bsns108 2011 2 time value of money i (four per page)

8/21/2019 BSNS108 2011 2 Time Value of Money I (Four Per Page)

1/7

BSNS108

1

Dr. Helen Roberts (Finance)

Key Concepts/SkillsWhat is the...

future value of an investment made now?

present value of cash to be received in the future?

2

interest rate?

timing of a cash flow/series of cash flows?

Interest Rate

Interest rate: an exchange rate betweenpresent values & future values. Examples:

saving (earn) or loan (pay) rates at bank

3

an actual realized return on an investment

cost of capital or discount rate in ProjectEvaluation

Future Value (FV)

Invest $1000 for 1 year at 5% per year. Whatis the FV of the investment in one year?

FV = Principal + Interest Principal = $1,000

4

Interest = 1,000 x 0.05 = 50

So, FV = $1,000 + $50 = $1,050,

or

FV = $1,000 x (1 + 0.05)

= $1,000 x 1.05 = $1,050


2/7

Future Value (FV)

leave the money invested for a second

year. What is the FV after 2 years?

FV = [balance after 1 year] x 1.05

5

= , x . x .

= $1,000 x (1.05)2

= $1,102.50

What if the interest rate = 7.5%?

FV: General Formula

FV = PV x (1 + r)t

r = interest rate per period t = number of periods

= + t

6

multiplies PV to give FV,

assumes r is constant

Need the xy (or ^ ) key on your calculator

Compounding

In previous example, your FV was:

FV= $1,102.50 = $1,000 + $50 + $50 +$2.50

FV= Principal + simple interest for two years,plus interest on the interest

7

ompoun n eres means you earn n ereson both principal and on earned interest

FV of $100 invested @ 10% for 5 years?

FV = $

How much of this is simple interest?

Figure 4.1

8


3/7

Future Value (FV)

Compounding implies that FV increases

as the number of periods increases.KiwiSaver Example #1:

9

$1,000 & gets $1,000 Kick-Start from IRD

Invests in ING Growth Fund; assume r=10%.

Make no other contributions

At age 18 this is worth FV=

Present Value (PV)

How much do I have to invest today to have a

quoted dollar amount in the future? Rearrange FV = PV x (1 + r)t

...to solve for PV = FV / (1 + r)t

10

When we talk about discounting, we meanfinding the PV of some future amount.

When we talk about the value of something, we

usually mean the PV unless we specificallyindicate that we want the FV or value at someother date.

Example (PV; 1-period)

You need $10,000 in one year to buy a car.

You can earn 7% per annum. How much doyou need to invest today to meet your goal?

11

, . , .

What is the PV if you can earn 12% p.a.?

PV= ...have a guess first!

Example (PV; t-period)It is 2009. My 2-year old wants to go to Harvard

(a private U.S. university) starting 2025 (in 16yrs). Estimated FV of costs is USD614,050.

I can earn 6% p.a., how big is the lump sum Ineed to deposit now in the U.S. to pay for this?

12

PV = USD614,050/ (1.06) =USD127,076.40

Quiz: what about a state school then:FV=USD171,564; PV=?

http://www.admissions.college.harvard.edu/financial_aid/cost.html : cost now (2009) = USD52,000. I assumed growth

at 7% per annum, and then paid for up front as simply 4 years times this future number..


4/7

PV Relationship I

What impact does timing t have on PV of a

future cash flow?

Given FV & interest rate r (both positive),

13

e.g., What is the value today of a $500 gift to be received in

5 or 10 years (discount rate 10%)?:

In 5 years: PV = $500 / (1.10)5 = $310.46

In 10 years: PV = $500 / (1.10)10 = $192.77

PV Relationship II

What impact does interest rate r have on

the deposit I need to reach a fixed goal?

For a given (positive) FV & time period t,

14

Goal of $500 in 5 years; two interest rates:

Rate = 10%: PV = $500 / (1.10)5 = $310.46

Rate = 15%; PV = $500 / (1.15)5 = $248.59

PV/FV Relation

FV = PV x (1+r)t

PV = FV / (1 + r)t

There are four parts to this equation

15

PV, FV, r, and t

If we know any three, we can solve forthe fourth

Draw a time line!

Discount Rate (r)

Often we want to know the impliedinterest rate for an investment

Take the FV equation: = + t

16

r = (FV / PV)1/t 1

Now need both the yx and the 1/x keys


5/7

Discount Rate (r) Example

An investment pays $1,200 in 5 years if

you deposit $1,000 today. What is theimplied rate of interest r per annum?

r = (FV / PV)1/t 1

17

= (1,200 / 1,000)1/5 1

= .03714

= 3.714%

Check! Use FV=PV x (1+r)t to confirm

Solve for Number of Periods (t)

Start with basic equation and solve for t

FV = PV(1 + r)t [divide both sides by PV]: (FV/PV) = (1 + r)t

take lo of both sides : ln FV/PV = ln 1 + rt

18

[recall property of logs: ln[ya]=a* ln(y)]

[so move t down]: ln(FV/PV)= t * ln(1 + r)

[now divide both sides by ln(1+r)]:

t = ln(FV / PV) / ln(1 + r)

Number of Periods (t) Example

You are willing to pay $20,000 for a car.You have $15,000. You can earn 10% p.a.How long until you can buy the car?

19

= ln(20,000 / 15,000) / ln(1.10)

= 3.02 years (just over 3 yrs)

Now check it!

Summary of TVM I

trPVFV )1( +=

1 ln FV

20

tr

FVPV

)1( += 1

=PV

r)1ln( r

t+

=

Always draw a time line!

Always have a guess!

Always write down the formula!

Always check your answer! END


6/7

Worked Example (PV)

A relative set up a trust fund for you 10

years ago. It is now worth $19,671.51. Ifthe fund earned 7% per year, how much

21

Draw a time line!

What do we know: FV, PV, r, t ?

PV = $19,671.51 / (1.07)10

= $10,000

Worked Example (r)

An investment will double your money in 6

years. You have $10,000 to invest. Whatis the implied interest or discount rate?

r = (FV / PV)1/t 1

22

= (20,000 / 10,000)1/6 1

= .122462

= 12.25% p.a.

Check! Use FV=PV x (1+r)t to confirm

Worked Example (r)US private college example again. Cost is

USD614,050 in 16 years. If I have onlyUSD50,000 to invest, what interest rate

p.a. will get me to my goal? r = (FV / PV)1/t 1

23

= (614,050/50,000)1/16 1

= 0.1697071

16.97%

What should you do now? Ch... !

Worked Example (t)...

You want to buy a $150,000 house sometime in the future. You need a 10% down

payment in cash plus 5% (of the loan) in

24

. ,in the bank, and you can earn 7.5% p.a.interest. How long until you have enoughcash for the down payment and closing

costs?


7/7

...Example (t) Continued

How much do you need to have in the future?

Down payment = .10 x (150,000) = $15,000 Closing costs = .05 x (150,00015,000) = $6,750

Total needed = $15,000 + $6,750 = $21,750

25

Using the formula

t = ln($21,750 / $15,000) / ln(1.075)

= 5.14 years (just over 5 years)

What should you do now? ...

Extra Problems (FV & PV)

Find the FV of a $2,000 deposit that earns

simple interest only at 12% p.a. for 3 years. What is the FV if the interest is compounded for 3years at 12% p.a.?

26

Find the PV of $10,000 to be received in 8 yearstime if the discount rate is 9% p.a. What if the payment is in 12 years?

At 8% interest p.a. how long does it take to tripleyour money?

Extra Problem

You have $10,000 to deposit. Whichoption gives you the best return if the

money is left for 9 years:

27

b) 12% compounded annually?

bsns108 2011 2 time value of money i (four per page)

Documents