Chapter 5 Time Value of Money

Annuity

A level periodic stream of cash flow. Examples include either paying out or receiving $800 at the end of each of the next 7 years.

Present Value of Ordinary Annuity

See image for formula

Single amount

A lump-sum amount either currently held or expected at some future date. Ex: receiving $1,000 today and $650 to be received at the end of 10 years.

Loan amortization schedule

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

Ordinary annuity

An annuity for which the cash flow occurs at the END of each period. Most of the time this is what we'll encounter

Mixed stream

A stream of unequal periodic cash flows that reflect no particular pattern.

Amortizing a loan actually involves creating an _________________ out of the present amount.

Annuity

How to find loan amortization on calculator

Calculator is similar to annuity, except here, you put a value into either PV or FV and try to calculate for the payment. Whereas with an annuity a value goes into the PMT and you're trying to find the PV or FV. Put in PV, N, I, solve for PMT

Present value of annuity due

Cash flows are discounted for one period less than in an ordinary annuity. See image for formula

How to find EAR in Casio FC-200V calculator

Click CNVR go to EFF

Time line

Depicts the cash flows associated with a given investment. It is a horizontal line on which time zero appears at the leftmost end and future periods are marked from left to right. Can be used to depict investment cash flows. (0 is equivalent to today, and every other number represents a new year corresponding with the relevant cash flow up to the 5th year)

Finding the interest rate of an annuity

Ex: Jan Jacobs can borrow $2,000 to be repaid in equal annual end-of-year amounts of $514.14 for the next 5 years. She wants to find the interest rate on this loan. Calculator & Spreadsheet use: Most calculators require either the PMT or the PV value to be input as a negative number to calculate an unknown interest rate on an equal-payment loan. Input PMT= -514.14 PV=2000, N=5, Solve for I Answer: 9%

T/F: the more frequently interest is compounded, the less the amount of money accumulated. This statement is true for any interest rate for any period of time.

False: it's greater

Example of future value of annuity due

Fran Abrams wishes to determine how much money she will have at the end of 5 years if she chooses annuity A, the annuity due. She will deposit $1,000 annually, at the beginning of each of the next 5 years, into a savings account paying 7% annual interest. Answer: $6,153.29

Future Value of Ordinary annuity example problem

Fran Abrams wishes to determine how much money she will have at the end of 5 years if she chooses annuity A, the ordinary annuity. She will deposit $1,000 annually, at the end of each of the next 5 years, into a savings account paying 7% annual interest.

Present value of a perpetuity

If a perpetuity pays an annual cash flow of CF, starting 1 year from now, the present value of the cash flow stream is:

Compound interest

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

quarterly compounding

Involves four compounding periods within the year. One-fourth of the stated interest rate is paid four times a year.

Continuous compounding

Involves the compounding of interest an infinite number of times per year at intervals of microseconds. See image for formula Note that: e = the exponent function so approx. 2.7183

Example of solving present value using equation:

Paul Shorter has an opportunity to receive $300 one year from now. If he can earn 6% on his investments in the normal course of events, what is the most he should pay now for this opportunity? To answer this question, Paul must determine how many dollars he would have to invest at 6% today to have $300 one year. Letting PV equal this unknown amount and using the same notation as in the future value discussion,

Example of finding interest or growth rate

Ray Noble purchased an investment 4 years ago for $1,250. Now it is worth $1,520. What compound annual rate of return has Ray earned on this investment? r = ($1,520 ÷ $1,250)(1/4) - 1 = 0.0501 = 5.01% per year Using a Calculator (Casio FC 200V): n = 4; PV= -1250; FV= 1520, Solve for I. Answer: 5.01%

Time value of money

Refers to the idea that money you have in hand today can be invested to earn a positive rate of return, producing more money tomorrow.

With annuity dues, what should you remember to do with your calculator?

Switch to begin mode

Principal

The amount of money on which interest is paid.

Discounting cash flows

The process of finding present values; the inverse of compounding interest.

T/F: Compounding more frequently than once a year results in a higher effective interest rate. As a result, the effective interest rate is greater than the nominal (annual) interest rate.

True

Mixed stream

a stream of unequal periodic cash flows that reflect no particular pattern.

Perpetuity

an annuity with an infinite life, providing continual annual cash flow. It is an annuity that never stops providing the holder with a cash flow at the end of each year. (Ex: the right to receive $500 at the end of each year, forever). Note: you will need to calculate perpetuities by hand

For any interest rate grater than zero, the future value is ____________ the he present value of $1.00

greater

The higher the interest rate the _______________ the future value

higher

What are some other terms that the discount rate is called?

- Opportunity cost - Required return - Cost of capital

Present value of mixed stream

Compute the present value of each cash flow at the specified present date and then add all the individual present values to find the total present value

Effective (true) annual rate (EAR)

The annual rate of interest actually paid or earned

Future value

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.

Future value of a mixed stream

To determine the future value of a mixed steam, we determine the future value of each cash flow at the specified future date and then add all the individual future values to find the total future value.

What is the point of the time value of money? Why find the present value or future value of an investment for example?

To make the correct investment decision, managers need to compare the cash flows at a single point in time. Typically, that point is either the end (future value) or the beginning (present value) of the investment's life.

T/F: Given a discount rate of 0 percent, the present value always equals the future value ($1.00).

True

T/F: The future value of an annuity due is always higher than the future value of an ordinary annuity.

True

T/F: For an interest rate of 0 percent, the future value always equals the present value ($1.00).

true

What are the two basic types of annuities?

1) Ordinary annuity 2) Annuity due

What are some special applications of the time value of money?

1.) Determining deposits needed to accumulate a future sum 2.) Loan amortization 3.) Finding interest or growth rates 4.) Finding the interest rate of an annuity (equal payment loan) 5.) Finding the number of periods of an annuity

What are the three basic patterns of cash flow

1.) Single amount 2.) Annuity 3.) Mixed stream

Annuities

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

Comparing ordinary annuity to an annuity due

Although the cash flows of both annuities in the figure total $5,000, the annuity due would have a higher future value than the ordinary annuity because each of its five annual cash flows can earn interest for 1 year more than each of the ordinary annuity's cash flows. In general, as will be demonstrated later in this chapter, the value (present or future) of an annuity due is always greater than the value of an otherwise identical ordinary annuity.

Annuity due

An annuity for which the cash flow occurs at the BEGINNING of each period.

Present value of annuity due example problem

Braden Company, a small producer of plastic toys, wants to determine the most it should pay to purchase (i.e. upfront) for a particular ordinary annuity. The annuity consists of cash flows of $700 at the beginning of each year for 5 years. The firm requires the annuity to provide a minimum return of 8%. Answer: $3,018.49

Present value of ordinary annuity Example problem

Braden Company, a small producer of plastic toys, wants to determine the most it should pay to purchase (i.e. upfront) for a particular ordinary annuity. The annuity consists of cash flows of $700 at the end of each year for 5 years. The firm requires the annuity to provide a minimum return of 8%. Answer: $2794.90

Cash flow signs

Cash inflows are indicated by entering positive values, and cash outflows are indicated by entering negative values. By entering the cash flows correctly, you are providing the financial calculator or electronic spreadsheet the calculation's time line.

Example of finding an unknown number of periods

Example: Ann Bates wishes to determine the number of years it will take for her initial $1,000 deposit, earning 8% annual interest, to grow to equal $2,500. Simply stated, at an 8% annual rate of interest, how many years, n, will it take for Ann's $1,000, PV, to grow to $2,500, FV n? Calculator & Spreadsheet Use: Using the calculator, we treat the initial value as the present value, PV, and the latest value as the future value, FV n . (Note: Most calculators require either the PV or the FV value to be input as a negative number to calculate an unknown number of periods. Input PV: -1000, FV: 2500, I: 8, CPT N Answer: 8.15 years

Determining deposits needed to accumulate a future sum

Example: Suppose you want to buy a house 5 years from now, and you estimate that an initial down payment of $30,000 will be required at that time. To accumulate the $30,000, you will wish to make equal annual end-of-year deposits into an account paying annual interest of 6 percent. The solution to this problem is closely related to the process of finding the future value of an annuity. You must determine what size annuity will result in a single amount equal to $30,000 at the end of year 5. On a calculator you input FV, N, I, and solve for PMT See image for formula

Semiannual compounding example

Fred Moreno has decided to invest $100 in a savings account paying 8% interest compounded semiannually. If he leaves his money in the account for 24 months (2 years), he will be paid 4% interest compounded over four periods, each of which is 6 months long.

Quarterly compounding example problem

Fred Moreno has found an institution that will pay him 8% interest compounded quarterly. If he leaves his money in this account for 24 months (2 years), he will be paid 2% interest compounded over eight periods, each of which is 3 months long.

Present value of mixed stream example problem

Frey Company, a shoe manufacturer, has been offered an opportunity to receive the following mixed stream of cash flows over the next 5 years. If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity? This situation is depicted on the following time line. If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity? Answer: $1,904.76

Present value

Is 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.

Finding interest or growth rates

It is often necessary to calculate the compound annual interest or growth rate (that is, the annual rate of change in values) of a series of cash flows. Examples include finding the interest rate on a loan, the rate of growth in sales, and the rate of growth in earnings. See Image for formula

Future value of annuity due

Remember that the cash flows of an annuity due occur at the start of the period. In other words, if we are dealing with annual payments, each payment in an annuity due comes 1 year earlier than it would in an ordinary annuity, which in turn means that each payment can earn an extra year's worth of interest. That is why the future value of an annuity due exceeds the future value of an otherwise identical ordinary annuity. See image for formula

Present value of a perpetuity example problem

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%. Answer: PV = $200,000 ÷ 0.10 = $2,000,000

Future value of a mixed stream example problem

Shrell Industries, a cabinet manufacturer, expects to receive the following mixed stream of cash flows over the next 5 years from one of its small customers. If the firm expects to earn at least 8% on its investments, how much will it accumulate by the end of year 5 if it immediately invests these cash flows when they are received? Answer: $83,608.15

Finding an unknown number of periods

Sometimes it is necessary to calculate the number of time periods needed to generate a given amount of cash flow from an initial amount. Here we briefly consider this calculation for both single amounts and annuities. This simplest case is when a person wishes to determine the number of periods, n, it will take for an initial deposit, PV, to grow to a specified future amount, FV n , given a stated interest rate, r.

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. Amortizing a loan actually involves creating an annuity out of a present amount. Lenders use loan amortization schedule to determine these payment amounts and the allocation of each payment of interest and principal. See image for formula

Graphical view of the present value (in regards to: discount rates, time, and present value of one dollar)

The figure below clearly shows that, everything else being equal: (1) the higher the discount rate, the lower the present value (2) the longer the period of time, the lower the present value. (3) Therefore, there is an inverse relationship between the discount (interest) rate and the present value, as well as the period of time and the present value. Also note that given a discount rate of 0 percent, the present value always equals the future value ($1.00). But for any discount rate greater than zero, the present value is less than the future value of $1.00.

Graphical view of Future value (in regards to: interest rates, time, and future value of one dollar)

The figure below shows how the future value depends on the interest rate and the number of periods that the money is invested. It shows that: (1) the higher the interest rate, the higher the future value, and (2) the longer the period of time, the higher the future value. (3) Therefore an increase in interest rate or an increase in the period of time increases the future value. Note that for an interest rate of 0 percent, the future value always equals the present value ($1.00). For any interest rate greater than zero, however, the future value is greater than the present value of $1.00.

Annual percentage rate (APR)

The nominal annual rate of interest, found by multiplying the periodic rate by the number of periods in one year, that must be disclosed to consumers on credit cards and loans as a result of "truth in lending laws"

Present value of a single amount

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: Note: similar to future value the present value depends largely on the interest rate and the point of time at which the amount is to be received.

Future value of an ordinary annuity

To calculate the future value of an ordinary annuity, it's possible to calculate the future value of each of the individual cash flows and then add up those figures. But that is extremely time consuming. Instead financiers use the following calculation Where: · r = interest rate · n = number of years over which the annuity is spread · CF = annual cash flow (amount of cash flow per year) · FVn =future value of ordinary annuity

The future value technique uses ____________ to find the future value of each cash flow at the end of the investment's life and then sums these values to find the investment's future value.

compounding

The present value uses ______________ · to find the present value of each cash flow at time zero and then sums these values to find the investment's value today.

discounting

With ordinary annuities, cash flow occurs at the ____________ of the period.

end

In general, the effective rate is (greater/less) than the nominal rate whenever compounding occurs more than once per year.

greater

The present value of an annuity due, is always ____________ than the present value of an ordinary annuity.

greater Because the cash flow of the annuity due occurs at the beginning of the period rather than at the end.

The longer the period of time, the ________________ the future value.

higher

An increase in interest rate or an increase in period of time _______________ the future value.

increases

There is an _____________ relationship between the discount (interest) rate and the present value, as well as the period of time and the present value.

inverse

Semiannual compounding

involves two compounding periods within the year. Instead of the stated interest rate being paid once a year, one-half of the stated interest rate is paid twice a year.

The higher the discount rate, the ____________ the present value

lower

The longer the period of time, the __________ the present value.

lower

With annuity dues cash flow occurs at the _____________ of the period.

start

Nominal (stated) annual rate

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

Annual percentage yield (APY)

this is the effective annual rate a savings product pays. For example, a savings account that pays 0.5 percent per month would have an APY of 6.17 percent [(1.005)12 - 1] truth and savings laws require, that banks quote this on their savings products.

What is the point of nominal and effective annual rates of interest?

· Both businesses and investors need to make objective comparisons of loan costs or investment returns over different compounding periods. To put interest rates on a common basis, so as to allow comparison, we distinguish between nominal and effective annual rates.

Compounding interest more frequently than annually

•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 See image for general formula Note: m = number of times compounding per year if the interest were compounded monthly, weekly, or daily, m would equal 12, 52, or 365, respectively.