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Transcript of FI3300 Corporate Finance Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance 1.
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FI3300Corporate Finance
Spring Semester 2010
Dr. Isabel TkatchAssistant Professor of Finance
1
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Learning Objectives
☺ Calculate the PV and FV in multi-period multi-CF time-value-of-money problems:
☺ General case☺ Perpetuity☺ Annuity
☺ Find the rate of return in multi-period (multi-CF) time-value-of-money problems
☺ Adjusting the rate of return:☺ The frequency of compounding☺ Inflation
☺ Loan amortization schedule
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Example
You plan to spend the next four summers abroad. The first summer trip, which is exactly one year away, will cost you $22,000, the second, $27,500, the third, $33,000 and the fourth $35,000.
1. How much should you deposit in your account today (pays 6% interest per annum) so that you will have exactly enough to finance all the trips?
2. If you borrow the money to finance those trips (at 6% interest per annum) and plan to repay it in 5 years when you get your trust fund, how much do you expect to pay?
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Present Value (PV) of a CF Stream
CF1 CF2 … CFt … CFT |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T
CFt = the cash flow on date t (end of year t)
r = the cost of capital for one period (one year)t = date index, t = 1,2,3,…,TT = the number of periods (number of years)
1 21 2
... ...(1 ) (1 ) (1 ) (1 )
t Tt T
CFCF CF CFPV
r r r r
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Future Value (FV) of a CF Stream
CF1 CF2 … CFt … CFT |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T
Step 1: calculate the present value of the CF stream
PV FV |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T
Step 2: use the PV-FV formula to calculate the future value of the CF stream: (1 )TFV PV r
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Perpetuity
You invest in a project that is expected to pay $1,200 a month, at the end of the month, forever. The monthly cost of capital is 1%. What is the present value of this CF stream?
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Present Value (PV) of a Perpetuity
CF CF … CF … |------------|-----------|--------- … -----|----- … --------> time0 1 2 t
CF = the SAME CF at the end of EVERY period (year)
First CF (start date): end of the first period (date 1)
We get the same CF FOREVER (T = , infinity)
r = the cost of capital for one period (one year) CF
PVr
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Perpetuity examples
1. Suppose the value of a perpetuity is $38,900 and the discount rate is 12% per annum. What must be the annual cash flow from this perpetuity?
2. An asset that generates $890 a year forever is priced at $6,000. What is the required rate of return?
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Annuity
You consider investing in real estate. You expect the property to yield (i.e., generate) rent CFs of $18,000 a year for the next twenty years, after which you will be able to sell it for $250,000.
Your required rate of return is 12% per annum. What is the maximum amount you’d pay for this CF stream?
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Present Value (PV) of an Annuity
CF CF … CF … CF |------------|-----------|--------- … -----|----- … --------|----> time0 1 2 t T
CF = the SAME CF at the end of EVERY period (year)
First CF (start date): end of the first period (date 1)
Last CF (end date): end of the last period (date T)
T = the number of periods (number of years)
r = the cost of capital for one period (one year)
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1
TCF
PVr r
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Annuity, find the FV
You open a savings account and deposit $20,000 today. At the end of each of the next 15 years, you deposit $2,500.
The annual interest rate is 7%. What will be the account balance 15 years from now?
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Annuity, find the PMT
You are trying to borrow $200,000 to buy a house on a conventional 30-year mortgage with monthly payments. The monthly interest rate on this loan is 0.70%. What is the monthly payment on the loan?
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Annuity, find the PMT: Challenge
You plan to retire in 30 years. Then you will need $200,000 a year for 10 years (first withdrawal at t=31). Ten years later you expect to go to a retirement home where you will stay for the rest of your life. To enter the retirement home, you will have to make a single payment of $1,000,000. You can start saving for your retirement in an account that pays 9% interest a year. Therefore, starting one year from now (end of the first year: t =1), you will make equal yearly deposits into this account for 30 years. In 30 years (on date t=30), you expect a deposit of $500,000 to your retirement account from your cash value insurance policy. What should be your yearly deposit into the retirement account?
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Adjusting the rate of return
☺The frequency of compounding:☺ Quoted (stated) rate☺ Effective rate☺ Always use the effective annual rate to discount
annual CFs, effective monthly rate to discount monthly CFs etc.
☺The case of inflation:☺ Nominal rate☺ Real rate☺ Always use the nominal rate to discount nominal
CFs and the real rate to discount real CFs.
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Effective to Effective: Example
The annual interest rate is 8%.What is the 2-year rate of return on $1?
2 1
(1 ) (2 )
2
( ,1 ) ( ,2 )
2
( ,2 )
( ,2 )
1 1
$1 1 $1 1
1 0.08 1 1.1664
1.16
year year
effective year effective year
ettective year
ettective year
FV PV r PV r
FV r r
r
r
64 1 0.1664 16.64%
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Effective to Effective: Formula
r(effective, 1-period) = 1-period effective rate
The return on $1 invested for 1 period
r(effective, n-period) = n-period effective rate
The return on $1 invested for n periods
( , ) ( ,1 )1 1n
effective n period effective periodr r
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Effective to Effective: n>1The effective monthly rate is 1%, what is the effective annual rate?
Since the effective monthly rate is known and there are n=12 months in one year
( , )
12
( , )
12
( , )
12
( ,12 ) ( ,1 )1
1% 0.01
1 1 0.01
1 0.01 1 0.1268 12.68
1
%
effective months effective m
effective monthly
effective annual
effective annua
o t
l
n hr r
r
r
r
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Effective to Effective: n>1The effective monthly rate is 1%, what is the effective quarterly rate?
Since the effective monthly rate is known and there are n=3 months in one quarter
3
( ,3 ) ( ,1
( , )
3
( , )
3
)
( , )
1% 0.01
1 1 0.01
1 0.01 1 0.0303 3.0
1 1
3%
effective months effective mon
effective monthly
effective quarterly
e
t
ffective quart
h
erly
r
r
r
r r
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Effective to Effective: n<1The effective monthly rate is 1%, what is the effective weekly rate?
Since the effective monthly rate is known and there are 4 weeks in one month or (1/4)=0.25 months in one week
4
( ,4 ) ( ,1 )
1
( , )
0.25
(
4( , ) ( , )
, )
1
1% 0.01
1 0.01 1 0.00
1
1
24 . %
1
9 0 249
effective weeks effective week
effective
effective monthly
effective weekly
weekly effective monthly
r r
r
r
r
r
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Effective to Effective: n<1The effective annual rate is 12%, what is the effective rate for 10 months?
Since the effective annual rate is known and there are (10/12)=0.8333 years in a period of 10 months we get
1012
1012
( , )
( ,10 )
0
( ,10 ) ( ,
.8333
( ,
12
0
)
1 )
10% 0.1
1 1 0.1
1 0.1 1 0.0827 8
1
.27%
1
effective annual
effective months
effective months
effective months effective monthsr r
r
r
r
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Quoted to Effective: Example
The offer: a credit card with 9% APR (annual percentage rate). 9% is the quoted (stated) annual interest rate.
The convention: since credit card payments are monthly, the frequency of compounding is monthly or m=12 times in one year.
The terminology: 9% a year, compounded monthly.
The problem: 9% is NOT the effective annual interest rate (9% is not the annual rate of return).
To compare this rate to other offers or discount annual CFs we need the effective rate.
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Example Continued
Since the frequency of compounding is monthly, start by finding the effective monthly rate:
( , )( ,1 )
( ,12
( , ) ( ,12 )
( ,1
)
)
)( ,1
9% 0.09
0.090.
1
0075 0.75%12
2
quoted m p
quoted annual quoted m
eriodseffective period
quoted monthseffectiv
onth
effecti
e month
ve month
rr
m
rr
r r
r
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Example Continued
Now, use the effective-to-effective formula to find any other effective interest rate.What is the effective annual interest rate?(Remember: there are n=12 months in one year)
12
(
( , )
, )
12
( , )
( , )
( , )
0.0075 0.75%
1 1 0.0075 1.093
1 1
8
1.0938 1 0.0938 9.38%
effective monthly
effective annual
effective
effective annual effective m
annu
on hly
al
t
r
r
r r
r
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Example Continued
What is the effective quarterly interest rate? (Remember: there are n=3 months in one quarter)
( , )
3
( , )
( ,
3
( , ) ( , )
)
0.0075 0.75%
1 1 0.0075 1.0227
1.0227 1 0.0227 2.2
1
7%
1 effective quarterly effective monthly
effective monthly
effective quarterly
effective annual
r
r
r
r r
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Example Continued
What is the effective weekly interest rate? (Remember: there are 4 weeks in one month or n=1/4=0.25 months in one week)
0.25
( , ) ( ,
( , )
0.25
( , )
( ,
)
)
0.0075 0.75%
1 1 0.0075 1.0019
1.0019 1 0.0019 0.19
1 1
%
effective monthly
effective weekly
effectiv
effective weekly effective mont
e ann
ly
ual
hr r
r
r
r
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Quoted to Effective: Formula
r(quoted, m-period) = quoted rate, compounded m times
r(effective, 1-period) = 1-period effective rate
( , )( ,1 )
quoted m periodseffective period
rr
m
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Quoted to Effective: Example
You plan to buy a car for $45,000. The dealer offers to finance the entire amount and requires 60 monthly payments of $950.
1. What is the effective monthly interest rate?
2. What annual interest rate will the dealer state (quote)?
3. What is the effective annual interest rate?
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Quoted to Effective: Example
Your bank states that the interest rate on a three month certificate of deposit (CD) is 4.68% per annum. 1. What is the quoted (stated) interest rate? 2. What is the frequency of compounding?3. What is the effective annual interest rate?
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Quoted to Effective: Example
You are trying to borrow $200,000 to buy a house on a conventional 30-year mortgage with monthly payments. Your bank is asking for 8.4% a year.
1. What is the quoted (stated) interest rate? 2. What is the frequency of compounding?3. What is the effective annual interest rate?4. What is the monthly payment on the loan?
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Inflation
The inflation rate (i): the rate of a general rise in prices over time. If i>0 then the same commodity becomes more expensive over time.
If you could buy a product for $100 in 2005, in 2006 you had to pay $103.23 for the same product. In 2007 you had to pay $106.17 and in 2008, $110.24. The implied inflation rates are:
Date Price Inflation rate - i
2005 $100.00
2006 $103.23 103.23 / 100.00 – 1 = 3.23%
2007 $106.17 106.17 / 103.23 – 1 = 2.85%
2008 $110.24 110.24 / 106.17 – 1 = 3.83%
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Interest rates and inflation
The real interest rate (rreal): the return (compensation) you demand for lending someone money and thus postponing consumption.
The nominal interest rate (rnominal): inflation-adjusted interest rate that represents compensation for both: inflation and postponing consumption.
In the real world, all the quoted rates are nominal rates (e.g., car loan, house loan, student loan).
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Nominal and real Interest rates
The nominal annual rate of return in 2006 was 4.91% (30 year US-Treasury bond), what is the real annual rate of return?
(1 + rnominal) = (1 + rreal) x (1 + i)
i = inflation raternominal = nominal interest rate
rreal = real interest rate
Note: all rates must be for the same period (say, one year).
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Interest rates and inflation
Given the nominal (annual) rate of return of a 30 year US-Treasury bond, find its real (annual) rate of return
(1 + rnominal) = (1 + rreal) x (1 + i)
Year i rnominal rreal
2005
2006 3.23% 4.91% (1.049 / 1.0323) – 1 = 1.6177%
2007 2.85% 4.84%
2008 3.83% 4.28%
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Examples
1. If the real interest rate is 8% and the inflation rate is 4%, what is the nominal interest rate?
2. If the nominal interest rate is 12.2% and that inflation rate is 3.6%, what is the real interest rate?
FI 3300 - Corporate Finance Zinat Alam
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Textbook Example: Annuity due
You own a property that you want to rent for 10 years.
Prospective tenant A promises to pay $12,000 per year with payments made at the end of each year.
Prospective tenant B promises to pay $12,000 per year with payments made at the beginning of each year.
Which is the better deal if the appropriate annual discount rate is 10%?
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An annuity pays $300 a year for three years
(ordinary) annuity:
annuity due:
T = 0
$300
T = 1
$300 $300
T = 2 T = 3
T = 0 T = 1
$300
T = 2 T = 3
$300 $300
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The Relation between(ordinary) annuity and annuity due
PV(annuity due)= PV(ordinary annuity) x (1 + r)
FV(annuity due)= FV(ordinary annuity) x (1 + r)
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Textbook example: loan amortization
You borrowed $8,000 from a bank and promised to repay the loan in five equal annual payments. The first payment is at the end of the first year. The annual interest rate is 10%. Write down the amortization schedule for this loan.
Compute the annual payment ($2,110.38)
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Textbook example: loan amortization
We separate each payment into two parts:
☺ Interest payment☺ Repayment of principal
For a fixed payment loan:☺ Total payment is fixed☺ Interest payment decreases over time☺ Principal repayment increases over time
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Amortization schedule table
DateBeginning Balance
Total Payment
Interest (10%) Principal
End Balance
0 8,000.00
1 8,000.00 2,110.38
2 2,110.38
3 2,110.38
4 2,110.38
5 2,110.38
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loan amortization: solution
First year:
Beginning balance = 8,000
Interest payment = 8,000 x 0.1 = 800
Principal repayment = 2,110.38 – 800 = 1,310.38
New principal balance = 8,000 – 1,310.38= 6,689.62
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Amortization schedule
DateBeginning Balance
Total Payment
Interest (10%) Principal
End Balance
0 8,000.00
1 8,000.00 2,110.38 800 1,310.38 6,689.62
2 6,689.62 2,110.38
3 2,110.38
4 2,110.38
5 2,110.38
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loan amortization: solution
Second year:
Beginning balance = 6,689.62
Interest payment = 6,689.62 x 0.1 = 668.96
Principal repayment = 2,110.38 – 668.96= 1,441.42
New principal balance = 6,689.62 – 1,441.42
= 5,248.20
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Amortization schedule
DateBeginning Balance
Total Payment
Interest (10%) Principal
End Balance
0 8,000.00
1 8,000.00 2,110.38 800.00 1,310.38 6,689.62
2 6,689.62 2,110.38 668.96 1,441.42 5,248.20
3 5,248.20 2,110.38
4 2,110.38
5 2,110.38
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Amortization schedule
DateBeginning Balance
Total Payment
Interest (10%) Principal
End Balance
0 8,000.00
1 8,000.00 2,110.38 800.00 1,310.38 6,689.62
2 6,689.62 2,110.38 668.96 1,441.42 5,248.20
3 5,248.20 2,110.38 524.82 1,585.56 3,662.64
4 3,662.64 2,110.38 366.26 1,744.12 1,918.53
5 1,918.53 2,110.38 191.85 1,918.53 0.00