Finance

Chapter 6Time value of money

Time lines & Future Value

Time Lines, pages 218-219

Time: 0 1 2 3 4 5

Time: 0 1 2 3 4 5

Cash flows: -100 Outflow ? Inflow

5%

Time lines & Future Value

CompoundingThe arithmetic process of determining the final value of a cash flow or series of cash flows when compound interest is applied.

Future value (FV)the amount cash flow(s) will grow over a given period of time when compounded at a given interest rate.

Time lines & Future Value

PV=present value (beginning amount).PV = $100

i = interest rate for one yeari = 5%, or i = 0.05

INT = dollars of interest earned during the yearINT = $100(0.5) = $5

FVn = the value n years into the futuren = number of periods in the analysis, n = 1

FVn = FV1 = PV + $105

Time lines & Future Value

Future value, pages 219-223

Time: 0 1 2 3 4 5

Cash flows: -100 FV1=? …………………………… FV5 =?

Interest earned: 5.00 5.25 5.51 5.79 6.08

Amount at the end 105.00 110.25 115.76 121.55 127.63

of each period

5%

Time lines & Future Value

FVN = PV(1 + i)n

The equation has 4 variables. If we know any 3 we can solve for the 4th.

Problem format:

Time: 0 1 2 3 4 5 5%

-100 FV=?

FVN = PV(1 + i)n = $100(1.05)5

Present Value

Opportunity cost rate The rate of return on the best available alternative

investment of equal risk, or the rate of return you could earn on an alternative

investment of similar risk. Present Value (PV)

The value today of a future cash flow or series of cash flows The $100 is defined as the present value (PV) of

$126.63 due in 5 years when the opportunity cost rate is 5%.

If an alternative security is less than $100, buy it If an alternative security is more than $100, ignore it

Present Value

Fair (Equilibrium) ValueThe price at which investors are indifferent between buying or selling a security

DiscountingThe process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding

Present Value

The present value of a cash flow due in n years is the amount, if in hand today, would grow to equal the future amount

Time: 0 1 2 3 4 5 5%

PV = ? 127.63

Present Value

Discounting equation Start with the future value equation and solve for PV:

FVN = PV(1 + i)n

PV = FVN / (1 + i)n

Time: 0 1 2 3 4 5 5%

-100= 105.00 110.25 115.76 121.55 127.63

/1.05 /1.05 /1.05 /1.05 /1.05

Annuities

AnnuityA series of payments of an equal amount (PMT) at fixed intervals for a specified number of periods

Ordinary (deferred) annuityPMT occur at the end of each period

Annuity duePMT occur at the beginning of each period

Perpetuitiesa stream of equal payments expected to continue forever

Interest rates

Nominal (Quoted, Stated, APR) interest rateThe contracted, or quoted, or stated interest rate

Effective (Equivalent) annual rate (EFF% or EAR)The actual rate of interest actually being earned, as opposed to the quoted rate. Also called “equivalent annual rate.” Used to convert any nominal rate to an equivalent

annual rate These two rates may differ.

Amortized loans

Amortized* loansA loan that is repaid in equal payments over its life.

Amortized scheduleA table that shows how a loan will be repaid, showing how much is interest and how much is principal repayment

Example: $1,000 loan, 6% interest on loan balance$1,000 represents the PV of an annuity of PMT dollars per year for n years, discounted at 6%

Partial amortization with a balloon paymentpage 248

*mors = Latin for “kill”