Transcript Time Value of Money

Chapter 2: Valuation of
Stocks and Bonds
2.1 Time Value of Money
A Rupee today is more worthy than a
Rupee a year hence. Why ?
• Individuals,
in
general
prefer
current
consumption to future consumption
• Reinvestment opportunity with rate of return
‘r’.
• In an inflationary period, a rupee today
represents a greater real purchasing power
than a rupee year hence.
2
Importance:
•
•
•
•
•
•
•
Valuing securities
Analyzing investment projects
Determining lease rentals
Choosing right financing instruments
Setting up loan amortization schedules
Valuing companies
Setting up sinking fund etc….
3
Time lines and notations
• Difference between period of time and point in
time.
• End of the year cash flow vs Beginning of the
year cash flow
• Positive cash flow = cash inflow
• Negative cash flow = cash outflow
4
3.1.1 Future value and compounding
• The phenomenon whereby the principle along with
interest are reinvested is called compounding.
FVn  PV FVIFr ,n  PV (1  r ) n
where,
r  discount rate
n  number of periods
FVIF  Future Value InterestFactor
FVIFr ,n  (1  r ) n
5
Example:
• Suppose you deposit Rs 1000 today in a bank
that pays 10 % interest compounded annually,
how much will the deposit grow after 8 years ?
Ans: Rs 2,144
6
Variation of FVIF with n and r
• Higher the interest rate, faster the growth rate.
• Higher the period, higher the FVIF
7
Compound and Simple Interest
Simple Interest:
FV  PV1  n  r   PV  PV n  r
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Power of compounding:
“ I don’t know what the seven wonders of the
world are, but I know the eighth – THE
COMPOUND INTEREST”
- Albert Einstein
9
Doubling period
How long would it take to double the amount at a
given rate of interest ?
Rule of 72 :
72
Doubling P eriod
InterestRate
Moreaccuraterule :
69
Doubling P eriod 0.35
InterestRate
10
Finding the growth rate
ABC ltd had revenues of $ 100 million in 1990
which increased to Rs 1000 million in the year
2000. What was the compound growth rate in
revenues ?
Ans: g = 26 %
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Future Value of streams
of cash flow
FVn  C1 (1  r)  C2 (1  r)
n
n 1
 ....... Cn
3.1.2 Present Value and Discounting
1
PV  FVn  PVIFr ,n  FVn 
(1  r ) n
where,
r  discountrate
n  number of periods
P VIF  P resentValue InterestFactor
1
PVIFr ,n 
(1  r ) n
13
Example:
What is the present value of $ 1,000 receivable
20 years hence if the discount rate is 8 % ?
Ans: $ 214
14
Variation of PVIF with r and n
• The PVIF declines as the interest rate rises and
as the length of time increases.
15
Present Value of uneven series of
cash flow
C1
C2
Cn
P Vn 

 ........
2
n
(1  r ) (1  r )
(1  r )
n
Ct

t
t 1 (1  r )
where,
C t  cash flow occuringat theend of year t
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3.1.3 Future Value of an Annuity
• An annuity is a stream of constant cash flow
occurring at the regular intervals of time.
• When the cash flows occur at the end of each
period, the annuity is called an ordinary
annuity or a deferred annuity.
• When the cash flows occur at the beginning of
each period, the annuity is called an annuity
due.
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Formula
FVA  A  FVIFAr,n
(1 r)  1
 A
r
n
where,
A  annuitycash flow
FVIFAr,n  Future Value Int erestFactorfor Annuity
(1  r)  1

r
n
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Applications
1. Knowing what lies in store for you.
Suppose you have decided to deposit Rs
30,000 per year in your PPF Account for 30
years. What will be accumulated amount in
your PPF Account at the end of 30 years if the
interest rate is 11 % ?
Ans: Rs 5,970,600
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Applications (contd…)
2. How much should you save annually ?
You want to buy a house after 5 years when it
is expected to cost Rs 2 million. How much
should you save annually if your savings earn
a compound return of 12 % ?
Ans: Rs 314,812
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Applications (contd…)
3. Annual Deposit in Sinking Fund
ABC ltd has an obligation to redeem Rs 500
million bonds 6 years hence. How much
should the company deposit annually in a
sinking fund account wherein it earns 14 %
interest ?
Ans: Rs 58.575 million
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Applications (contd…)
4. Finding the Interest Rate
A finance company advertises that it will pay
a lump sum of Rs 8,000 at the end of 6 years
to investors who deposit annually Rs 1,000
for 6 years. What interest rate is implicit in
this offer ?
Ans: 8.115 %
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Applications (contd…)
5. How long should you wait ?
You want to take up a trip to the moon which
costs Rs 1 million – the cost is expected to
remain unchanged in nominal terms. You can
save annually Rs 50,000 to fulfill your desire.
How long will you have to wait if your savings
earn an interest of 12 % ?
Ans: 10.8 years
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Present Value of an Annuity
(1 r)n  1
P VA  A  P VIFAr,n  A 
n
r (1 r)
where,
A  annuitycash flow
P VIFAr,n  Present Value InterestFactorfor Annuity
(1 r)n  1

n
r (1 r)
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Applications
1. How much can you borrow for future need ?
After reviewing your budget, you have
determined that you can afford to pay Rs
12,000 per month for 3 years towards a new
car. You call a finance company and learn
that the going rate of interest on car finance
is 1.5 % per month for 36 months. How much
can you borrow ?
Ans: Rs 332,400
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Application (contd…)
2. Period of Loan amortization
You want to borrow Rs 1,080,000 to buy a flat.
You approach a housing finance company
which charges 12.5 % interest. You can pay
Rs 180,000 per year toward loan amortization.
What should be the maturity period of the
loan ?
Ans: 11.76 years
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Application (contd….)
3. Determining the Loan Amortization Schedule
• Most loans are repaid in equal periodic
installments (monthly, quarterly, or annually),
which cover interest as well as principal
repayment. Such loans are referred to as
amortized loans.
•
For amortized loans, we would like to know:
a) The periodic installment payment and
b) The loan amortization schedule showing the
breakup of installment between the interest
component
and
the
principal
repayment
component.
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Loan Amortization Schedule
A firm borrows Rs 1,000,000 at an interest rate
of 15 % and the loan is to be repaid in 5 equal
installments payable at the end of each of the
next 5 years.
What is the annual installment payment ?
Ans: Rs 298,312
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Loan Amortization Schedule
Year
Beginning
Annual
Amount
Installment
Interest
Principal
Repayment
Remaining
Balance
5
(1)
(2)
(3)
(2)-(3)=(4)
(1)-(4)=(5)
1
2
3
4
5
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Loan Amortization Schedule
Beginning
Amount
Annual
Installment
Interest
Principal
Repayment
Remaining
Balance
(1)
(2)
(3)
(2)-(3)=(4)
(1)-(4)=(5)
1
1,000,000
298,312
150000
148312
851688
2
851,688
298,312
127753
170559
681129
3
681,129
298,312
102169
196143
484987
4
484,987
298,312
72748
225564
259423
5
259,423
298,312
38913
259399
24*
Year
* Rounding off error
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Applications (contd….)
4. Determining the Periodic Withdrawal
A father deposits Rs 300,000 on retirement in
a bank which pays 10 % annual interest. How
much can be withdrawn annually for a period
of 10 years ?
Ans: Rs 48,819
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Applications (contd…)
5. Finding the Interest Rate
Someone offers you the following financial
contract: If you deposit Rs 10,000 with him to
pay Rs 2,500 annually for 6 years. What
interest rate do you earn on this deposit ?
Ans: 13 %
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Present Value of a Growing annuity
If ,
A(1 g)  cash flow at t heend of 1st year
A(1 g) 2  cash flow at t heend of 2nd year
A(1 g) n  cash flow at t heend of nt h year
 (1  r)n  (1  g ) n 
P V of growing annuit y A(1 g) 

n
(
r

g
)
(1

r)


T hisis t rue for g  r and g  r but not for g  r in t he
case of which, P V shall be only nA.
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Example:
Suppose you have the right to harvest a teak
plantation for the next 20 years over which you expect
to get 100,000 cubic feet of teak per year. The current
price per cubic feet of teak is Rs 500, but it is expected
to increase at a rate of 8 % per year. The discount rate
is 15%. What is the present value of the teak that you
can harvest ?
 (1 0.15)20  (1  0.08) 20 
PV of teak Rs 500100,000(1 0.08)
20 
(
0
.
15

0
.
08
)
(1

0.15)


 Rs 551,736,68
3
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Annuity Due
• Annuity which occur at the beginning of the
period are called annuity due.
• Eg: monthly lease rentals in apartments
Annuity due value = Ordinary annuity value * (1+r)
• This applies to both, present and future value.
• Two steps are involved:
– Calculate the PV or FV as though it were an ordinary
annuity
– Multiply your answer by (1+r)
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Present Value of a Perpetuity
A perpetuity is an annuity of infinite duration.
PV = A * (1/r)
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3.1.4 Intra-Year Compounding and
Discounting
• So far we assumed that compounding is done
annually.
• Now we shall consider the case, where
compounding is done more frequently within a
year.
Suppose you deposit Rs 1,000 with a finance
company which advertises that it pays 12 %
interest semi-annually – this means that the
interest is paid every six months.
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Semi-Annual Compounding (Example)
• First Six months:
– Principal at the beginning = Rs 1,000
– Interest for 6 months = Rs 1,000*0.06 = Rs 60
– Principal at the end = Rs 1,060
• Second Six months:
– Principal at the beginning = Rs 1,060
– Interest for 6 months = Rs 1,060 * 0.06 = Rs 63.6
– Principal at the end = Rs 1,123.6
• Note: If compounding is done annually, the
principal at the end of one year would be Rs
1,120
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Intra-Year Compounding
T hegeneralformulafor thefuture value of a single
cash flow aftern years when compounding is done
m timesa year is :
r

FVn  P V1  
 m
where,
m n
m  frequncyof compounding per year
n  number of periodsin years
r  nominal(annual)discount rate
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Example
Suppose you deposit Rs 5,000 in a bank for 6
years. If the interest rate is 12 % and the
compounding is done quaterly, then you deposit
after 6 years will be ……….. ?
Rs 10,164
40
Effective versus Nominal Interest Rate
• Note the example of semiannual compounding
with 12 % interest rate for Rs 1000.
• At the end of a year, it grew to Rs 1,123.6
• That means Rs 1,000 grows at the rate of 12.36
% per annum.
• This figure of 12.36 % is called effective
interest rate.
• And 12 % interest rate is called nominal
interest rate.
• 12.36 % under annual compounding produces
the same result as that produced by an interest
rate of 12 % under semi-annual compounding.
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Relationship:
m
 NominalInterestRate
EffectiveInterestRate  1 
1

m


Where,
m  frequencyof compounding per year
For our example,
2
 0.12
EffectiveInterestRate  1 
 1  0.1236i.e.12.36%

2 

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Comparing Rates: The effect of
compounding.
• Interest Rates are quoted in different ways.
• Sometimes the way a rate is quoted is the
result of tradition.
• Sometimes it’s the result of legislation.
• At time, they are quoted deliberately in
deceptive ways to mislead borrowers and
investors.
• Lets make sure that we never fall victim of
such deception.
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Effective Annual Rates (EAR) and
compounding
• A rate is quoted as 12% compounded semiannually.
• What it means is that the investment actually
pays 6 % every six months.
• Is 6 % every six months the same thing as 10
% a year ?
NO
• If you invest $ 1 at 12 % per year, you’ll have $
1.12 at the end of the year.
• If you invest at 6 % every six months, then
you’ll have $ 1.1236 at the end of the year.
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EAR and the effect of compounding
• 12 % compounded semi-annually is actually
equivalent to 12.36 % per year.
• In
other
words,
12%
compounded
semiannually is equivalent to 12.36 %
compounded annually.
• In this example, 12 % is called STATED, OR
QUOTED OR NOMINAL INTEREST RATE.
• 12.36 % is EFFECTIVE ANNUAL RATE (EAR)
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Lets not get decieved…
• You’ve researched and come up with following
three rates:
– Bank A : 15 % compounded daily.
– Bank B: 15.5 % compounded quarterly.
– Bank C: 16 % compounded annually.
• Which of these is the best if you are thinking of
opening a savings account ? Which of these is
best if they represent loan rates ?
• Find out EAR for each.
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Answer:
• Bank A – 16.18 %
• Bank B – 16.42 % - Good for savers
• Bank C – 16 % - Good for borrowers
• Inference:
– The highest quoted rate is not necessarily the best.
– Compounding during a year can lead to a significant
difference between the quoted rate and the effective
rate.
– Remember, EAR is what you get or what you pay.
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Annual Percentage Rate (APR)
• It is the interest rate charged per period multiplied by
the number of periods per year.
• If a bank quotes a car loan as 1.2 % per month, then
the APR that must be reported is 1.2 % * 12 = 14.4 %.
• If a bank quotes a car loan at 12 % APR, is the
consumer actually paying 12 % interest ?
• i.e. IS APR and EAR ?
NO.
• APR of 12 % is actually 1 % per month.
• EAR on such loan is 12.68 %.
• Hence APR is actually Stated or quoted or nominal rate
in the sense we’ve been discussing.
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Continuous compounding
EffectiveInterestRate  e  1
where,
e  base of naturallogarithm
r  statedinterestper year
r
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Compounding Frequency and Effective
Interest Rate
Frequency
Nominal Int rate
%
m
Effective Int rate
%
Annual
12
1
12.00
Semiannual
12
2
12.36
Quarterly
12
4
12.55
Monthly
12
12
12.68
Weekly
12
52
12.73
Daily
12
365
12.75
Continuous
12
inf
12.75
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The effect of increasing the
frequency of compounding is
not as dramatic as some
would believe it to be – the
additional gains dwindle as
the frequency of compounding
increase
Intra-Year Discounting
T hegeneralformulafor thepresent value when discounting
periodis shorter


 1 
P Vn  FV 
r 
1  
 m
where,
mn
m  frequncyof discounting per year
n  number of periodsin years
r  nominal(annual)discount rate
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3.1.5 Loan Types and Loan
Amortization
•
•
There might be unlimited number of
possibilities to the way the principal and
interest of loan are repaid.
Three basic types of loans are :
1. Pure Discount Loans
2. Interest-Only Loans
3. Amortized Loans
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1. Pure Discount Loans
• Borrower receives money today and repays a
single lump sum at some time in the future.
• Very common when the loan term is short.
• However,
they’ve
become
increasingly
common for much longer period recently.
• Eg. Treasury Bill (T-bills)
• If a T-bill promises to repay $ 10,000 in 12
months and the market interest rate is 7 %,
how much will the bill sell for in the market ?
Ans: $ 9,345.79
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2. Interest-Only Loans
• Borrower pays interest each period and repays
the entire principal at some point in the future.
• If there’s only one period, a pure discount loan
and an interest-only loan are the same thing.
• For eg, a 50- year interest-only loan would call
for the borrower to pay interest every year for
next 50 years and then repay the principal.
• Most corporate bonds are interest-only loan.
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3. Amortized Loan
• The process of providing for a loan to be paid
off my making regular principal reductions is
called amortizing the loan.
• A simple way of amortizing a loan is to have
the borrower pay the interest each period plus
some fixed amount as the principal repayment.
• This approach is common with medium-term
business loans.
• Almost all consumer loans and mortgages
work this way.
56
Partial Amortization or “Bite the Bullet”
• A common arrangement in real state lending might
call for a 5-year loan, with say 15-year
amortization.
• What this means is that the borrower makes a
payment every month of a fixed amount based on
a 15 year amortization.
• However, after 60 months, the borrower makes a
single, much larger payment called a “balloon” or
“bullet” to pay off the loan.
• Because the monthly payments don’t fully pay off
the loan, the loan is said to be partially amortized.
57
Example
Suppose we have a $ 100,000 commercial
mortgage with a 12 % annual percentage rate
and a 20-year amortization (240 months).
Further, suppose the mortgage has a five-year
balloon. What will the monthly payment be ?
How big will the balloon payment be ?
• Here, monthly interest = 12 % / 12 = 1 % per
month.
• The monthly payment can be calculated based
on an ordinary annuity with a PV = $ 100,000.
58
Solution:
$100,000  A  P VIFA1%,240
(1 0.01)  1
 A
240
0.01(1 0.01)
240
 A  90.8194
A  $1,101.09
• That means, for 60 months i.e. 5 years we have
to pay $ 1,101.09
• Remaining amount shall be paid in lump-sum
balloon. What shall be that balloon payment ?
59
Solution (Contd….)
After60 months,we have240- 60  180 monthloan.
P aymentis still $ 1,101.09per month,and interestrate
is still1 % per month.
T heloan balanceis thus theP V of theremainingpayments:
Loan Balance  $ 1,101.09 P VIFA1%,180
(1 0.01)180  1
 $ 1,101.09
0.01(1 0.01)180
 $ 91,744.69
The balloon payment is $ 91,744.
Why is it so large ?
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End of section:
2.1: Time value of Money