1 chapter 5 discounted cash flow valuation. 2 overview important definitions finding future value of...
TRANSCRIPT
1
Chapter 5
Discounted Cash Flow Valuation
2
Overview
Important Definitions Finding Future Value of an Ordinary Annuity Finding Future Value of Uneven Cash Flows Finding Present Value of an Ordinary Annuity Finding Present Value of Uneven Cash Flows Valuing Level Cash Flows: Annuities and
Perpetuities Comparing Rates: The Effect of
Compounding Periods Loan Types and Loan Amortization
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Important Definitions
Annuity: A level stream of cash flows for a fixed period of time.
Ordinary Annuity: Cash flows take place at the end of each period.
Annuity Due: Cash flows takes place at the beginning of each period.
Perpetuity: A level stream of cash flows forever. Note: For now computations will be shown using
formulations and calculator entries. I recommend focusing on calculator entries because it is much more efficient in getting answers. Formulations are provided for demonstration purposes. You should also review Excel file to see how excel functions can be used.
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Finding Future Value of an Ordinary Annuity
20.210,12
10.0
110.01000,2
11 5
r
rCFV
t
t
N I/Y P/Y PV PMT FV MODE
5 10 1 0 2,000 -12,210.20
5
Finding Future Value of Uneven Cash Flows
Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?
FV can be computed for each cash flow at a common future period (in this case 5 years from today)
FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97
100
0 1 2 3 4 5
300
136.05
349.92
485.97
6
Finding Future Value of Uneven Cash Flows – Continued
Calculator solution of the previous problem would involve determining FV of each payment in five years from today and then adding them
Total FV = 136.05 + 349.92 = 485.97
N I/Y P/Y PV PMT FV MODE
4 8 1 -100 0 -136.05
2 8 1 -300 0 -349.92
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Finding Present Value of an Ordinary Annuity
37.212,4
06.006.01
11
000,11
11 5
rr
CPVt
t
N I/Y P/Y PV PMT FV MODE
5 6 1 -4,212.37
1,000 0
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Finding Present Value of Uneven Cash Flows
Find the PV of each cash flow and add them Year 1 CF: 200 / (1.12)1 = 178.57 Year 2 CF: 400 / (1.12)2 = 318.88 Year 3 CF: 600 / (1.12)3 = 427.07 Year 4 CF: 800 / (1.12)4 = 508.41 Total PV = 178.57 + 318.88 + 427.07 +
508.41 Total PV = 1,432.93
Years 0 1 2 3 4Cash Flow 200 400 600 800
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Finding Present Value of Uneven Cash Flows – Continued
0 1 2 3 4
200 400 600 800178.57
318.88
427.07
508.41
1,432.93
0 1 2 3 4
200 400 600 800178.57
318.88
427.07
508.41
1,432.93
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Finding Present Value of Uneven Cash Flows – TI BA II PLUS
CF
CF0 0 ENTER
C01 200 ENTER F01 1.00 ENTER
C02 400 ENTER F02 1.00 ENTER
C03 600 ENTER F03 1.00 ENTER
C04 800 ENTER F04 1.00 ENTER
NPV 12 ENTER CPT 1,432.93
When using CF make sure to clear previous entries CF 2ND CLR WORK
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Example You are offered the opportunity to put some
money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%?
This cash flow set has two parts PV of annuity payments 39 years from today PV of a single payment in 39 years See the timeline below
0 1 2 … 39 40 41 42 43 44
0 0 0 … 0 25K 25K 25K 25K 25K
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Example – Continued
PV of annuity 39 years from today
PV of single payment in 39 years
41.119,90
12.012.01
11
000,251
11 5
rr
CPVt
t
71.084,1
12.01
41.119,9039
tPV
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PV of annuity 39 years from today
PV of single payment in 39 years
Example – Continued – TI BA II PLUS
N I/Y P/Y PV PMT FV MODE
5 12 1 -90,119.4
1
25,000
0
N I/Y P/Y PV PMT FV MODE
39 12 1 -1,084.7
1
0 90,119.41
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Example – Continued – TI BA II PLUS
CF
CF0 0 ENTER
C01 0 ENTER F01 39.00 ENTER
C02 25,000 ENTER F02 5.00 ENTER
NPV 12 ENTER CPT 1,084.71
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Finding Future or Present Value of an Annuity Due
Annuity Due cash flows take place at the beginning of each period – this means that there is a payment now
The immediate payment requires that we account its impact on FV and PV correctly
The basic adjustments Treat the annuity as if it were an ordinary annuity and determine
FV or PV Then multiply the answer by (1 + r) FV or PV of Annuity Due = (FV or PV of Ordinary Annuity) x (1 + r)
If you are computing PMT then the adjustment is slightly different
PMT of Annuity Due = (PMT of Ordinary Annuity) / (1 + r) This is because immediate payment with annuity due reduces
amount borrowed leading to lower overall payment I suggest using calculator settings for PMT, N, and I/Y
computations
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Finding Future or Present Value of an Annuity Due – Continued
Consider the above annuity due. This is a five year annuity due. When you compute FV it is computed at the end of year 5. Assume 10% rate.
95.667,110.0110.0
10.01
11
40011
11 5
r
rr
CPVt
t
24.686,210.0110.0
110.014001
11 5
r
r
rCFV
t
t
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Finding Future or Present Value of an Annuity Due – TI BA II PLUS
To compute FV and PV of an annuity due you need to change MODE to BGN
2ND BGN 2ND SET 2ND QUIT Once complete you will see BGN on the display To return to END follow the same steps END mode is not displayed
N I/Y P/Y PV PMT FV MODE
5 10 1 -1,667.9
5
400 0 BGN
N I/Y P/Y PV PMT FV MODE
5 10 1 0 400 -2,686.24 BGN
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Example
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
N I/Y P/Y PV PMT FV MODE
30 5 1 -5,124,150.2
9
333,333.33
0
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Non Annual Interest Earnings (Compounding of Interest Rate) and Payments
Remember that FVt = PV0 x (1 + r)t
When dealing with non annual interest and payment you should make two adjustment
Interest rate should be periodic Number of periods should reflect the total number of periods
given number of years and frequency of interest earnings in a year
Formulations should be adjusted as follows If “m” is the number of compounding in a year then where
you see “r” divide it by “m” and multiply “t” by “m” This assumes that “t” is the number of years
FVt = PV0 x (1 + r/m)t X m
Excel adjustment are same as formulations that can be made in a function dialog box or data cells
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Non Annual Interest Earnings (Compounding of Interest Rate) and Payments – Continued
FVt = PV0 x (1 + r/m)t X m
Entering the above equation into TI BA II PLUS Method 1:
Keep P/Y = 1 (so is C/Y = 1) N = t X m (Number of periods) I/Y = r/m (Periodic interest rate) Rest of the information is entered as before
Method 2: Change P/Y = m (so is C/Y = m) N = t X m (Number of periods) I/Y = r (Annual interest rate) Rest of the information is entered as before Note that P/Y has to be checked for consistency every time
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ExampleBuying a Car
Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly. If you take a 4-year loan, what is your monthly payment?
26.488
12
08.012
08.01
11
000,20
1
11
48124
CC
m
rm
r
CPV
mt
t
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ExampleBuying a Car – TI BA II PLUS
Note the adjustment
N I/Y P/Y PV PMT FV MODE
48 8 12 -20,000 488.26 0
2ND P/Y12 ENTER2ND QUIT
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ExampleCredit Card
Note: Finding interest rate and number of periods in annuity formulations are relatively time consuming. By this point you should be more comfortable with your calculator. From now on, solutions will be shown based on calculator entries.
You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month (18% per year compounded monthly). How long will you need to pay off the $1,000.
N I/Y P/Y PV PMT FV MODE
93.11
18 12 -1,000 20 0
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ExampleBorrowing Money from Unlikely Resources
Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the annual interest rate compounded monthly?N I/Y P/Y PV PMT FV MODE
60 9.00 12 -10,000 207.58 0
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Present Value of a Perpetuity Suppose you will receive a fixed payment every
period (month, year, etc.) forever. This is an example of a perpetuity
You can think of a perpetuity as an annuity that goes on
r
C PV
r
rr
CPV
t
t
toreducesequation PV theTherefore
zero. approaches 1
1 term then thelarge gets t If
1
11
:as PVfor equation theknow We
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ExamplePerpetuity
What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?
PV = $10,000 / 0.08 = $125,000
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Effective Annual Rate (EAR) vs. Annual Percentage Rate (APR)
Effective Annual Rate (EAR) This is the actual rate paid (or received) after accounting for
compounding that occurs during the year If you want to compare two alternative investments with different
compounding periods you need to compute the EAR and use that for comparison.
Sometimes it is also called Annual Percentage Yield (APY) or Effective Rate
Annual Percentage Rate (APR) This is the annual rate that is quoted by law By definition APR = periodic rate times the number of periods per
year Consequently, to get the periodic rate we rearrange the APR
equation: Periodic rate = APR / number of periods per year Sometimes it is also called Stated Rate, Quoted Rate or Nominal
Rate You should NEVER divide the effective rate by the number of
periods per year – it will NOT give you the periodic rate
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Rate Conversions
APR to EAR
EAR to APR
Calculator convention NOM = APR EFF = EAR C/Y = m
11 m
m
APR EAR
1
11 - m EAR m APR
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ExampleRate Conversion
You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?
Compare EAR and choose the larger one since you are investing. If you were borrowing then you would choose the lower EAR alternative
5.25% compounded daily 2nd ICONV NOM 5.25 ENTER (EFF) C/Y 365 ENTER (EFF) CPT 5.39
5.30% compounded semiannually 2nd ICONV NOM 5.30 ENTER (EFF) C/Y 2 ENTER (EFF) CPT 5.37
Choose 5.25% compounded daily
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ExampleRate Conversion – Continued
Let’s verify the choice. Suppose you invest $100,000 in each account. How much will you have in each account after 5 years? Daily Compounded Account:
Semiannually Compounded Account:
You have more money in the first account.
N I/Y P/Y PV PMT FV MODE
1,825 5.25 365 -100,000 0 130,015.19
N I/Y P/Y PV PMT FV MODE
10 5.30 2 -100,000 0 129,894.13
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Amortized Loan with Fixed Payment
Each payment covers the interest expense plus reduces principal
Consider a 4-year loan with annual payments. The interest rate is 8% and the principal amount is $5,000.
N I/Y P/Y PV PMT FV MODE
4 8 1 -5,000 1,509.60 0
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Amortization Table
Computations: Interest Paid = Beginning Balance X Interest Rate Principal Paid = Total Payment – Interest Paid Ending Balance = Beginning Balance – Principal Paid
Year Beginning Balance
Total Payment
Interest Paid
Principal Paid
Ending Balance
1 5,000.00 1,509.60 400.00 1,109.60 3,890.40
2 3,890.40 1,509.60 311.23 1,198.37 2,692.02
3 2,692.02 1,509.60 215.36 1,294.24 1,397.78
4 1,397.78 1,509.60 111.82 1,397.78 0.00
Totals 6,038.42 1,038.41 4,999.99
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Amortization Table – Continued
If you want to determine the outstanding balance of the loan after 2 years: 2nd AMORT 2 ENTER 2 ENTER
This will allow you to see loan information at that point in time.
You can change P1 and P2 to get the data for the specified payment range