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Time Value of Money
All time lines were drawn and discussed in class
Removal of errors and omissions, if any, in thisppt are your responsibility
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Agenda
TimelinePresent Value - Future Value
One period PV, NPV
PV - different rates and time periods
FV - different rates and time periods
Multi-Period Case (incl. power of compounding)
Annuities, Perpetuities - Simplifications
Special CasesEffective Annual Rate
APR and Loan Balance
Risk and CF
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Timeline
Main issue is what value to place on a cash flow
When or time component (early part of thecourse)
Uncertainty (risk-return latter part)
Importance is because we need to make
apples-to-apples comparisonsIt is only possible to compare or combinevalues at the same point in time
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Timeline
A timeline is a linear representation of the timing ofpotential cash flows
Drawing a timeline of the cash flows will help youvisualize the financial problem
Usually cash flows are assumed to be at the end of
the period unless o/w stated
By convention, inflows are positive and outflows are
negative
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Present Value
Key insight is that a Rupee today is worth morethan a Rupee tomorrow the Rupee today can beinvested to earn interest which can be obtainedalong with the original Rupee invested (tomorrow)
Conversely one could ask how much is it worth tome today to have a Rupee tomorrow
orHow much do I have to put away today to get aRupee tomorrow
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Present Value
The PV of a delayed payment is obtained bymultiplying the payoff by the discount factor
Consider the following example
Suppose you receive Rs.1 tomorrow and the bankpays an interest rate of 15% (reward for waiting) ,what is the PV?
X (1+r)1 = 1
X = 1/(1+r)1 = 1/(1.15) = 0.869
PV = Discount factor * payment to be received
Discount factor (DF)
r is the rate of return,hurdle rate,opportunity cost ofcapital
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PV & FV
Present Value
Value today of afuture cash flow
Future ValueAmount to which aninvestment will growafter earning interest
Discount RateInterest rate used to
compute presentvalues of future cash
flows
Discount FactorPresent value of a
Rs.1 future
payment
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One Period: PV
Consider the following example
Suppose you are offered an investment that paysRs.12,000 in three years. If you expect to earn a 13%return, what is the value of this investment?
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One Period: PV
Consider the following example
John Doe wishes to find the present value ofRs.1,700 that will be received 8 years from now;Johns opportunity cost is 8%
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One Period: Net Present Value (NPV)
The Net Present Value (NPV) of an investment is thepresent value of the expected cash flows, less thecost of the investment.
Consider the following exampleSuppose an investment that promises to payRs.15,500 in one year is offered for sale forRs.12,500. Your interest rate is 8%. Should you buy?
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PV- different rates and time periods
Interest rate, time period and PV
0
10
20
30
40
50
60
70
80
90
100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0% 5% 10% 15% 20%
As t increases PV decreases
As r increases PV decreases
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FV- different rates and time periods
Interest rate, time period and FV
0
100
200
300
400
500
600
700
800
900
1000
1 3 5 7 9 11 13 15 17 19 21 23 25
0% 5% 10% 15% 20%
As t increases FV increases
As r increases FV increases
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Multi-period Case
What happens when the number of period
increases?
FV = X* (1+r)t
PV = X/(1+r)t
FV = PV * (1+r)t or PV = FV /(1+r)t
Consider a simple example
Suppose r = 10% & you want to compare two planA: receive Rs.400 4 years from nowB: Receive Rs.600 12 years from now
Which one would you prefer?
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Multi-period Case
Consider the following example
Suppose you have a choice between receivingRs.5,000 today or Rs.10,000 in five years. Youbelieve you can earn 10% on the Rs.5,000 today.What should you do?
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Multi-period Case
Consider the following example
Suppose that Brealey invested in the initial public
offering of the Myers company. Myers pays a
current dividend of Rs.1.15, which is expected to
grow at 40% per year for the next five years. Whatwill the dividend be in five years?
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Multi-period Case
Consider the following example
Suppose we plan to save Rs.1000 today, andRs.1000 at the end of each of the next two years. Ifwe can earn a fixed 10% interest rate on our
savings, how much will we have three years fromtoday?
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Multi-period Case
Consider the following example (PV case)
You have an opportunity to invest in a business that will payRs.200,000 in year one, Rs.400,000 in year two, Rs.600,000in year three and Rs.800,000 in year four. You can earn12% per year compounded annually on a mutual fund that
has similar risk. If it costs Rs.1.2 million to start thisbusiness, should you invest?
0 1 2 3 4
| | | |
1.2 mil 200K 400K 600K 800K
Discount rate = 12%
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Multi-Period Case
In general when you have multiple cash flows C0..CN, the PV of those Cash flows is
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Multi-period Case
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PV & FV
Only values at the same point in time can be
combined or compared (time travel)When cash flows occur at different points in timethey must be discounted or compoundedappropriately
To move cash flow forward compound itTo compound cash flows, multiply the amount by(1 + r)nwhere r is the periodic interest rate and nis the number of compounding periods
To move cash flow backward discount it
To discount cash flows, divide the amount by
(1 + r)
n
where rand nare as defined previously
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Annuities, Perpetuities - Simplifications
Annuity - A stream of constant cash flows that lastsfor a fixed number of periods
Growing annuity - A stream of cash flows that grows
at a constant rate for a fixed number of periods
Perpetuity - A constant stream of cash flows thatlasts forever
Growing perpetuity - A stream of cash flows thatgrows at a constant rate forever.
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Annuities, Perpetuities - Simplifications
Annuity - A constant stream of cash flows with afixed maturity
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Annuities, Perpetuities - Simplifications
Consider the following example
If you can afford a Rs.400 monthly car payment,what is the maximum price of the car you can buy ifthe interest rates is 7% on 36-month loan
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Annuities, Perpetuities - Simplifications
Consider the following example
What is the present value of a four-year annuity ofRs.100 per year that makes its first payment twoyears from today if the discount rate is 9%?
297.05
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Annuities, Perpetuities - Simplifications
Consider the following example (spreadsheet)Braden Company, a small producer of plastic toys,wants to determine the most it should pay to purchase aparticular annuity. The annuity consists of cash flows of
Rs.1000 at the end of each year for 10 years. Therequired return is 9%.
6417.66
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Annuities, Perpetuities - Simplifications
Future Value of an Annuity
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Annuities, Perpetuities - Simplifications
Consider the following example
Ellen is 35 years old today and wants to begin savingfor retirement from next year. At the end of each year
until she is 65 she will save Rs.10,000 each year in aretirement account. If the account earns 10% peryear how much would Ellen have saved at age 65?
1644940.27
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Annuities, Perpetuities - Simplifications
Growing annuity - A growing stream of cash flows
with a fixed maturity
The PV of a growing annuity with the initial cash flow c, growth rate g, andinterest rate ris
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Annuities, Perpetuities - Simplifications
Consider the following example
In the Ellen example Suppose Ellen expects to
salary to increase and so she plans to save 5% more
each year - how much will she have saved by at age65? All other details remain the same
2625491.98
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Annuities, Perpetuities - Simplifications
Consider the following example PV of a lottery
You have won the Rs. 30 Million state lottery. You
have two options to claim the prize
(a) 30 payments of 1 million starting today or(b)15 million lump-sum paid today
If the interest rate is 8% which is the option you
should choose (you are rational!)
a 11.26
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Annuities, Perpetuities - Simplifications
In an annuity due all that happens is that thetimeline changes. So how does the value change..
An annuity due is worth (1+r) times an ordinary
annuity
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Annuities, Perpetuities - Simplifications
Perpetuity - A constant stream of cash flows thatlasts forever
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Annuities, Perpetuities - Simplifications
Consider the following example
What is the value of a British consol that promisesto pay 15 each year, every year until the sun turnsinto a red giant and burns the planet to a crisp?
The interest rate is 10-percent
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Annuities, Perpetuities - Simplifications
Consider the following exampleHow much would I have to deposit today in order towithdraw Rs.2,500 each year forever if I can earn9% on my deposit?
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Annuities, Perpetuities - Simplifications
Consider the following example
You want to endow a graduation party at IIMK. Youwant the event to be memorable and plan Rs.5,000
per year for ever for the party. If IIMK can invest at7% year and the first party is a year from now howmuch will you need to donate to endow the party?
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Annuities, Perpetuities - Simplifications
Growing perpetuity - A growing stream of cash flowsthat lasts forever
The PV of a growing perpetuity with the initial cash flow c, growth rate g,and interest rate ris
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Annuities, Perpetuities - Simplifications
Consider the following example
Theexpected dividend next year is Rs.1.30 anddividends are expected to grow at 5% forever;
If the discount rate is 10%, what is the value of
this dividend stream?
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Annuities, Perpetuities - Simplifications
Consider the following example
If in the MBA graduation party endowment caseyou realize that the inflation rate will be 4% peryear after the first year and that will also need to be
factored in in your original donation amount whatdo you now have to donate. All other detailsremain the same.
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Annuities, Perpetuities - Simplifications
Consider the following example
Your firm is about to make its initial public offeringof stock and your job is to estimate the correctoffering price. Forecast dividends are as follows
If investors demand a 10% return on investments forstocks of similar risk, what price will they be willingto pay?
32.81
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Annuities, Perpetuities - Simplifications
Conceptually, a firm should be worth the presentvalue of the firms cash flows.
The tricky part is determining the size, timing and
riskof those cash flows.
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Special Cases
So far, we have computed either the PV or the FVof a stream of CF (unequal or equal) for a givennumber of time periods and interest rate
Sometimes we know the present value or futurevalue, but do not know one of the other variableswe have previously been given as an input
For example, when you take out a loan you mayknow the amount you would like to borrow, but maynot know the loan payments that will be required to
repay it
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Special Cases
Consider the following example
Your firm plans to buy a warehouse for Rs.500,000.The bank requires you to pay 20% down paymentand will lend Rs.400,000 to you for a 30-year time
period at 9% annual rate of interest. What is yourannual payment amount?
38934.54
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Special Cases
At times it may be necessary to compute the r (also
called IRR) more on this later in the session butlet us do an example
Jane has just graduated and is offered a fantasticjob at ABC investment corporation. She pondersbut decides to set up her own business and asksthem for funding. ABC lends her Rs.2 million with
the agreement that she will pay back Rs.200,000 atthe end of each year for the next 30 years.Assuming she fulfills her agreement what is therate at which ABC has lent Jane the money?
9.5%
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Special Cases
Suppose ABC corporation gives Jane another
option pay 200,000 in the first year and 4% morein perpetuity all other details remain the same.What is the rate that ABC is lending at (IRR)?
14%
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Special Cases
Consider the following example
Assume the total cost of a college education will beRs.100,000 when Ms.Xs child enters college in 12
years. She has Rs.50,000 to invest today. Whatrate of interest must she earn on the investment to
cover the cost of her childs education?
6%
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Special Cases
Calculating time T (or N)
If we deposit Rs.20,000 today in an account paying10%, how long does it take to grow to Rs.100,000?
17
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Effective Annual Rate
Compounding an investment mtimes a year for Tyears provides for future value of wealth
C0 = initial investment or PV of the CF
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Effective Annual Rate
A reasonable question to ask in the above example
is what is the effective annualrate of interest onthat investment
The Effective Annual Interest Rate (EAR) is theannual rate that would give us the same end-of-investment wealth after 3 years
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Effective Annual Rate
So, investing at 12.36% compounded annually isthe same as investing at 12% compoundedsemiannually
Eff i A l R
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Effective Annual Rate
Find the Effective Annual Rate (EAR) of an 18% APR(annual % rate) loan that is compounded monthly.
Eff i A l R
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Effective Annual Rate
What is the difference between the EAR of interestand stated rate of interest in the following cases:
Case A: Stated rate of interest is 8 percent and thefrequency of compounding is six times a year.
Case B: Stated rate of interest is 10 percent and thefrequency of compounding is four times a year.
Case C: Stated rate of interest is 12 percent and thefrequency of compounding is twelve times a year.
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Eff ti A l R t
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Effective Annual Rate
The nominal interest rate is the stated or contractualrate of interest charged by a lender or promised by aborrower
The effective interest rate is the rate actually paid orearned
In general, the effective rate > nominal rate whenevercompounding occurs more than once per year
Eff ti A l R t
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Effective Annual Rate
Example of an actual automobile loan agreement
The ad says
12 month car loans - Only 9%!
What is actually happening.
APR d L B l
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APR and Loan Balance
Ten years ago your firm borrowed $3 million topurchase an office building using a loan with a7.80% APR, and monthly payments for 30 years
How much do you owe on the loan today
How much interest was paid on the loan in the lastyear
C = 21596
Owe(10) = 2,620,759
Owe(9) = 2,673,247
Interest = 215896*12 - 52488
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Problem Solving
Mini case see word doc
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Problem Solving
Mini case see word doc
What are the four questions to be answered
How much money would Vikas need 20 years from
now? How much money should Vikas save each year for the
next 20 years to be able to meet his investmentobjective?
How much money would Vikas need when he reachesthe age of 60 to meet his donation and bequeathalobjective?
What is the present value of Vikass life time earnings?
P bl S l i
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Problem Solving
Ms. Ann Chen receives an annuity of $550,payable
once every two years The annuity stretches out over20 years. The first payment occurs at date 2, that is,two years from today. The annual interest rate is 8%
2596
Ri k d CF
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Risk and CF
A safe Rupee is worth more than a risky one
Generally, investors do not like risk
In order to induce the investors to invest in riskyprojects, a higher rate of return is needed
Higher rate of return causes lower PVs
Consider a Lottery vs. bank deposit
H d hi
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How do you master this
Practice, practice, practice
It is also easy to watch what is done in class andconvince yourself that you can do it, if needed
There is no substitute for getting out the calculator,paper and pen and working out many, many
problems from the book until you can do themcorrectly and quickly
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Thank you !