Chapter 4. The Time Value of Money and Discounted Cash Flow
Analysis
Sections 4.1, 4.2, 4.3, 4.5, and 4.6
Problems P1, P2, P4, P16
4.1 Compounding
FV = PV (1+i)n
P1.
PV = $1000
i = 10%
n = 20
FV = $1000 (1 + 10%)20 = $1000 (1 + 0.10)20 = $1000 (1.1)20 = $6,727.50
Explanation:
In this problem you have invested an initial amount of $ 1000. This amount earns interest at the rate of
10% each year. However, there are no withdrawals at anytime during the life of the investment (20
years). The interest that is earned on the initial amount of $ 1000 is added back to the outstanding
principal at the end of each year. The first year’s principal is $ 1000. The second year’s principal is
$1000 plus interest earned during the first year ($100). The principal for the 2nd year is therefore $
1100. We earn 10% interest on this amount in the 2nd year; which is $ 110 . At the end of the 2nd year
this interest of $ 110 gets added to the outstanding principal of $ 1100. The principal amount at the
beginning of the 3rd year is therefore $1210 ( $1100 + 110). This process continues till the end of 20
years. The process I have just described is the compounding process.
1000
1100
Today 1year
1210
2 years
1331
1464.1
3 years
4 years
1610.51
5 years
1771.56
……
6727.50
6 years ……
20 years
FV = PV [(1 + i) (1+ i) …………………..(1+i)]
FV = PV ( 1 + i )n
In our case PV = 1000 , i = 0.10, n = 20 years
Substituting these values in the formula gives you 6727.50 at the end of 20 years.
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Quiz
In Quick Check 4-2 on page 108, how much would the Native Americans have had in 1996 if they
could invest the money at 7%, 8%, 9%, and 10%.
4.2 The Frequency of Compounding
Federal regulations require that all rates be quoted in terms of an Annual Percentage
Rate (APR) with a given frequency of compounding.
Let m be the number of compounding periods in one year, and Y the number of years
the money is invested. Two rules for using the APR are given as:
Rule 1
i = APR/m = interest rate over the compounding period
Rule 2
n = mY = total number of compounding periods
Example 1.
A pawnbroker gives you $1000 loan for one year at 240% APR with monthly compounding.
Guess how much would you owe him at the end of the year?
You think close to $3,400 – WRONG!
Lets apply the two rules for using the APR. Because compounding is monthly we need the
monthly interest rate given as:
i = APR/m = 240%/12 = 20% = 0.20
Also, the total number of periods is given as:
n = mY = 12 X 1 = 12
Hence, FV = PV(1+i)n = $1,000 (1.20)12 = $8916.10
You owe the pawnbroker $8,916.10
EFFECTIVE ANNUAL RATE
The effective annual rate (EAR) is the equivalent interest rate if compounding was done
annually.
In the previous example, the EAR would be computed as follows:
$8916.1 = $1,000 (1 + EAR), or
2
1+ EAR = 8916.1/1000 = 8.916, or
EAR = 8.916 – 1 = 7.916 = 791.6%
In general EAR = (1 + APR/m)m - 1
P16.
First we determine the frequency of compounding (m). For monthly the frequency is 12, for annually
it is 1, and for daily it is 365.
a. EAR = (1 + 0.12/12)12 - 1 = ______
b. EAR = (1 + 0.10)1 - 1
= ______
c. EAR= (1 + 0.06/365)365 – 1 = ______
4.3 Present Value and Discounting
We know that FV = PV (1+i)n. Dividing both sides by (1+i)n we get:
PV = FV/(1+i)n
Example 2.
The present value of $1000 to be received in five years at 10% interest rate is equal to:
PV = $1000/(1.1)5 = $620.92
4.5 Multiple Cash Flows
4.5.1 TIME LINE
0
-100
1
10
2
20
3
30
4.5.2 FUTURE VALUE OF A STREAM OF CASH FLOWS
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4.5.3 PRESENT VALUE OF A STREAM OF CASH FLOWS
4.5.4 INVESTING WITH MULTIPLE CASH FLOWS
4.6 Annuities
ORDINARY ANNUITIES
An n-period Ordinary Annuity is given as:
0
1
PMT
2
PMT
3
PMT
...
...
n
PMT
PRESENT VALUE OF ORDINARY ANNUITIES
The present value of an n-period Ordinary Annuity is given as:
PVOA = PMT/(1+i) + PMT/(1+i)2 + PMT/(1+i)3 + … + PMT/(1+i)n
PVOA = PMT [1 – 1/(1+i)n]/i
FUTURE VALUE OF ORDINARY ANNUITIES
The future value of an n-period Ordinary Annuity is given as:
FVOA = PMT (1+i)n-1 + PMT (1+i)n-2 + PMT (1+i)n-3 + … + PMT/(1+i)1 + PMT
FVOA = PMT [(1+i)n - 1]/i
IMMEDIATE ANNUITIES
An n-period Immediate Annuity is given as:
0
PMT
1
PMT
2
PMT
3
...
...
4
n-1
PMT
n
PRESENT VALUE OF IMMEDIATE ANNUITIES
The present value of an n-period Immediate Annuity is given as:
PVIA = PVOA (1+i)
FUTURE VALUE OF IMMEDIATE ANNUITIES
The future value of an n-period Immediate Annuity is given as:
FVIA = FVOA (1+i)
P2.
a. This is an ordinary annuity given as:
0
1
100
2
100
3
100
...
...
20
100
i = interest rate = 10% = 0.10
FVOA = PMT [(1+i)n - 1]/i
FVOA = 100 [(1.1)20 - 1]/0.10
FVOA = 100 X 57.275 = $5,727.50
b. In this part, the future value of the annuity is given as $50,000. You have to find out what
PMT at the end of every year will be required.
0
1
PMT
2
PMT
3
PMT
...
...
i = interest rate = 10% = 0.10
FVOA = $50,000 = PMT [(1+i)n - 1]/i
50,000 = PMT [(1.1)20 - 1]/0.10 = PMT X 57.275
PMT = 50,000/57.275 = $872.98
5
20
PMT
P4.
This problem is an ordinary annuity given as:
0
1
1000
2
1000
3
1000
4
1000
i = interest rate = 10% = 0.10
PVOA = PMT [1 – 1/(1+i)n]/i
PVOA = 1000 [1 – 1/(1.1)4]/0.10
PVOA = 1000 X 3.16987 = $3,169.87
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Chapter 4. The Time Value of Money and Discounted Cash Flow
Analysis – Part 2
Sections 4.4, 4.7, 4.8, 4.9, 4.10, and 4.11
Problems P5, P6, P30, P39
ANNUITY PROBLEMS CONTINUED P5.
(a)
This problem is an ordinary annuity given as:
0
1
PMT
2
PMT
3
PMT
...
...
12 months
PMT
We know that the initial loan amount equals $1,000. This must equal the present value of all
the payments made in the next twelve months, or the PVOA, the present value of ordinary
annuity.
Lets apply the two rules for using the APR. Because compounding is monthly we need the
monthly interest rate given as:
i = APR/m = 12%/12 = 1% = 0.01
Also, the total number of periods is given as:
n = mY = 12 X 1 = 12
The present value of ordinary annuity is given as follow:
PVOA = PMT [1 – 1/(1+i)n]/i
$1,000 = PMT [1 – 1/(1+ 0.01)12]/0.01
$1,000 = PMT [11.255077]
PMT = $88.85
The PMT value obtained is the monthly installment that has to be paid each month for 12
months ($88.85). The PMT amount includes part of the principal as well interest on the
outstanding principal. Remember the principal you owe the bank declines each month over
the life of the loan (See Loan Amortization – Section 4.8).
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(b)
The principal amount of the loan or the PV is $1000. We know that the monthly payment on
the loan is $88.85. For 12 months this amounts to $88.85 X 12 = $1,066.20.
Hence, the total interest over the 12 months must equal $1,066.20 - $1,000 = $66.20
P6.
(a)
This problem uses the same principles described in problems 4 and 5, above. Here again we
are paying fixed installments on a loan over a period of time. The loan is for 25 years and
payment is monthly. Therefore, using the two rules of APR we have:
i = APR/m = 16%/12 = 1.333% = 0.01333
Also, the total number of periods is given as:
n = mY = 12 X 25 = 300
PVOA = PMT [1 – 1/(1+i)n]/i
$100,000 = PMT [1 – 1/(1+ 0.01333)300]/0.01333
$100,000 = PMT [ 0.981191931 / 0.01333 ]
$100,000 = PMT [73.591234]
PMT = $1,358.89
(b)
In this part we know that monthly PMT = $1,000. The unknown variable is the PV OA or the
amount of loan you can get today, if you make monthly payments of $ 1000 for 25 years at an
annual interest rate of 16 %.
Using the same formula again and substituting the known variables ‘n’ = 300, ‘i’= .01333 and
PMT = $1,000 we get a PVOA of $73,590.
PVOA = $1,000 [1 – 1/(1+0.01333)300]/0.01333
PVOA = $1,000 [1 – 0.018808068]/0.01333
PVOA = $1,000 [0.981191931 / 0.01333]
PVOA = $73,590
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(c)
Here we need to calculate the number of time periods ‘n’ (months) given the values of PMT ($1500),
PVOA ($100,000) and ‘i’ (1.3333% per month).
PVOA = PMT [1 – 1/(1+i)n]/i
$100,000 = 1,500 [ 1 – 1/( 1+0.01333) n ] / 0.01333
0.888866666 = 1 – 1/( 1+0.01333) n
1/( 1+0.01333) n = 0.111133334
Taking reciprocals, we get:
( 1.01333) n = 8. 998200306
Mathematical rule: log a n = n log a
Taking natural logs on both sides
n log 1.01333 = log 8.998200306
n (0.013244897) = 2.197025
n = 165.91 or 166 approximately.
(d)
Here we need to calculate the interest rate given all the other values. Note that the interest rate
you get will be a monthly rate since ‘n’ is number of months.
PVOA = PMT [1 – 1/(1+i)n]/i
$100,000 = $1,500 [ 1 – 1 / (1+i )300 ] / i
This is a non-linear equation that cannot be solved analytically for i. However, the “Solver”
function in Excel “Tools” can solve it.
1
2
3
4
A
1500
0.01
300
B
=A1*(1-1/(1+A2)^A3)/A2
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We want to solve for the value of “i” which will make the right hand side of the following
equation equal to the left-hand-side value of $100,000.
$100,000 = $1,500 [ 1 – 1 / (1+i )300 ] / i
Enter the inputs to the right hand side formula into cells A1, A2, and A3. You may put any
reasonable initial guess in cell A2 for “i”, say 0.01. Then do the following steps:
Step 1:
Step 2:
Step 3:
Step 4:
Go into Tools and then Solver.
SET TARGET CELL: B1
EQUAL TO: 100000
BY CHANGING CELL: A2
Then hit “Solve” and you should get the right answer in cell A2.
1
2
3
4
A
1500
0.014818
300
B
100000
The answer is 0.014818 or 1.482% per month. Hence, the APR is 1.482% X 12 = 17.782%
P30.
For alternative 1, the PV of the loan amount is $20,000. This entire amount is financed at 4%
APR for 36 months with monthly payments. The concept is the same as that of annuity, which
we have dealt with in earlier problems. Therefore, using the two rules of APR we have:
i = APR/m = 4%/12 = 0.333% = 0.00333
Also, the total number of periods is given as:
n = mY = 12 X 3 = 36
PVOA = PMT [1 – 1/(1+i)n]/i
Given the terms of the loan we need to calculate the PMT (monthly payment).
$20,000 = PMT [(1 – 1/(1 +0.003333 ) 36 ]/0.00333
PMT = 590.48
For alternative 2, we get $1,500 cash back upfront. The effective loan amount to be financed
is $18,500, which is financed at an APR of 9.5%.
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i = APR/m = 9.5%/12 = 0.7917% = 0.007917
Hence,
$18,500 = PMT [(1 – 1/(1 +0.007917) 36 ]/0.007917
PMT = $ 592.61.
Since we have a lower payment per month when we go with alternative 1, we choose
alternative 1.
4.4 Alternative Discounted Cash Flow Decision Rules
NPV Rule:
Definition: The NPV is the present value of all current and future cash inflows
minus the present value of all current and future cash outflows.
Accept a project if the NPV is positive.
Reject a project if the NPV is negative.
For the NPV calculation of any project we use the opportunity cost of capital as the
relevant discount rate.
The opportunity cost of capital or the discount rate is the rate of interest we could have earned
elsewhere in projects of similar risk, if we did not invest in the project under evaluation.
Example 1.
If you could buy a piece of land for $10,000 and after five years sell it for $20,000, should you invest
in the land if the discount rate is 8%? How does your answer change by using alternative discount
rates of 5%, 10%, and 15%?
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IRR RULE
Intuitive Definition: Internal rate of Return (IRR) is the rate of return that you earn on
your investment.
Example 2.
Compute the rate of return on your investment in land in Example 1?
Technical Definition: The Internal Rate of Return (IRR) on an investment is the
discount rate that makes the present value of current and future cash inflows equal to
the present value of current and future cash outflows. It is the discount rate at which
the NPV=0.
Accept a project if the IRR > Discount Rate
Reject a project if the IRR < Discount Rate
When you have to choose between several alternatives, choose the investment with the
highest NPV.
4.7 Perpetual Annuities
A perpetuity (perpetual annuity) is a stream of cash flows that lasts forever.
PV of a Level Perpetuity = C / i
Where C is the periodic payment and I is the interest rate expressed as a decimal
fraction.
4.7.1 Investing in Preferred Stock
Price = Annual Dividend / Yield on Preferred Stock
Yield on Preferred Stock = Annual Dividend / Price
4.7.2 Investing in Common Stock
Wait until Chapter 9
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4.8 Loan Amortization
See Table 4.6 and Table 4.7 (pages 124-125)
The interest paid decreases and the principal paid increases with time.
4.9 Exchange Rates and Time Value of Money
Rule to be observed while making financial decisions with different currencies:
In any time value of money calculation, the cash flows and the interest rate must
be denominated in the same currency
P39.
(a)
Option 1
If you invest $ 1 today in dollar denominated bonds you will get $ 1.07 one year from now.
Options 2
You could convert $1 into 0.66667 pounds and invest that amount in pound – denominated
bonds to have 0.726667 pounds 1 year from now ( pound interest rate is 9 %) .
Next you need to convert this amount after a year back to US$. The expected exchange rate 1year from now is $1.4 per pound. So you would get 0.726667 X 1.4 = $1.017.
Clearly you get back less if you use the 2nd option. So you would choose to invest in US
dollar bonds.
(b)
In order to break even, the 0.726667 pounds you received after a year ( if you invested in
pound denominated bonds using option 2), should get you US$ 1.07 when you convert it back
to US dollars 1 year from now. For this the exchange rate needs to be 1.07/0.726667 or
$1.4725 per pound.
For a future exchange rate (1 year from now) lower than $1.4725 per pound, the US dollar
would give a better return.
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4.10 Inflation and Discounted Cash Flow Analysis
1 + Real Rate = ( 1 + Nominal Interest Rate ) / ( 1 + Rate of Inflation)
This can be approximately given as:
Real Rate = Nominal Rate – Inflation Rate
If cash flows are real cash flows, use real rate for discounting
If cash flows are nominal cash flows, use nominal rate for discounting
4.11 Taxes and Investment Decisions
After-tax rate = ( 1 – tax rate) * Before tax Interest rate
Invest so as to maximize the net present value of your after-tax cash flows
More on this in Chapter 6
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