Chapter 5
DISCOUNTED CASH FLOW VALUATION
and
MULTIPLE CASH FLOWS
The basic PV and FV techniques can be extended to handle any number of cash flows.
Example: PV with multiple cash flows:
Suppose you need $500 one year from now, $1000 two years from now, and $1,500 three years from now. If you can earn 10% on your money, how much will you have to invest today to exactly cover these amounts in the future?
Example: FV with multiple CF
Suppose you invest $750 today in an account that earns 7%. In 1 year, you will invest another
$750. How much will you have at the end of 2 years?
VALUING LEVEL CASH FLOWS: ANNUITIES
Many situations involve multiple cash flows of the same amount. (consumer loans, home mortgages, etc.)
Annuity: A level stream of cash flows for a fixed period of time.
Ordinary annuity: Multiple, identical cash flows occurring at the end of each period, for a fixed number of periods.
Annuity DUE: Multiple, identical cash flows occurring at the beginning of each period, for a fixed number of periods.
Which type of annuity (an annuity due, or an ordinary annuity) would be worth more in present value terms?
PRESENT VALUE OF ANNUITY CASH FLOWS
Methods:
1) Discount each cash flow to present value and sum them up.
2) Use the Annuity Present Value Formula
=> let r = rate per period, t = # of periods
C = annuity cash flow APV = Annuity Present Value
APV = $C * [ ( , ) 1PVIF r t ]
r => APV = $C * [ ( ) ]
1 1
1
r r
t
3) (Recommended) Use TVM keys on your financial calculator.
Example:
Suppose you can make 36 monthly payments of $100, at 1.5% interest per month. What size loan can you obtain?
Loan amount = present value of 36-period annuity of $100, at 1.5% per period.
Solve for PV on your financial calculator:
N I/Y PV PMT FV
36 1.5 ? -100 0
$2,766.07
FINDING THE PAYMENT, C, GIVEN APV, rate and t:
Example:
Suppose you borrow $400 with the agreement to repay in 4 monthly installments at 1% per month. What is the amount of each payment?
You could solve for the payment amount C in the annuity present value formula:
(
( ) )
( )
( )
Or use the TVM keys on your calculator:
Another example: You wish to take a Caribbean vacation in three years, and it will cost $5,000 at that time. You have $1200 now. If you can earn 6% annually on your investments, what amount will you have to put into savings at the end of each year to have the five grand available at the end of the third year? Answer: You must deposit $1,121.62 at the end of each year.
What if you make the deposits at the beginning of each year, starting today? That is, today is the beginning of Year 1. What will be the amount of each deposit when it’s an annuity due?
Answer: Set your calculator to BEGIN mode and solve. The amount is $1,058.13.
Set your calculator back to END mode right away!
N I/Y PV PMT FV
4 1 400 ? 0
-102.51
FINDING THE NUMBER OF PAYMENTS, GIVEN APV, r, AND C
Use your financial calculator here! (Note: this can also be solved with logarithms) Example:
How many $100 payments are required to pay off a $5000 loan at 1% per period.?
FINDING THE RATE, R
Example:
A finance company offers to loan you $1000 today in return for 48 "low" monthly payments of
$32.60. What is the implicit interest rate on the loan?
This is a rate per what time period? (Per month? Per year? Per day?)
FUTURE VALUE FOR ANNUITIES
Question: How much will an annuity of $X grow to over a specified number of periods?
Method 1: Solve using the Annuity Future Value (AFV) formula
let r = rate per period, t = # of periods
C = annuity cash flow AFV = Annuity Future Value
AFV = $C * ( , ) 1]
[ r
t r FVIF
= $C * (1 ) 1]
[ r
r t
Method 2 (Recommended): Use the TVM keys on your financial calculator.
Example:
Suppose you make 20 deposits of $1000 at 10% per period. How much will be in your account at the end of the 20th period?
Suppose you make no further deposits, but you leave the funds in the account, at 10% per period, for 4 more periods. Now how much is in the account?
COMPARING RATES WITH DIFFERENT COMPOUNDING PERIODS
To compare rates that differ in their number of compounding periods, we must convert them to a comparable basis (an Effective Annual Rate).
Stated or quoted rate: The rate before considering any compounding effects.
e.g. 10% compounded quarterly (the stated or quoted rate is 10%).
The stated rate required on consumer loans is called the Annual Percentage Rate (A.P.R.).
APR is the periodic compounding rate times the number of periods in a year.
Quoted rates usually cannot be used in TVM calculations. Must use a rate that accounts for compounding: the periodic compounding rate. So change the APR to R: Divide by the number of periods in one year (m).
Key terminology:
APR = Quoted rate = Stated rate
Effective annual interest rate (E.A.R.): The rate, on an annual basis, that reflects compounding effects.
EAR is the rate per year which gives the same Future Value as a rate compounded on some other period (monthly, quarterly, etc.).
(
) Important equation
Steps:
1) Divide the quoted rate (A.P.R.) by the # of compounding periods per year (m).
2) Add 1 (to step 1) and raise to the power of m (# of compounding periods) 3) Subtract 1
Example:
10% compounded quarterly = an effective annual rate of ____%.
( (
))
Note that these two rates DO give the same FV at the end of one year, so they are effectively the same rate. Start with $100, for example:
Quarterly rate = 10% compounded quarterly
10% compounded quarterly means R = 10 ÷ 4 = 2.5% per quarter.
FV of $100 after one year (four quarters) = 100 × 1.0254 = __________
Annual rate = 10.38% per year
FV of $100 after one year = 100 × 1.10381 = ____________
Example:
Which option would you choose if you were getting a loan?
A) 10% compounded monthly B) 10.2% compounded quarterly C) 10.3% compounded annually
Compare by converting each to an annual rate (EAR), as follows:
A) EAR of 10% compounded monthly = B) EAR of 10.2% compounded quarterly = C) EAR of 10.3% compounded annually = Which rate is lower?
Example:
Which is lower, the E.A.R. of 15% compounded monthly, or the E.A.R. of 16% compounded yearly?
IMPLICATIONS FOR SOLVING FOR PV AND FV
When we solve for present or future value, we must either use the E.A.R. with years or the periodic rate (APR ÷ m) and the appropriate number of periods.
FV = PV * (1 q) m
mt ; PV = FV * [
( )
1 ] 1 q
m
mt
; PV = FV q m (1 )mt
Example:
What is the present value of $100 to be received in 2 years at 10% compounded quarterly?
Method 1: Use a quarterly rate and the total number of quarters
Method 2: Use the E.A.R. and the number of years.
RATE ADJUSTMENTS FOR
MULTIPLE COMPOUNDING PERIODS
Whenever the compounding periods differ from once per year (e.g. semiannually, quarterly, monthly, etc.) you must make the appropriate adjustments to all formulas!
t (n) = the # of years * # compounding periods per year = no. years × m R = stated rate / # compounding periods per year = APR ÷ m
With annuities, you must be sure to match the appropriate rate to your cash flows! On the calculator: N and I/Y and PMT all must use the same time period.
Example:
Suppose you borrow $10,000 to purchase a car and agree to repay the loan over 5 years of monthly payments. The A.P.R. (stated rate) is 12% per year (compounded annually). What is the amount of the payment monthly?
Example:
Suppose you have just purchased a new washer and dryer for $700. You have financed the purchase and agreed to make annual payments starting in 1 year, for the next 5 years. The interest rate on the loan is 1% per month. What is the amount of the annual payment?
Example:
Your car loan requires monthly payments of $432.86. The amount borrowed was $20,000, for five years. What is the APR on this loan? What is the EAR?
Example:
A corporation borrows $800,000. The loan will be repaid over 12 years of quarterly payments.
The interest rate on the loan is 7.2% compounded monthly. What is the amount of each quarterly payment? Answer: $25,087.68
PERPETUITIES
Perpetuity: An annuity in which the cash flows continue forever.
Perpetuity Present value: C / r , where C is the amount of the perpetuity per period and r is the periodic interest rate (as a decimal).
Example:
Suppose that starting at the end of one year from now, you will receive a perpetuity of $100 each year, forever. What is the present value of the series of cash flows at a 10% yearly interest rate?
Example:
Suppose that starting 4 years from now, you will start receiving a perpetuity cash flow of
$5,000 per year. What is the present value of these cash flows assuming a 12% yearly interest rate?
AMORTIZATION SCHEDULES
An amortization schedule is a repayment schedule for a typical consumer loan that shows:
1) the number of payments 2) the amount of each payment 3) the interest paid per period
4) the reduction in principal per period 5) the remaining loan balance
Example:
Suppose you borrow $4,000 and agree to repay the loan in five equal installments over a 5-year period. Payments are made at the end of each year. The interest rate on the loan is 10% per year.
Step 1: Solve for payment using the TVM keys.
NOTE: The payment will be the same amount in every period. It’s an annuity.
Step 2: Complete the amortization table as follows:
Period Beginning Balance
Periodic Payment
Interest Charge
Reduction in Principal
Ending Balance
Calculations:
Initial loan amount;
then ending balance from prior period.
PMT Beginning balance
× periodic interest rate PMT– interest charge Beginning balance – reduction in principal
1 $4,000.00
2 3 4 5
