


5

Example 1

7/25/2019 Fbf 2015 Lect 02 Tvm 1

6/34

A dollar amount today,

present value

and an interest rate,

and a period of time,

gives a dollar amount in the future

future value


6

Variables

7/25/2019 Fbf 2015 Lect 02 Tvm 1

7/34

Time Value of Money has its own language:

PV = present value, or principal

i = interest rate, later we use r

n = number of periods, later we use t

FV = future value

PMT = periodic payment

This week, we learn about applications thatrequire three variables to then determine a fourth


7

Terminology

7/25/2019 Fbf 2015 Lect 02 Tvm 1

8/34

TimeLine


8

start of period 2start of period 1

end of period 1end of period 2

| | | |

0 1 2 3

Drawn as

Interest for the period of time

7/25/2019 Fbf 2015 Lect 02 Tvm 1

9/34

Calculated on the original principal

Takes no account of changes in principal

Sometimes called Flat rate interest

Used in the valuation of short-term financialinstruments traded in the money market

Term is under 12 months

Bills of exchange


9

Simple Interest

7/25/2019 Fbf 2015 Lect 02 Tvm 1

10/34

INT = PV i n

i = simple interest rate per year

n = number of years

FV = PV + INT FV = future value at end of term

PV = principal value at beginning

INT = interest amount over the time period FV = PV + PV i n

= PV(1 + i n)


10

Future value with simple interest

7/25/2019 Fbf 2015 Lect 02 Tvm 1

11/34

Present Value with simple interest

The formula can be rearranged to calculatepresent values

PV = FV / (1 + i n)

This can also be used to price short-termfinancial instruments

This is covered more in lecture 4


11

7/25/2019 Fbf 2015 Lect 02 Tvm 1

12/34

What is the future value of $100,000 invested for180 days at 10% pa simple interest?

Solution

FV = PV(1 + i n) FV = 100,000(1 + 10%180/365)

= 100,000(1 + 0.0493)

= 104,930

Note the annual interest rate is adjusted for the numberof days the funds are invested during the year


12

Example 2

7/25/2019 Fbf 2015 Lect 02 Tvm 1

13/34

Compound Interest

Interest is added to the principal each period

Interest on interest

Called compounding

The compounding period can be any designatedlength of time

yearly, half-yearly, monthly

Simple interest is calculated only on the originalamount


13

7/25/2019 Fbf 2015 Lect 02 Tvm 1

14/34

Applying the formula many times gives

FV = PV(1+i1)(1+i1)..... (1+i1)

which is equivalent to:

FV = PV(1 + i)n

where

i = the per period interest rate

n = the number of compounding periods PV = the original principal


14

Future Value

7/25/2019 Fbf 2015 Lect 02 Tvm 1

15/34

Example 3


15

Mavis deposits $1,000 today in a savingsaccount that pays interest once a year.

How much will Mavis have in three years time ifthe interest rate is 12% p.a.?

1000|_______|________|________|0 1 2 3

To answer the question you first need to identifywhat information has been given Interest rate; term; PV or FV

7/25/2019 Fbf 2015 Lect 02 Tvm 1

16/34

Example 3: Using a table


16

YEAR OPENING INTEREST CLOSING

BALANCE BALANCE

1 1000.00 120.00 1120.00

2 1120.00 134.40 1254.40

3 1254.40 150.53 1404.93

Interest = Opening balance multiplied by interest

rate i.e., 1,000 x 12% = 120.00

7/25/2019 Fbf 2015 Lect 02 Tvm 1

17/34

FV = PV ( 1 + i )n

= 1000(1 + 0.12)3

= 1404.93

Or, expressed another way

FV = 1000(1.12)(1.12)(1.12)

= 1120(1.12) (1.12)

= 1254.40(1.12)

= 1404.93


17

Example 3: Using the formula

7/25/2019 Fbf 2015 Lect 02 Tvm 1

18/34

Example 4

Freda deposits $5,317 today in a savingsaccount with an interest rate of 5% pa

What is the value of Freda's deposit in four years

time?

|______|_______|________|________|

0 1 2 3 4What have you been asked to calculate?


18

7/25/2019 Fbf 2015 Lect 02 Tvm 1

19/34

What information have you been given?

PV = 5,317

i = 5%

n = 4

FV = PV ( 1 + i )n

FV = 5,317 ( 1 + 0.05 )4 = $6,462.85


19

Example 4 Solution

7/25/2019 Fbf 2015 Lect 02 Tvm 1

20/34

Present Value


20

Rearranging the future value formula gives theformula for the present value

PV = FV (1 + i)n

or

PV = FV(1 + i)-n

or

ni)(1FVPV

7/25/2019 Fbf 2015 Lect 02 Tvm 1

21/34

You own a bank fixed term deposit thatguarantees to pay you $230,000 in six yearstime, however you are not prepared to wait.

What amount of cash would you receive today ifsomeone will buy the fixed term deposit today?

The buyer applies a discount rate of 20% pa

0 1 2 3 4 5 6


21

Example 5

FV

7/25/2019 Fbf 2015 Lect 02 Tvm 1

22/34

Example 5 Solution


FV = 230,000

i = 20%

n = 6

PV = FV ( 1 + i )-n

= 230,000 (1 + 0.20)-6

= $77,026.53


22

7/25/2019 Fbf 2015 Lect 02 Tvm 1

23/34

You estimate that you will need $2,500,000 in 20years to fund your retirement.

Your superannuation fund will generate a return

of 9% p.a.What amount of money would you need to invest

today to have $2,500,000 when you retire?


23

Example 6

7/25/2019 Fbf 2015 Lect 02 Tvm 1

24/34

Example 6 Solution


FV = 2,500,000

i = 9%

n = 20

PV = FV ( 1 + i )-n

= 2,500,000 (1 + 0.09)-20

= $446,077.22


24

7/25/2019 Fbf 2015 Lect 02 Tvm 1

25/34

Frequency of Compounding

Interest rates are normally quoted as per annum

But the compounding frequency is not alwaysannual

A nominal rate is the rate you can observe in themarket

If the compounding period is not annual the ratemust be qualified

16% p.a. compounded monthly This nominal rate is not the same as 16% return pa


25

7/25/2019 Fbf 2015 Lect 02 Tvm 1

26/34

What is the compounding rate for each timeperiod for a 18% nominal annual interest ratewith monthly compounding?

Solution The number of compounding periods each year

is 12

Rate per period = 18% 12 = 1.5%


26

Example 7

7/25/2019 Fbf 2015 Lect 02 Tvm 1

27/34

Effective Annual Rates (EAR)


27

An effective rate is an interest rate thatcompounds annually

To convert a nominal rate to an effective rate

EAR = (1 + i)m- 1 where

m = number of compounding periods per year

i = interest rate per period

7/25/2019 Fbf 2015 Lect 02 Tvm 1

28/34

Example 8


28

Which interest rate is higher?

12.5% annual interest rate, compounded half-yearly, or

12.3% annual interest rate compounding monthly Convert both nominal rates to EARs

EAR = (1+ i)m- 1 EAR = (1 + i)m- 1

= (1+.0625)2-1 = (1 + .01025)121 = 12.89% = 13.02%

7/25/2019 Fbf 2015 Lect 02 Tvm 1

29/34

Example 9

A company has the opportunity to buy an assettoday for $70,000

The company expects to be able to sell this

asset in three years for $87,500 The appropriate return is 9.5% pa.

Should the firm buy the asset?


29

7/25/2019 Fbf 2015 Lect 02 Tvm 1

30/34

Example 9 Solution


30

70,000 87,500|______|______|______|

0 1 2 3

i = 9.5%

PV= 87500/(1+0.095)3= 66,644.71

The firm should not buy the asset

7/25/2019 Fbf 2015 Lect 02 Tvm 1

31/34

Example 10


31

A forest company is offering investors anopportunity to invest in a rubber plantation

The cost to invest is $6,000 in 2013

Projected returns from this investment are $3,000 in 2016 and

$7,500 in 2021

The required return is 15%pa

Is it worth investing in this project? (think in current day dollar terms PV)

7/25/2019 Fbf 2015 Lect 02 Tvm 1

32/34

Example 10 Solution


32

-6000 3000 7500 |___|___|___|___|___|___|___|___| 13 14 15 16 17 18 19 20 21

i = 15% PV3000= 3000/(1+0.15)

3= 1972.55

PV7500= 7500/(1+0.15)8= 2451.76

Value in 2013 = $4,424.31 Not worth investing in this project

7/25/2019 Fbf 2015 Lect 02 Tvm 1

33/34

Bert and Beryl want to establish a fund that willpay $10,000 to each of their two grandchildren,Peter and Mary, when they turn 21.

Mary turns 21 in ten years time and Peter turns21 in twelve-and-a-half years time.

If the funds interest rate is 8% pa. compounded

half-yearly, what amount should Bert and Beryldeposit today?


33

Example 11

7/25/2019 Fbf 2015 Lect 02 Tvm 1

34/34

$10,000 $10,000

0 20 25

i = 8%/2 = 4%

FV = $10,000 in both cases

PV of Marys payment = 10,000/(1.04)20

PV of Peters payment = 10,000/(1.04)25

PV = 4,563.87 + 3,751.17 = $8,315.04


34

Example 11 Solution

Documents

Fbf 2015 Lect 02 Tvm 1