Introduction To Valuation: The Time Value Of Money

Chapter 4

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Topics1. How To Determine The Future

Value Of An Investment2. How To Determine The Present

Value Of Cash To Be Received At A Future Date

3. How To Find The Return On An Investment

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Fundamental Truth in Finance: A dollar earned now is worth more than a dollar earned later. This is true because of the ability of individuals to earn interest.

Today 1 year from today $1 received today $1 received 1 year from today

Today (at 10% simple rate) 1 year from today

$1.00 grows to $1.10

Present Value (Interest going backwards)

The $1 received today is worth more than a dollar received 1 year from now because you can invest the dollar and earn interest.

Future Value (Interest going forward)

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Fundamental Financial Concept A dollar received today is worth

more than a dollar received later This is because of interest This is because of the discount rate

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Definitions: Simple interest

Interest earned only on the original principal amount invested

Compound interest Interest earned on both the initial principal

and the interest reinvested from prior periods Interest on interest

Interest earned on the reinvestment of pervious interest payments

Compounding The process of accumulating interest in an

investment over time to earn more interest

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Variables for the Financial Functions Defined: An = Annuity = Regular Payments (PMT) Made at Regular Time Intervals

Made at End of Period

LS = Lump Sum Payment = Payment Made Once

FV=Future Value (Lump Sum Value in the Future)

PV=Present Value (Lump Sum Value in the Present)

PMT = Regular Payment Made at Regular Time Intervals

i = Annual Interest Rate

n = Number of Compounding Periods per Year

x = Years

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Future value: The amount an investment is worth after

one or more periods

n*x

LS LS

iFV = PV * 1+

n

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Future Values: (Textbook formula)

Textbook FV = PV(1 + r)t

FV = future value PV = present value r = period interest rate, expressed as a decimal T = number of periods

Future value interest factor = (1 + r)t

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2

1*2

*

100 bucks invested @ 10%

compounded yearly for 2 years

100*(1+.10)=110

110*(1+.10)=121

100*(1+.10)*(1+.10)=121

100*(1+.10) =121

.10100* 1+ 121

1

* 1n x

LS LS

iFV PV

n

10

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How To Determine The Future Value Of An Investment

Suppose you invest $1000 for one year at 5% per year, compounded yearly. What is the future value in one year? Interest = 1000(.05) = 50 Value in one year = principal + interest = 1000 +

50 = 1050 Future Value (FV) = 1000(1 + .05) = 1050

Suppose you leave the money in for another year. How much will you have two years from now? FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

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Effects of Compounding

Simple interest Compound interest Consider the previous example

FV with simple interest = 1000 + 50 + 50 = 1100 FV with compound interest = 1102.50 The extra 2.50 comes from the interest of .05(50) =

2.50 earned on the first interest payment

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Future Values – Example 2

Suppose you invest the $1000 from the previous example for 5 years, compounded yearly. How much would you have? FV = 1000(1.05)5 = 1276.28

The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.)

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Future Values – Example 3

Suppose you had a relative deposit $10 at 5.5% interest 200 years ago , compounded yearly. How much would the investment be worth today? FV = 10(1.055)200 = 447,189.84

What is the effect of compounding? Simple interest = 10 + 200(10)(.055) = 210.55 Compounding added $446,979.29 to the value of

the investment

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Present Value How much should you put in the bank

today in order to receive a future value amount after one or more periods

The current value of future cash flows discounted at the appropriate rate

LSLS n*x

FVPV =

i1+

n

If you know the future amount you would like, assume an interest rate, and take all the interest that you will need to earn out of the future value amount

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Present Values (Textbook formula)

How much do I have to invest today to have some amount in the future? Textbook FV = PV(1 + r)t

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

When we talk about discounting, we mean finding the present value of some future amount.

When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

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Definitions: Discount Rate

The rate used to calculate the present value of future cash flows

Discount Calculate the present value of some

future amounts Discounted Cash Flow (DCF) valuation

Calculating the present value of future cash flows to determine its value today

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How To Determine The Present Value Of Cash To Be Received At A Future Date

You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, compounded yearly, how much do you need to invest today? PV = 150,000 / (1.08)17 = 40,540.34

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Present Values – Example 2

Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, compounded yearly, how much did your parents invest? PV = 19,671.51 / (1.07)10 = 10,000

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PV – Important Relationship I

For a given interest rate – the longer the time period, the lower the present value What is the present value of $500 to be received in

5 years? 10 years? The discount rate is 10%, compounded yearly.

5 years: PV = 500 / (1.1)5 = 310.46 10 years: PV = 500 / (1.1)10 = 192.77

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PV – Important Relationship II

For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5

years if the interest rate is 10%? 15%? (both compounded yearly). Rate = 10%: PV = 500 / (1.1)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.58

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Figure 4.3

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How To Find The Return On An Investment If you invest $100 today in an account

that compounds interest yearly and in 8 years you have $200, what is the interest rate?

Rule of 72 = A reasonable estimate for the required rate to have an investment double = 72/(i/n) = Number of periods

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How To Find The Number Of Periods Required For An Investment If you want to buy an asset that cost

$100,000 and you have $50,000 to invest now, at a rate of 12%, compounded annually, how many years must you wait?

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Example: Spreadsheet Strategies

Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv)

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Real assets/ Financial assets Present value and future value are

fundamental to finance Most instruments:

Real assets Buildings, trucks

Financial assets Debt, Equity, Preferred stock,

derivatives

Can be analyzed using DCF valuation techniques