Chapter 6: Discounted Cash Flow Valuation Flashcards
Based on this example, there are two ways to calculate future values for multiple cash flows:
(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
Present Value with Multiple Cash Flows Example
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
A Note on Cash Flow Timing
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.
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
Almost all consumer loans (such as car loans and student loans) and home mortgages feature equal payments, usually made each month.
Annuity
A level stream of cash flows for a fixed period of time.
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?
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
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
Annuity Present Value = C x {[1 - 1 / (1 + r)^t] / r}
Present Value Factor (PVIFA)
(1 - Present Value Factor / r)
or
(1 - {1 - [ 1 / ( 1 + r )^t ] } / r
Present Value Factor equation
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
The only way to find a value for r is to:
1) Use a calculator
2) Use a table
3) Trial and Error
Annuity FV Factor Equation
( ( 1+ r ) ^ t - 1) / r
Ordinary annuity
Remember that with an ordinary annuity, the cash flows occur at the end of each period
Annuity Due
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
Annuity Due Value =
Equation
Ordinary Annuity Value x (1 + r)
This works for both present and future values, so calculating the value of an annuity due involves two steps:
(1) calculate the present or future value as though it were an ordinary annuity, and (2) multiply your answer by (1 + r)
Perpetuity
An annuity in which the cash flows continue forever
Consol
A type of perpetuity.
PV for a perpetuity =
Equation
C / r
PV for a perpetuity =
Example
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
C =
Cash Amount
Growing Perpetuity
A constant stream of cash flows without end that is expected to rise indefinitely
Present Value of Growing Perpetuity
PV = C / r - g
There are three important points concerning the growing perpetuity formula:
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
Growing Annuity
A finite number of growing annual cash flows.
Present Value of Growing Annuity
PV = C / r - g [1 - (1 + g / 1 + r)^t]
If a rate is quoted as 10 percent compounded semiannually, it means that
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.
Stated interest rate (aka quoted interest rate)
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.
Effective annual rate (EAR)
The interest rate expressed as if it were compounded once per year.
Which rate it best?
Bank A: 15 percent compounded daily
Bank B: 15.5 percent compounded quarterly
Bank C: 16 percent compounded annually
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
Remember that the effective rate is
what you get or what you pay.
Effective Annual Rate Equation
EAR Equation
EAR = [1 + (Quoted rate / m)^m] - 1
m = the number of times the interest is compounded during the year (quarterly, semiannually, daily, weekly, etc.)
_______ are a very common example of an annuity with monthly payments
Mortgages
To understand mortgage calculations, keep in mind two institutional arrangements
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
Annual Percentage Rate (APR)
The interest rate charged per period multiplied by the number of periods per year.
Cost of borrowing disclosure regulations (part of the Bank Act) in Canada require that lenders disclose an ______ on virtually all consumer loans.
APR
This rate must be displayed on a loan document in a prominent and unambiguous way.
By law, the APR is simply equal to the interest rate per period multiplied by the number of periods in a year
Interest Rate x Number of Periods
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
1.2%x12= 14.4%
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:
= 12.6825%
.12 / 12 + 1 ^12 - 1 = 0126825
There is a limit to the EAR
EAR = e^q - 1
e = 2.71828
Principal
The original loan amount
Three (3) basic types of loans
1) Pure discount loans
2) Interest-only loans
3) Amortized Loans
Pure discount loans
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.
Treasury Bills (aka T-bills)
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.
Interest-Only Loans
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.
Amortized Loans
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
APR is similar to _________ whereas EAR is similar to ________
Simple Interest
Compound Interest
What is the APR is the monthly rate is 0.5%?
6%!! Because 12 x 0.5 = 6%
APR Equation
APR = m [ (1 + EAR) ^ 1/m -1]
Annuity future Value factor (FVIFA)
{ [ ( 1 + r )^t - 1] / r }
