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FINC3131Business Finance

Chapter 5: Time Value of Money: The Basic Concepts

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Time value of money: Practical relevance

Examples

1. Retirement

2. Mortgage payment

3. Price of a stock

4. Helping your company to decide which project to undertake

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Learning objectives1. Understand the difference between a nominal

interest rate and a real interest rate

2. Recall the present value and future value formulas

3. Solve 1-period time value of money problems

involving one cash flow.

4. Solve 2-period time value of money problems

involving one cash flow.

5. Solve 2-period time value of money problems

involving 2 cash flows.

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Question

You are asked to choose from the following:

1. Receive $100 today

2. Receive $100 one year from now

Would you choose 1 or 2?

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1. Money has time value • People prefer to receive (and spend) it

sooner rather than later.

2. You prefer receiving $100 today to receiving $100 one year from now

• You place a LOWER value on cash flow received at a later date. The worth of $100 received one year from now is less than $100 received today.

Important observations 1

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3. You cannot simply add sums of money received at different points in time.

• In any decision involving amounts received (or paid) at different times, the amounts must first be converted to a common basis: the same point in time

Important observations 2

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Interest rates 1

1. Real rate of interest (rreal) : the return (compensation) you demand for lending someone money and thus postponing consumption.

2. In a world with NO inflation, we only need to work with the real rate of interest.

Inflation: general rise in prices. Same commodity becomes more expensive over time.

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Interest rates 2

1. In a world with inflation, when you lend money to people, you want to adjust your interest rate for inflation.

2. Inflation-adjusted interest rate is known as Nominal rate of interest (rnominal).

In the real world, all the quoted rates are nominal rates (e.g., car loan, house loan, student loan)

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Interest rates 3 If there is no inflation, is the real interest

rate equal to the nominal interest rate? If there is inflation, are the two rates

equal? What is the relation between real and

nominal rates?

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Preparing BAII Plus for use

1. Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’.

2. Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’.

3. Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.

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Time Value BasicsDeposit problem: If you put $100 in a bank deposit account earning 10% annually, how much will be in the account after one year?

100 110 + 10 =

Principal Future Value+ Interest =

100 110 + 100(.10) =

100 110 x (1+.10) =

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Time Value Basics

What would it be worth after two years?

110 121 x 1.1 =

But, since 110 = 100(1+.10)…

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

or

100 121 x (1+.10)2 =

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The Formula for Future Value

Future Value

Present Value

Number of periods

Rate of return or discount rate or interest rate or

growth per period

Right now we look at n = 1, n = 2nrPVFV )1(

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The Formula for Future Value

1. The formula lets you convert a current cash flow into its future value.

2. This process is called compounding.

What about the reverse process? How do we convert future cash flows into their present values?

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The Formula for Present ValueFrom before, we know that

Solving for PV, we get Unless otherwise stated, r stated on an annual basis.

nrPVFV 1

nr

FVPV

)1(

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1. The formula lets you convert future cash flows into their present values.

2. This process is called discounting.

The Formula for Present Value

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More observations

We assume the discount rate, r, is positive. With that assumption, we can say:

1. PV is always less than FV.

2. 1/(1 + r) is always less than one.

3. (1 + r) is always greater than one.

Do on your own: go back to deposit problem (slide 11) and verify statements 1,2,3.

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Discount rate and PV If FV is fixed, then as the discount rate

increases, PV decreases.

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80

90

100

110

Discount rate (%)

Presen

t v

alu

e

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FV and PV formulas1. These are the basic building blocks which we will use

to construct bigger and more complex ideas & concepts.

2. Not surprisingly, we will use these basic building blocks to solve complicated TVM problems.

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Types of TVM problemsSingle-period (think of 1 year)

Two-period

(think of 2 years)

Two-period, interim cash flow

Find FV, given PV & r

Find FV, given PV & r

Find FV, given cash flows & r

Find PV, given FV & r

Find PV, given FV & r

Find PV, given cash flows & r

Find r, given FV and PV

Find r, given FV and PV

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1-period, find FV

You require $1,700 to buy a computer and the bank is offering a loan at an interest rate of 14 percent. If you plan to repay the loan after one year, how much will you have to pay the bank?

Use FV = PV(1 + r)

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1-period, find PV

What is the present value of $16,000 to be received at the end of one year if the discount rate is 10 percent?

Use PV = FV/(1 + r)

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1-period, find r

If the bank promises to give you $28,400 after one year if you deposit $27,000 today, what is the interest rate that the bank is giving you?

Use r = (FV/PV) – 1

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2-period, find FV

You plan to loan $11,000 to your friend at an interest rate of 8 percent per year compounded annually. The loan is to be repaid in two years. How much will your friend pay you at t = 2?

Use FV = PV(1+r)2

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2-period, find PVYou are offered the chance to buy into an investment that promises to pay you $28,650 at the end of two years. If your required rate of return is 12 percent, what is the maximum amount that you would pay for this investment?

Use PV = FV/(1+r)2

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2-period, find rYou are offered an investment opportunity which requires you to pay $31,500 today and promises to give you $39,700 at the end of two years. What is the rate of return on this investment?

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Value additivity principleYou make two investments. The first investment will give you $5,500 after one year and the second investment will give you $12,100 after one year. If your required rate of return is 10 percent, what is the current value of your investments?

Current value means present value.

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Value additivity principle

This problem illustrates the

value additivity principle

which says:You can add values only at the SAME point in

time.

This means you can find the PV or FV of multiple cash flows!

Now, we apply this principle to solve 2-period, 2 cashflows TVM problems.

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2-period, 2 cash flows, find FVYou deposit $5,000 in a bank account today. You make another deposit of $4,000 into the account at the end of the first year. If the bank pays interest at 6 percent compounded annually, how much will you have in your account at the end of two years, that is, at t = 2?

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Time line: visualizing cash flows 1. For a TVM problem with 2 or more periods, a

time line helps you to understand the problem better.

2. A time line is a graphical representation of a TVM problem. For the previous problem, the time line would look like this:

t = 0

-$ 5,000

t = 1 t = 2

-$ 4,000 ?

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2-period, 2 cash flows, find PVYou want to withdraw $3,200 from your account at the end of one year and $7,300 at the end of the second year. How much should you deposit in your account today so that you can make these withdrawals? Your account pays 6 percent p.a.

What’s the time line?

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Summary

1. Time value of money

2. Real vs. nominal interest rate

3. 1-period problems (find FV, PV, r)

4. 2-period problems (find FV, PV, r)

5. Value additivity principle

6. 2-period, 2 cash flows problems (find FV, PV)