Download Corporate Finance

Corporate Finance
Lecture Four – The Time Value of
Money (Pt 2)
Learning Objectives
1.
2.
3.
4.
5.
6.
7.
8.
9.
Compute the future value of multiple cash flows.
Determine the future value of an annuity.
Determine the present value of an annuity.
Adjust the annuity equation for present value and future value for an annuity
due and understand the concept of a perpetuity.
Distinguish between the different types of loan repayments: discount loans,
interest-only loans and amortized loans.
Build and analyze amortization schedules.
Calculate waiting time and interest rates for an annuity.
Apply the time value of money concepts to evaluate the lottery cash flow
choice.
Summarize the ten essential points about the time value of money.
4.1 Future Value of Multiple Payment Streams
• With unequal periodic cash flows, treat each
of the cash flows as a lump sum and calculate
its future value over the relevant number of
periods.
• Sum up the individual future values to get the
future value of the multiple payment streams.
Figure 4.1 The time line of a nest
egg
4.1 Future Value of Multiple Payment
Streams (continued)
Example 1: Future Value of an Uneven Cash Flow Stream:
Jim deposits $3,000 today into an account that pays 10% per
year, and follows it up with 3 more deposits at the end of each
of the next three years. Each subsequent deposit is $2,000
higher than the previous one. How much money will Jim have
accumulated in his account by the end of three years?
4.1 Future Value of Multiple Payment Streams (Example 1
Answer)
FV = PV x (1+r)n
FV of Cash Flow at T0 = $3,000 x (1.10)3 = $3,000 x 1.331= $3,993.00
FV of Cash Flow at T1 = $5,000 x (1.10)2 = $5,000 x 1.210 = $6,050.00
FV of Cash Flow at T2 = $7,000 x (1.10)1 = $7,000 x 1.100 = $7,700.00
FV of Cash Flow at T3 = $9,000 x (1.10)0 = $9,000 x 1.000 = $9,000.00
Total = $26,743.00
4.1 Future Value of Multiple Payment Streams
(Example 1 Answer)
ALTERNATIVE METHOD:
Using the Cash Flow (CF) key of the calculator, enter the respective cash flows.
CF0=-$3000;CF1=-$5000;CF2=-$7000;
CF3=-$9000;
Next calculate the NPV using I=10%; NPV=$20,092.41;
Finally, using PV=-$20,092.41; n=3; i=10%;PMT=0;
CPT FV=$26,743.00
4.2 Future Value of an Annuity Stream
• Annuities are equal, periodic outflows/inflows., e.g. rent, lease, mortgage,
car loan, and retirement annuity payments.
• An annuity stream can begin at the start of each period (annuity due) as is
true of rent and insurance payments or at the end of each period,
(ordinary annuity) as in the case of mortgage and loan payments.
• The formula for calculating the future value of an annuity stream is as
follows:
FV = PMT * (1+r)n -1
r
• where PMT is the term used for the equal periodic cash flow, r is the rate
of interest, and n is the number of periods involved.
4.2 Future Value of an Annuity Stream
(continued)
Example 2: Future Value of an Ordinary Annuity Stream
Jill has been faithfully depositing $2,000 at the end of each
year since the past 10 years into an account that pays 8% per
year. How much money will she have accumulated in the
account?
4.2 Future Value of an Annuity
Stream (continued)
Example 2 Answer
Future Value of Payment One = $2,000 x 1.089 =
$3,998.01
Future Value of Payment Two = $2,000 x 1.088 =
$3,701.86
Future Value of Payment Three = $2,000 x 1.087 =
$3,427.65
Future Value of Payment Four = $2,000 x 1.086 =
$3,173.75
Future Value of Payment Five = $2,000 x 1.085 =
$2,938.66
Future Value of Payment Six = $2,000 x 1.084 =
$2,720.98
Future Value of Payment Seven = $2,000 x 1.083 =
$2,519.42
Future Value of Payment Eight = $2,000 x 1.082 =
$2,332.80
Future Value of Payment Nine = $2,000 x 1.081 =
$2,160.00
Future Value of Payment Ten = $2,000 x 1.080 =
$2,000.00
Total Value of Account at the end of 10 years $28,973.13
4.2 Future Value of an Annuity Stream
(continued)
Example 2 (Answer)
FORMULA METHOD
FV = PMT * (1+r)n -1
r
where, PMT = $2,000; r = 8%; and n=10.
FVIFA [((1.08)10 - 1)/.08] = 14.486562,
FV = $2000*14.486562  $28,973.13
USING A FINANCIAL CALCULATOR
N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13
4.2 Future Value of an Annuity Stream
(continued)
USING AN EXCEL SPREADSHEET
Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13
Rate, Nper, Pmt, PV,Type
Type is 0 for ordinary annuities and 1 for annuities due
USING FVIFA TABLE (A-3)
Find the FVIFA in the 8% column and the 10 period row; FVIFA =
14.486
FV = 2000*14.4865 = $28.973.13
FIGURE 4.3 Interest and principal growth with
different interest rates for $100-annual payments.
4.3 Present Value of an Annuity
To calculate the value of a series of equal
periodic cash flows at the current point in time,
we can use the following simplified formula:
 
1
1  
n


1

r

PV  PMT  
r



The last portion of the equation, is the
Present Value Interest Factor of an Annuity (PVIFA).
Practical applications include figuring out the nest egg needed
prior to retirement or lump sum needed for college expenses.
FIGURE 4.4 Time line of present value of annuity
stream.
4.3 Present Value of an Annuity
(continued)
Example 3: Present Value of an Annuity.
John wants to make sure that he has saved up enough
money prior to the year in which his daughter begins
college. Based on current estimates, he figures that
college expenses will amount to $40,000 per year for 4
years (ignoring any inflation or tuition increases during
the 4 years of college). How much money will John need
to have accumulated in an account that earns 7% per
year, just prior to the year that his daughter starts
college?
4.3 Present Value of an Annuity
(continued)
Example 3 Answer
Using the following equation:
  1 

1 
n 
 1 r  

PV  PMT 
r
1. Calculate the PVIFA value for n=4 and r=7%3.387211.
2. Then, multiply the annuity payment by this factor to get the PV,
PV = $40,000 x 3.387211 = $135,488.45
4.3 Present Value of an Annuity
(continued)
Example 3 Answer—continued
FINANCIAL CALCULATOR METHOD:
Set the calculator for an ordinary annuity (END mode) and
then enter:
N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45
SPREADSHEET METHOD:
Enter =PV(7%, 4, 40,000, 0, 0); Output = $135,488.45
Rate, Nper, Pmt, FV, Type
4.3 Present Value of an Annuity
(continued)
Example 3 Answer—continued
PVIFA TABLE (APPENDIX A-4) METHOD
For r =7% and n = 4; PVIFA =3.3872
PVA = PMT*PVIFA = 40,000*3.3872
= $135,488 (Notice the slight rounding error!)
4.4 Annuity Due and Perpetuity
A cash flow stream such as rent, lease, and
insurance payments, which involves equal periodic
cash flows that begin right away or at the beginning
of each time interval is known as an annuity due.
Figure 4.5 An ordinary annuity versus an annuity
due.
4.4 Annuity Due and Perpetuity
PV annuity due = PV ordinary annuity x (1+r)
FV annuity due = FV ordinary annuity x (1+r)
PV annuity due > PV ordinary annuity
FV annuity due > FV ordinary annuity
Can you see why?
Financial calculator
Mode  BGN for annuity due
Mode END for an ordinary annuity
Spreadsheet
Type” =0 or omitted for an ordinary annuity
Type = 1 for an annuity due.
4.4 Annuity Due and Perpetuity
(continued)
Example 4: Annuity Due versus Ordinary Annuity
Let’s say that you are saving up for retirement and decide
to deposit $3,000 each year for the next 20 years into an
account which pays a rate of interest of 8% per year. By
how much will your accumulated nest egg vary if you
make each of the 20 deposits at the beginning of the
year, starting right away, rather than at the end of each
of the next twenty years?
4.4 Annuity Due and Perpetuity
(continued)
Example 4 Answer
Given information: PMT = -$3,000; n=20; i= 8%; PV=0;


1 r   1
FV  PMT 
n
r
FV ordinary annuity = $3,000 * [((1.08)20 - 1)/.08]
= $3,000 * 45.76196
= $137,285.89
FV of annuity due = FV of ordinary annuity * (1+r)
FV of annuity due = $137,285.89*(1.08) = $148,268.76
4.4 Annuity Due and Perpetuity
(continued)
Perpetuity
A Perpetuity is an equal periodic cash flow
stream that will never cease.
The PV of a perpetuity is calculated by using
the following equation:
PMT
PV 
r
4.4 Annuity Due and Perpetuity
(continued)
Example 5: PV of a perpetuity
If you are considering the purchase of a consol that pays $60
per year forever, and the rate of interest you want to earn is
10% per year, how much money should you pay for the
consol?
Answer:
r=10%, PMT = $60; and PV = ($60/.1) = $600
$600 is the most you should pay for the consol.
4.5 Three Loan Payment Methods
Loan payments can be structured in one of 3
ways:
1) Discount loan
• Principal and interest is paid in lump sum at end
2) Interest-only loan
• Periodic interest-only payments, principal due at end.
3) Amortized loan
• Equal periodic payments of principal and interest
4.5 Three Loan Payment Methods
(continued)
Example 6: Discount versus Interest-only versus Amortized loans
Roseanne wants to borrow $40,000 for a period of 5 years.
The lenders offers her a choice of three payment structures:
1) Pay all of the interest (10% per year) and principal in one lump sum at the end of 5
years;
2) Pay interest at the rate of 10% per year for 4 years and then a final payment of
interest and principal at the end of the 5th year;
3) Pay 5 equal payments at the end of each year inclusive of interest and part of the
principal.
Under which of the three options will Roseanne pay the least interest and why?
Calculate the total amount of the payments and the amount of interest paid under
each alternative.
4.5 Three Loan Payment Methods
(continued)
Method 1: Discount Loan.
Since all the interest and the principal is paid at the end of 5
years we can use the FV of a lump sum equation to calculate the
payment required, i.e.
FV
= PV x (1 + r)n
FV5
= $40,000 x (1+0.10)5
= $40,000 x 1.61051
= $64, 420.40
Interest paid = Total payment - Loan amount
Interest paid = $64,420.40 - $40,000 = $24,420.40
4.5 Three Loan Payment Methods
(continued)
Method 2: Interest-Only Loan.
Annual Interest Payment (Years 1-4)
= $40,000 x 0.10 = $4,000
Year 5 payment
= Annual interest payment + Principal payment
= $4,000 + $40,000 = $44,000
Total payment = $16,000 + $44,000 = $60,000
Interest paid = $20,000
4.5 Three Loan Payment Methods
(continued)
Method 3: Amortized Loan.
n = 5; I = 10%; PV=$40,000; FV = 0;CPT PMT=$10,551.86
Total payments = 5*$10,551.8 = $52,759.31
Interest paid
= Total Payments - Loan Amount
= $52,759.31-$40,000
Interest paid = $12,759.31
Loan Type
Total Payment Interest Paid
Discount Loan $64,420.40
$24,420.40
Interest-only Loan $60,000.00
$20,000.00
Amortized Loan
$52,759.31
$12,759.31
4.6 Amortization Schedules
Tabular listing of the allocation of each loan payment
towards interest and principal reduction
Helps borrowers and lenders figure out the payoff
balance on an outstanding loan.
Procedure:
1) Compute the amount of each equal periodic payment
(PMT).
2) Calculate interest on unpaid balance at the end of
each period, minus it from the PMT, reduce the loan
balance by the remaining amount,
3) Continue the process for each payment period, until
we get a zero loan balance.
4.6 Amortization Schedules
(continued)
Example 7: Loan amortization schedule.
Prepare a loan amortization schedule for the
amortized loan option given in Example 6
above. What is the loan payoff amount at the
end of 2 years?
PV = $40,000; n=5; i=10%; FV=0;
CPT PMT = $10,551.89
Year
4.6 Amortization Schedules
(continued)
Beg. Bal
Payment
Interest
Prin. Red
End. Bal
1
40,000.00
10,551.89
4,000.00
6,551.89
33,448.11
2
33,448.11
10,551.89
3,344.81
7,207.08
26,241.03
3
26,241.03
10,551.89
2,264.10
7,927.79
18,313.24
4
18,313.24
10,551.89
1,831.32
8,720.57
9,592.67
5
9,592.67
10,551.89
959.27
9,592.67
The loan payoff amount at the end of 2 years is
$26,241.03
0
4.7 Waiting Time and Interest
Rates for Annuities
Problems involving annuities typically have 4 variables, i.e. PV
or FV, PMT, r, n
If any 3 of the 4 variables are given, we can easily solve for the
fourth one.
This section deals with the procedure of solving problems
where either n or r is not given.
For example:
– Finding out how many deposits (n) it would take to reach a retirement
or investment goal;
– Figuring out the rate of return (r) required to reach a retirement goal
given fixed monthly deposits,
4.7 Waiting Time and Interest
Rates for Annuities (continued)
Example 8: Solving for the number of
annuities involved
Martha wants to save up $100,000 as soon as
possible so that she can use it as a down
payment on her dream house. She figures that
she can easily set aside $8,000 per year and
earn 8% annually on her deposits. How many
years will Martha have to wait before she can
buy that dream house?
4.7 Waiting Time and Interest
Rates for Annuities (continued)
Example 8 Answer
Method 1: Using a financial calculator
INPUT
?
8.0
0
-8000
100000
TVM KEYS N
I/Y
PV
PMT
FV
Compute 9.00647
Method 2: Using an Excel spreadsheet
Using the “=NPER” function we enter the following:
Rate = 8%; Pmt = -8000; PV = 0;
FV = 100000; Type = 0 or omitted;
i.e.
=NPER(8%,-8000,0,100000,0)
The cell displays 9.006467.
4.8 Solving a Lottery Problem
In the case of lottery winnings, 2 choices
1) Annual lottery payment for fixed number of
years, OR
2) Lump sum payout.
How do we make an informed judgment?
Need to figure out the implied rate of return
of both options using TVM functions.
4.8 Solving a Lottery Problem
(continued)
Example 9: Calculating an implied rate of
return given an annuity
Let’s say that you have just won the state
lottery. The authorities have given you a
choice of either taking a lump sum of
$26,000,000 or a 30-year annuity of
$1,500,000. Both payments are assumed to be
after-tax. What will you do?
4.8 Solving a Lottery Problem
(continued)
Example 9 Answer
Using the TVM keys of a financial calculator, enter:
PV=26,000,000; FV=0; N=30; PMT = -1,625,000;
CPT I = 4.65283%
4.65283% = rate of interest used to determine the 30-year annuity of
$1,625,000 versus the $26,000,000 lump sum pay out.
Choice: If you can earn an annual after-tax rate of return higher than
4.65% over the next 30 years, go with the lump sum.
Otherwise, take the annuity option.
4.9 Ten Important Points about
the TVM Equation
1. Amounts of money can be added or subtracted only if they
are at the same point in time.
2. The timing and the amount of the cash flow are what
matters.
3. It is very helpful to lay out the timing and amount of the
cash flow with a timeline.
4. Present value calculations discount all future cash flow back
to current time.
5. Future value calculations value cash flows at a single point in
time in the future
4.9 Ten Important Points about
the TVM Equation (continued)
6. An annuity is a series of equal cash payments at
regular intervals across time.
7. The time value of money equation has four variables
but only one basic equation, and so you must know
three of the four variables before you can solve for
the missing or unknown variable.
8. There are three basic methods to solve for an
unknown time value of money variable:
(1) Using equations and calculating the answer;
(2) Using the TVM keys on a calculator;
(3) Using financial functions from a spreadsheet.
4.9 Ten Important Points about
the TVM Equation (continued)
9. There are 3 basic ways to repay a loan:
(1) Discount loans,
(2) Interest-only loans, and
(3) Amortized loans.
10. Despite the seemingly accurate answers from the time
value of money equation, in many situations not all the
important data can be classified into the variables of
present value, i.e., time, interest rate, payment, or future
value.
Additional Problems with Answers Problem 1
Present Value of an Annuity Due. Julie has just been
accepted into Harvard and her father is debating whether he
should make monthly lease payments of $5,000 at the
beginning of each month, on her flashy apartment or to
prepay the rent with a one-time payment of $56, 662.If Julie’s
father earns1% per month on his savings should he pay by
month or take the discount by making the single annual
payment?
Additional Problems with Answers Problem 1
(Answer)
P/Y = 12; C/Y = 12; MODE = BGN
INPUT
12
-56,662 5,000
TVM KEYS
N
I/Y
PV
PMT
OUTPUT
12.70%
0
FV
Monthly rate = 12.7%/12 = 1.0583%
If he can get 1% interest per month...then his annual rate is 12% and he can generate
$4,984.51 per month with the $56,662 it would take to pay off the rent. He is ahead $15.49
per month by making the one time payment.
INPUT
TVM KEYS
OUTPUT
12
N
12
I/Y
-56,662
PV
0
PMT
4,984.51
FV
Additional Problems with Answers Problem 2
Future Value of Uneven cash flows. If Mary deposits $4000 a
year for three years, starting a year from today, followed by 3
annual deposits of $5000, into an account that earns 8% per
year, how much money will she have accumulated in her
account at the end of 10 years?
Additional Problems with Answers Problem 2
(Answer)
Future Value in Year 10 = $4000*(1.08)9 + $4000*(1.08)8 + $4000*(1.08)7 +
$5000*(1.08)6 + $5000*(1.08)5 + $5000*(1.08)4
=$4000*1.999+$4000*1.8509+
$4000*1.7138+$5000*1.5868+
$5000*1.4693+$5000*1.3605
=$7,996+$7,403.6+$6,855.2+
$7,934+ $7,346.5+6,802.5
=$44,337.8
Additional Problems with Answers
Problem 2 (Answer) (continued)
ALTERNATIVE METHOD:
Using the Cash Flow (CF) key of the calculator, enter the respective cash
flows.
CF0=0;CF1=-$4000;CF2=-$4000;CF3=-$4000;
CF4=-$5000; CF5=-$5000; CF6=-$5000
Next calculate the NPV using I=8%; NPV=$20,537.30;
Finally, using PV=-$20,537.30; n=10; i=8%; PMT=0; CPT FV$44,338
Additional Problems with Answers Problem 3
Present Value of Uneven Cash Flows: Jane
Bryant has just purchased some equipment
for her beauty salon. She plans to pay the
following amounts at the end of the next five
years: $8,250, $8,500, $8,750, $9,000, and
$10,500. If she uses a discount rate of 10
percent, what is the cost of the equipment
that she purchased today?
Additional Problems with Answers Problem 3
(Answer)
$8,250 $8,500 $8,750 $9,000 $10,500
PV 




2
3
4
(1.10) (1.10) (1.10) (1.10)
(1.10) 5
 $7,500  $7,024.79  $6,574  $6,147.12  $6,519.67
 $33,765.58
Additional Problems with Answers Problem 4
Computing Annuity Payment: The Corner Bar & Grill is in the
process of taking a five-year loan of $50,000 with First Community
Bank. The bank offers the restaurant owner his choice of three
payment options:
1) Pay all of the interest (8% per year) and principal in one lump
sum at the end of 5 years;
2) Pay interest at the rate of 8% per year for 4 years and then a
final payment of interest and principal at the end of the 5th
year;
3) Pay 5 equal payments at the end of each year inclusive of
interest and part of the principal.
Under which of the three options will the owner pay the least
interest and why?
Additional Problems with Answers Problem 4
(Answer)
Under option 1: Principal and Interest Due at end
Payment at the end of year5 = FVn = PV x (1 + r)n
FV5 = $50,000 x (1+0.08)5
= $50,000 x 1.46933
= $73,466.5
Interest paid = Total payment - Loan amount
Interest paid = $73,466.5 - $50,000 = $23,466.50
Additional Problems with Answers Problem 4
(Answer) (continued)
Under option 2: Interest-only Loan
Annual Interest Payment (Years 1-4)
= $50,000 x 0.08 = $4,000
Year 5 payment = Annual interest payment + Principal
payment
= $4,000 + $50,000 = $54,000
Total payment = $16,000 + $54,000
= $70,000
Interest paid = $20,000
Additional Problems with Answers Problem 4
(Answer) (continued)
Option 3: Amortized Loan.
To calculate the annual payment of principal and interest we can use the
PV of an ordinary annuity equation and solve for the PMT value using n =
5; I = 8%; PV=$50,000, and FV = 0.
PMT  $12,522.82
Total payments = 5*$12,522.82 = $62,614.11
Interest paid = Total Payments - Loan Amount
= $62,614.11-$50,000
Interest paid = $12,614.11
Additional Problems with Answers Problem 4
(Answer) (continued)
Comparison of total payments and interest paid under
each method
Loan Type
Total Payment
Interest Paid
Discount Loan
$73,466.5
$23,466.50
Interest-only Loan
$70,000.00
$20,000.00
Amortized Loan
$62,614.11
$12,614.11
So, the amortized loan is the one with the lowest interest
expense, since it requires a higher annual payment, part
of which reduces the unpaid balance on the loan and thus
results in less interest being charged over the 5-year term.
Additional Problems with Answers Problem 5
Loan amortization. Let’s say that the restaurant owner in
Problem 4 above decides to go with the amortized loan option
and after having paid 2 payments decides to pay off the
balance. Using an amortization schedule calculate his payoff
amount.
Amount of loan = $50,000; Interest rate = 8%; Term = 5 years;
Annual payment = $12,522.82
Additional Problems with Answers Problem 5
(Answer)
AMORTIZATION SCHEDULE
Year
Beg. Bal.
Payment
Interest
1 50,000.00
12,522.82
2 41,477.18
12,522.82
3 2,272.53
12,522.82
4 22,331.51
12,522.82
5 11,595.21
12,522.82
Prin. Red.
End Bal.
4,000.00 8,522.82
41,477.18
3,318.17 9,204.65 32,272.53
2,581.80 9,941.02 22,331.51
1,786.52 10,736.30
11,595.21
927.62 11,595.21
0
The loan payoff amount at the end of 2 years is
$32,272.53
Figure 4.2 The time line of a $1,000-per year nest egg.