chapter 5-time valueofmoney (1)

Financial ManagementFIN 3300

Weeks 2 and 3

Corporate Finance

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Review

What are the purposes of following types of ratios?• Market value

• Profitability

• Leverage

• Liquidity

• Efficiency

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Time Value of Money

Future values – compounding Present values – discounting Multiple cash flows Perpetuities Annuities Interest rates

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Future Value

Equation:

• FV = future value

• PV = present value

• r = interest rate, discount rate or cost of

capital per period

• t = number of time periods

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tr1PVFV

Future Value Examples

What will $100 be worth in one year, assuming you can invest at 2% interest per year?

What will $100 be worth in five years, assuming you can invest at 2% interest per year (assume interest reinvested - compounding)?

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Present Value

Equation:

• FV = future value

• PV = present value

• r = interest rate, discount rate or cost of

capital per period

• t = number of time periods

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tr1

FVPV

Present Value Examples

If you will receive $102 in one year, what is it worth to you today? Assume you can invest now at 2% interest per year (opportunity cost of capital).

If you will receive $110.41 in five years, what is it worth to you today assuming 2% interest per year (assume interest reinvested)?

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More Problems

Implied interest rates: You buy a new recliner and can pay $600 now or $800 in one year. How should you pay if you can get a one year 25% loan?

Internal rate of return: (compound annual growth rate - CAGR): What is the internal rate of return if you invest $100 and get back $1,000 in ten years?

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More Problems

Time needed to save: If you have $1,000 now and want to put it in a savings account to grow to $2,000, how many years will you need to wait assuming you get 2% interest per year?

Comparing future cash flows: Which is worth more, $1,500 in 1 year or $2,379 in 5 years? Let r = 12%.

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Another Problem

The value of free credit: You have the option of paying for a new one-wheeled motorcycle with cash now for $10,000 or pay $12,500 in two years. If a car loan would cost you 12% interest per year, should you pay now or in two years?

Present Value of Multiple Cash Flows

Equation:

• PV = present value

• C1 = future cash flow in one period

• C2 = future cash flow in two periods

• r = interest rate, discount rate or cost of

capital per period

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....PV 22

11

)r1(

C

)r1(

C

Present Value of Multiple Cash Flows Example

Draw time-lines to organize cash flows Discount each cash flow separately

Find the combined present value of getting $1,000 in one year and $1,500 in two years if r = 10%?

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Multiple Cash Flows Problem

Choose the less expensive option to buy a car if your cost of money is 8%:• Pay $15,500 cash now

• Pay $8,000 now and $4,000 at the end of each of the next two years

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Perpetuity A stream of level cash payments that starts one period into the future and

never ends. Equation:

• PV = present value

• C = periodic cash payment

• r = discount rate or interest rate per period rC

PV

Perpetuity Example

To create an endowment for a new charity which will pay $100,000 per year forever starting next year, how much money must you invest today if the interest rate will be 10%?

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Perpetuity Example (continued)

If you need the first perpetuity payment to start today, how much money do you invest now?

If you need the first perpetuity payment to start in three years, how much money do you invest now?• This is a delayed perpetuity and is covered further at the end of

this presentation.

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Annuity A stream of level cash payments that starts one period into the future and continues

for t periods. Equation:

• PV = present value

• C = periodic cash payment

• r = discount or interest rate per period

• t = number of payment periods

tr1r

1r1

CPV

Annuity Example

You purchase a TV by paying $1,000 per year at the end of the next three years. What is the price you are paying if the interest rate is 10%?

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Annuity Problems

You plan to save $4,000 every year for 20 years and then retire. Given a 10% interest rate, what will be the value of your savings at retirement?

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Annuity Problems

You purchase a $200,000 condominium with 100% financing over 30 years at 10% interest per year. If you make annual payments, what will they be?

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Interest Rates

Simple and compound interest Annual percentage rate (APR) Effective annual interest rate (EAR) Inflation: nominal and real interest rates

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Simple and Compound Interest

Simple• Interest earned only on original investment

• No interest on interest

• Invest $100 at 16% simple interest and have $132 after two years

Compound• Interest earned on interest by reinvesting

• Time value of money method

• Invest $100 at 16% compounded interest and have $134.56 after two years

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Annual and Effective Interest Rates

Many interest rates are expressed as daily or monthly interest rates• To compare rates over one year, we annualize them

Annual percentage rate (APR)• Interest rate that is annualized using simple interest

• APR = (periodic rate) x (number of periods in a year)

Effective annual interest rate(EAR)• Interest rate that is annualized using compound interest

• EAR = (1 + periodic rate)(number of periods in a year) - 1

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APR and EAR Examples

Find the APR and EAR for a 2% monthly interest rate.

What is the EAR for a car loan requiring quarterly payments at an 8% APR?

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Inflation rate

• Rate at which prices of goods increase

• Consumer price index (CPI) Nominal interest rate

• Rate at which an investment grows Real interest rate

• Rate at which the purchasing power of an investment grows Equation:

(1 + real interest rate) = (1 + nominal interest rate)

(1 + inflation rate)

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Interest Rates and Inflation

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Interest Rate Example

If the interest rate on one year government bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate?

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Inflation History

-15

-10

-5

0

5

10

15

201900

1920

1940

1960

1980

2000

Ann

ual I

nfla

tion

%100 Years of Inflation

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Nominal versus Real for Time Value of Money Problems

Normally use nominal cash flows with nominal interest rates

If you need to use real data, use real cash flows with real interest rates

Both should give the same answer if done properly

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Delayed Perpetuity A stream of level cash payments that starts “t” periods into the future and never ends. Equation:

• PV = present value

• C = periodic cash payment

• r = discount rate or interest rate per period

• n = number of periods until first payment

11

1nrr

CPV

Delayed Perpetuity Example

To create an endowment for a new charity which will pay $100,000 per year forever starting in three years, how much money must you invest today if the interest rate will be 10%?

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Delayed Annuity A stream of level cash payments that starts “n” periods into the future and continues for t periods. Equation:

• PV = present value

• C = periodic cash payment

• r = discount or interest rate per period

• t = number of payment periods

• n = number of periods until first payment

11

1

1

11nt rrrr

CPV

Delayed Annuity Example

Starting in 3 years, you will $4,000 every year for 17 years. Given a 10% interest rate, what is the present value?

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