2Time Value of Money

7/31/2019 2Time Value of Money

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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


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


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


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


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


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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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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Annuity - A constant stream of cash flows with afixed maturity


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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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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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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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Future Value of an Annuity


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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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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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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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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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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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Perpetuity - A constant stream of cash flows thatlasts forever


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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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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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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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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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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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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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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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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


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


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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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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So, investing at 12.36% compounded annually isthe same as investing at 12% compoundedsemiannually

Eff i A l R


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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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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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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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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 !

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2Time Value of Money