Chapter 6: Discounted Cash Flow Valuation

Question 1

Based on this example, there are two ways to calculate future values for multiple cash flows:

Answer 1

(1) compound the accumulated balance forward one year at a time or
(2) calculate the future value of each cash flow first and then add them up.

Both give the same answer, so you can do it either way

Question 2

Present Value with Multiple Cash Flows Example

Answer 2

Suppose you need $1,000 in one year and $2,000 more in two years. If you can earn 9% on your money, how much do you have to put up today to exactly cover these amounts in the future? In other words, what is the present value of the two cash flows at 9%?

Answer: Total PV = 1,683.36 +917.43 = 2,600.79

Question 3

A Note on Cash Flow Timing

Answer 3

In working present and future value problems, it is implicitly assumed that the cash flows occur at the end of each period

Unless you are very explicitly told otherwise, you should always assume that this is what is meant.

Question 4

A very common type of loan repayment plan calls for the borrower to repay the loan by making a series of equal payments over some length of time

Answer 4

Almost all consumer loans (such as car loans and student loans) and home mortgages feature equal payments, usually made each month.

Question 5

Annuity

Answer 5

A level stream of cash flows for a fixed period of time.

Question 6

Suppose we were examining an asset that promised to pay $500 at the end of each of the next three years. The cash flows from this asset are in the form of a three-year, $500 annuity. If we wanted to earn 10% on our money, how much would we offer for this annuity?

Answer 6

From the previous section, we know that we can discount each of these $500 payments back to the present at 10% to determine the total present value:

Year 1 = 500 / 1.1
Year 2 = 500 / 1.1^2
Year 3 = 500 / 1.1^3

Add up all the values and it should equal:
= $1,243.43

Question 7

Annuity Present Value Equation

For when the value each period (month, year, etc) is always equal such as your car insurance, mortgage, loan repayments, etc

Answer 7

Annuity Present Value = C x {[1 - 1 / (1 + r)^t] / r}

Question 8

Present Value Factor (PVIFA)

Answer 8

(1 - Present Value Factor / r)

or

(1 - {1 - [ 1 / ( 1 + r )^t ] } / r

Question 9

Present Value Factor equation

Answer 9

1 / ( 1 + r)^t

In our example from the beginning of this section, the interest rate is 10% and there are three years involved. The usual present value factor is thus:

PVF = 1 / 1.1^3 
PVF = 0.75131

Question 10

The only way to find a value for r is to:

Answer 10

1) Use a calculator
2) Use a table
3) Trial and Error

Question 11

Annuity FV Factor Equation

Answer 11

( ( 1+ r ) ^ t - 1) / r

Question 12

Ordinary annuity

Answer 12

Remember that with an ordinary annuity, the cash flows occur at the end of each period

Question 13

Annuity Due

Answer 13

An annuity for which the cash flows occur at the beginning of the period.

Such as apartment rent is due on the 1st of each month

Question 14

Annuity Due Value =

Equation

Answer 14

Ordinary Annuity Value x (1 + r)

Question 15

This works for both present and future values, so calculating the value of an annuity due involves two steps:

Answer 15

(1) calculate the present or future value as though it were an ordinary annuity, and (2) multiply your answer by (1 + r)

Question 16

Perpetuity

Answer 16

An annuity in which the cash flows continue forever

Question 17

Consol

Answer 17

A type of perpetuity.

Question 18

PV for a perpetuity =

Equation

Answer 18

C / r

Question 19

PV for a perpetuity =

Example

Answer 19

For example, an investment offers a perpetual cash flow of $500 every year. The return you require on such an investment is 8%. What is the value of this investment? The value of this perpetuity is:

Perpetuity PV = C / r
$500 / 0.08 = $6,250

Question 20

C =

Answer 20

Cash Amount

Question 21

Growing Perpetuity

Answer 21

A constant stream of cash flows without end that is expected to rise indefinitely

Question 22

Present Value of Growing Perpetuity

Answer 22

PV = C / r - g

Question 23

There are three important points concerning the growing perpetuity formula:

Answer 23

1) The numerator. The numerator is the cash flow one period hence, not at date 0
2) The interest rate and the growth rate: The interest rate r must be greater than the growth rate g for the growing perpetuity formula to work
3) The timing assumption: Cash generally flows into and out of real-world firms both randomly and nearly continuously. However, our growing perpetuity formula assumes that cash flows are received and disbursed at regular and discrete points in time

Question 24

Growing Annuity

Answer 24

A finite number of growing annual cash flows.

Question 25

Present Value of Growing Annuity

Answer 25

PV = C / r - g [1 - (1 + g / 1 + r)^t]

Question 26

If a rate is quoted as 10 percent compounded semiannually, it means that

Answer 26

the investment actually pays 5 percent every six months

As our example illustrates, 10 percent compounded semiannually is actually equivalent to 10.25 percent per year.

Question 27

Stated interest rate (aka quoted interest rate)

Answer 27

The interest rate expressed in terms of the interest payment made each period.

It is simply the interest rate charged per period multiplied by the number of periods per year.

Question 28

Effective annual rate (EAR)

Answer 28

The interest rate expressed as if it were compounded once per year.

Question 29

Which rate it best?

Bank A: 15 percent compounded daily
Bank B: 15.5 percent compounded quarterly
Bank C: 16 percent compounded annually

Answer 29

This example illustrates two things. First, the highest quoted rate is not necessarily the best. Second, the compounding during the year can lead to a significant difference between the quoted rate and the effective rate

Question 30

Remember that the effective rate is

Answer 30

what you get or what you pay.

Question 31

Effective Annual Rate Equation

EAR Equation

Answer 31

EAR = [1 + (Quoted rate / m)^m] - 1

m = the number of times the interest is compounded during the year (quarterly, semiannually, daily, weekly, etc.)

Question 32

_______ are a very common example of an annuity with monthly payments

Answer 32

Mortgages

Question 33

To understand mortgage calculations, keep in mind two institutional arrangements

Answer 33

First, although payments are monthly, regulations for Canadian financial institutions require that mortgage rates be quoted with semiannual compounding. Thus, the compounding frequency differs from the payment frequency

Second, financial institutions offer mortgages with interest rates fixed for various periods ranging from 6 months to 25 years. As the borrower, you must choose the period for which the rate is fixed

Question 34

Annual Percentage Rate (APR)

Answer 34

The interest rate charged per period multiplied by the number of periods per year.

Question 35

Cost of borrowing disclosure regulations (part of the Bank Act) in Canada require that lenders disclose an ______ on virtually all consumer loans.

Answer 35

APR

This rate must be displayed on a loan document in a prominent and unambiguous way.

Question 36

By law, the APR is simply equal to the interest rate per period multiplied by the number of periods in a year

Answer 36

Interest Rate x Number of Periods

Question 37

For example, if a bank is charging 1.2 percent per month on car loans, then the APR that must be reported is ________. So, an APR is, in fact, a quoted or stated rate in the sense we’ve been discussing

Answer 37

1.2%x12= 14.4%

Question 38

For example, an APR of 12 percent on a loan calling for monthly payments is really 1 percent per month. The EAR on such a loan is thus:

Answer 38

= 12.6825%

.12 / 12 + 1 ^12 - 1 = 0126825

Question 39

There is a limit to the EAR

Answer 39

EAR = e^q - 1

e = 2.71828

Question 40

Principal

Answer 40

The original loan amount

Question 41

Three (3) basic types of loans

Answer 41

1) Pure discount loans
2) Interest-only loans
3) Amortized Loans

Question 42

Pure discount loans

Answer 42

Simplest form of loan

With such a loan, the borrower receives money today and repays a single lump sum at some time in the future

Pure discount loans are very common when the loan term is short, say, a year or less

Example: A one year, 10% pure discount loan, for example, would require the borrower to repay $1.10 in one year for every dollar borrowed today.

Question 43

Treasury Bills (aka T-bills)

Answer 43

A T-bill is a promise by the government to repay a fixed amount at some future time, for example, in 3 or 12 months.

T-bills are originally issued in denominations of $1 million. Investment dealers buy T-bills and break them up into smaller denominations, some as small as $1,000, for resale to individual investors.

Question 44

Interest-Only Loans

Answer 44

The borrower to pay interest each period and to repay the entire principal (the original loan amount) at some point in the future

Notice that if there is just one period, a pure discount loan and an interest-only loan are the same thing

Most bonds issued by the Government of Canada, the provinces, and corporations have the general form of an interest-only loan

Similarly, a 50-year interest-only loan would call for the borrower to pay interest every year for the next 50 years and then repay the principal.

In the extreme, the borrower pays the interest every period forever and never repays any principal.

Question 45

Amortized Loans

Answer 45

The lender may require the borrower to repay parts of the principal over time. The process of providing for a loan to be paid off by making regular principal reductions is called amortizing the loan.

For example, suppose a student takes out a $5,000, five-year loan at 9% for covering a part of her tuition fees. The loan agreement calls for the borrower to pay the interest on the loan balance each year, reducing the loan balance each year by $1,000

For example, the interest in the first year will be $5000 x .09 = $450 . The total payment will be $1,000 + 450 = $1,450. In the second year, the loan balance is $4,000, so the interest is $4000 x 0.9 = $360, and the total payment is $1,360

Question 46

APR is similar to _________ whereas EAR is similar to ________

Answer 46

Simple Interest

Compound Interest

Question 47

What is the APR is the monthly rate is 0.5%?

Answer 47

6%!! Because 12 x 0.5 = 6%

Question 48

APR Equation

Answer 48

APR = m [ (1 + EAR) ^ 1/m -1]

Question 49

Annuity future Value factor (FVIFA)

Answer 49

{ [ ( 1 + r )^t - 1] / r }