BUSFIN 4211: 1.1 Time Value of Money

PP

0

Time Value of Money

• How do we value cash flows received in the future?

  • Present value, annuities and perpetuities

Interest rates

• The price of money.

We want to value every choice in the same terms:

cash today

Interest rates convert

the value of cash across time

Multiple period calculation must

• account for compounding

  • Interest from the first year earns interest in the second year

We will want to value cash flows in

present value terms (i.e., cash value today)

Valuing a stream of cash flows simply involves

adding present values!

You discount future cash flows because of

alternative costs

A perpetuity is

a security that offers a fixed payment C forever.

For simplicity, it will often be useful to

value perpetual streams of cash flows that

grow at a constant rate forever

There is a debate whether

𝑟 > 𝑔 or not

An annuity is a

security that offers a fixed payment C at regular intervals for a

fixed time period T. Most car loans and mortgages are annuities.

Compounding Intervals

• (E.g., daily, monthly, yearly)

Effective Annual Rate (EAR) aka Annual Percentage Yield (APY)

• The total amount of interest that will be earned at the end of one year

When computing present or

future values, you should

adjust the discount rate to match the

time period of the cash flows

Annual Percentage Rates (APR):

• Indicates the amount of simple interest earned in one year; that is the

amount of interest without the effect of compounding

Simple Interest

the amount of interest without the effect of compounding

Because it does not include the effect of compounding, the APR quote is

typically lower than the actual amount of interest you will earn

Because the APR does not reflect the true amount you will earn over one

year, the APR itself

cannot be used as a discount rate

Converting the APR to a Discount Rate STEPS

1. Divide the APR by 12 to get the monthly rate, 𝑟𝑀𝑜𝑛𝑡ℎ𝑙𝑦

2. Then use formula for present value of annuity:

Nominal Interest Rate:

• The rate at which your money will grow if invested for a certain period

Real Interest Rate:

• The rate of growth of your purchasing power, after adjusting for inflation

When the costs of an investment (cash out) occurs before future expected benefits (cash in),

an increase in the interest rate will make the investment less attractive

FED: Lower interest rates =

spur economic investment and “risk-taking”

FED: Raise interest rates =

slow down spending to curb inflation and economic investment

BOOK

0

Time Value of Money

• Def: A dollar in hand today is worth more than a dollar to be received in the future because if you had it now, you could invest it, earn interest, and own more than a dollar in the future.

Future Value FV

• Def: The amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate.

• The ending amount, of your account after N periods. Whereas PV is the value now, or the present value, is the value N periods into the future, after the interest earned has been added to the account.

Present Value PV

• Def: The value today of a future cash flow or series of cash flows.

• Present value, or beginning amount.

Compounding

• Def: The arithmetic process of determining the final value of a cash flow or series of cash flows when compound interest is applied.

• Going from PV to FV.

Annuities

• A series of equal payments at fixed intervals for a specified number of periods.

• when we use the term annuity in this book, assume that the payments occur at the ends of the periods unless otherwise noted.

• There are ordinary annuities (payments end of year) and annuity due (payment beginning of year).

Perpetuities

• Def: A stream of equal payments at fixed intervals expected to continue forever.

• an annuity with an extended life.

Interest Rates

• the price an entity pays for borrowing money or the fee they charge for lending it, expressed as a percentage. INVESTOPEDIA

Compounding

• Def: The arithmetic process of determining the final value of a cash flow or series of cash flows when compound interest is applied.

• the repeated addition of interest payments to the principal invested over a period of time. INVESTOPEDIA

  • The principal grows exponentially as each new payment of interest is added to it. INVESTOPEDIA

Discounting

• Def: The process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding.

Effective Annual Rate (EAR) or Annual Percentage Yield (APY)

• Def: The annual rate of interest actually being earned, as opposed to the quoted rate.

• The real annual return on a savings account or any interest-paying investment when the effects of compounding over time are taken into account. INVESTOPEDIA

Annual Percentage Rates (APR)

• In previous section.

Inflation

• Def: A rise in prices, which can be translated as the decline of purchasing power over time.

Nominal Interest Rate

• Def: The contracted interest rate.

• The interest rate before taking inflation into account. INVESTOPEDIA

Real Interest Rate

• Def: The observed market interest rate adjusted for the effects of inflation. INVESTOPEDIA