Chapter 5 - Time Value of Money

EAR calculation

(r + 1/m)^m - 1

The following equation calculates the equal periodic loan payments (CF) necessary to provide a lender with a specified interest return and to repay the loan principal (PV) over a specified period:

CF = (PV x r)/ [1 - 1/(1 + r)^n]

Calculating Continuous Compounding

FV = PV x ( e ^rXn)

Find the value at the end of 2 years (n = 2) of Fred Moreno's $100 deposit (PV = $100) in an account paying 8% annual interest (r = 0.08) compounded continuously.

FV2 (continuous compounding) = $100 x e ^0.08 x 2 = $100 x 2.7183 ^0.16 = $100 x 1.1735 = $117.35

Jane Farber places $800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years.

FV5 = $800 (1 + 0.06)^5 = $800 (1.33823) = $1,070.58

You can calculate the future value of an ordinary annuity that pays an annual cash flow equal to CF by using the following equation:

FVn = CF x {[(1+r)^n - 1]/ r}

You can calculate the future value of an annuity due that pays an annual cash flow equal to CF by using the following equation:

FVn = CF x {[(1+r)^n - 1]/ r} x (1 + r)

The general equation for the future value at the end of period n

FVn = PV (1 + r)^n

A general equation for compounding more frequently than annually:

FVn = PV x (1 + r/m)^mXn

We use the following notation for the various inputs:

FVn = future value at the end of period n PV = initial principal, or present value r = annual rate of interest paid. (Note: On financial calculators, I is typically used to represent this rate.) n = number of periods (typically years) that the money is left on deposit

The present value, PV, of some future amount, FVn, to be received n periods from now, assuming an interest rate (or opportunity cost) of r, is calculated as follows:

FVn/ (1+r)^n

Compound Interest

interest that is earned on a given deposit and has become part of the principal at the end of a specified period

Continuous compounding

involves the compounding of interest an infinite number of times per year at intervals of microseconds

Future Value of Money

is the value at a given future date of an amount placed on deposit today and earning interest at a specified rate. Found by applying compound interest over a specified period of time

Principal

the amount of money on which interest is paid

effective (true) annual rate (EAR)

the annual rate of interest actually paid or earned. In general, > nominal rate whenever compounding occurs more than once per year

nominal (stated) annual rate

the contractual annual rate of interest charged by a lender or promised by a borrower

The following equation calculates the annual cash payment (CF) that we'd have to save to achieve a future value (FVn):

............

A single amount

A lump sum amount either held currently or expected at some future date

The three basic patterns of cash flows include:

A single amount An annuity A mixed stream

A mixed stream

A stream of unequal periodic cash flows

If Fred Moreno places $100 in a savings account paying 8% interest compounded annually, how much will he have at the end of 1 year? If Fred were to leave this money in the account for another year, how much would he have at the end of the second year?

Future value at end of year 1 = $100 (1 + 0.08) = $108 Future value at end of year 2= $100 (1 + 0.08) (1 + 0.08) = $116.64

Paul Shorter has an opportunity to receive $300 one year from now. If he can earn 6% on his investments, what is the most he should pay now for this opportunity?

PV (1 + 0.06) = $300 PV = $300/(1 + 0.06) = $283.02

Pam Valenti wishes to find the present value of $1,700 that will be received 8 years from now. Pam's opportunity cost is 8%. What is their PV?

PV = $1,700/(1 + 0.08)8 = $1,700/1.85093 = $918.46

Ross Clark wishes to endow a chair in finance at his alma mater. The university indicated that it requires $200,000 per year to support the chair, and the endowment would earn 10% per year. To determine the amount Ross must give the university to fund the chair, we must determine the present value of a $200,000 perpetuity discounted at 10%.

PV = $200,000 ÷ 0.10 = $2,000,000

You can calculate the present value of an ordinary annuity that pays an annual cash flow equal to CF by using the following equation:

PVn = (CF/r) x 1/(1 + r)^n

loan amortization schedule

a schedule of equal payments to repay a loan. It shows the allocation of each loan payment to interest and principal

An annuity

a stream of equal periodic cash flows, over a specified time period. These cash flows can be inflows of returns earned on investments or outflows of funds invested to earn future returns

An annuity due

an annuity for which the cash flow occurs at the beginning of each period will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period

An ordinary (deferred) annuity

an annuity for which the cash flow occurs at the end of each period

Perpetuity

an annuity with an infinite life, providing continual annual cash flow. PV = CF/r

Ray Noble purchased an investment four years ago for $1,250. Now it is worth $1,520. What compound annual rate of return has Ray earned on this investment? Plugging the appropriate values into Equation 5.20, we have:

r = ($1,520 ÷ $1,250)(1/4) - 1 = 0.0501 = 5.01% per year

The following equation is used to find the interest rate (or growth rate) representing the increase in value of some investment between two time periods

r = (FVn/PV)^(1/n) - 1

Compounding more frequently than once a year

results in a higher effective interest rate because you are earning on interest on interest more frequently. As a result, the effective interest rate is greater than the nominal (annual) interest rate. Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

Present Value of Money

the current dollar value of a future amount—the amount of money that would have to be invested today at a given interest rate over a specified period to equal the future amount. It is based on the idea that a dollar today is worth more than a dollar tomorrow - used when making investment decisions

Loan amortization

the determination of the equal periodic loan payments necessary to provide a lender with a specified interest return and to repay the loan principal over a specified period. process involves finding the future payments, over the term of the loan, whose present value at the loan interest rate equals the amount of initial principal borrowed

Discounting cash flows

the process of finding present values; the inverse of compounding interest often also referred to as the opportunity cost, the required return, or the cost of capital

Time Line

used to make the right investment decision; managers need to compare the cash flows at a single point in time.