Lecture 2Lecture 2

Managerial Finance

FINA 6335

Ronald F. Singer

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In Lecture 1 we concluded that the object of a manager is to attempt to make an individual's " wealth" as great as possible.

WealthWealth

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Consumption next year After Investment242522501000

500

0 500 1000 1800 1940Notice: The investment increases wealth by $140.

WealthWealth

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If Benefits > Costs

P.V. Future > Current Reduction Income In income

P.V. Future > Investment Income

P.V. Future - Investment > 0 Income NPV > 0

Net Present ValueNet Present Value

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Net Present Value = Present Value - Required Investment

= Additional x Discount - Required

Future Amount Factor Investment Example:

Suppose there is a project requiring a $500 initial investment returning $800 in one year's time. What is the Net Present Value of that investment?

= 800 x 1 - 500

1.25

= 800 x [0.80] - 500 = 640 - 500 = 140 The individual's wealth increases by the net present value of

investment Alternatively, Net Present Value shows the change in investor’s wealth

Net Present ValueNet Present Value

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To maximize Stockholders' Wealth, take all Projects that increase Stockholders' Wealth, and reject all Projects that decrease Stockholders' wealth.

Positive Net Present Value:

  Projects increase Stockholders' Wealth Negative Net Present Value projects

  Projects decrease Stockholders' Wealth Take all positive Net Present Value projects Reject all positive Net Present Value

This is called the Net Present Value Rule

Net Present Value RuleNet Present Value Rule

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Separation PrincipleSeparation Principle

Now consider the Separation Principle

Regardless of their individual preferences, All stockholders will agree on the Best Investment Decisions

The Best Investment Decision is

one that maximizes the stockholders' wealth. Given his/her maximum wealth each stockholder can

borrow and lend in the capital markets to arrive at the most preferred consumption pattern

The Investment Decision is separate from the consumption decision - Thus the Separation Principle

Why do we care about the Separation Principle?

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Basic Concepts of Time Value of Basic Concepts of Time Value of MoneyMoney What is the time value of money? If I offered you either $6,000 or $6,500 which one would you

choose? If I offered you $6,000 today or $6,500 in two years, which one

would you choose? The first problem is easy: It involves two different amounts

received at the same time. The second problem is more difficult, as it involves different

amounts at different periods of time.

The interesting part of finance is that it involves cash flows that are received at different points in time. We must devise a way of "comparing" these two different amounts to be able to make a choice between them, (or to add them up).

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Present Value of a Lump Sum Present Value of a Lump Sum and the Discounting Processand the Discounting ProcessThere are three ways of computing the Present Value: 1. Use the Formula: Where R is the discount rate T is the number of periods to wait for the Cash

Flow 2. Use the Present Value Table (Discount Factor

Table) The table (Appendix Table 1 in Brealey and Myers)

gives the discount factor to be applied to the (future) cash flow to get its present value.

3. Use a Financial Calculator

)R+(1

Flow Cash=Value Present T)R+(1

Flow Cash=Value Present T

R).tor(T,iscountFacCashFlowxD=uePresentVal

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Future Value of a Lump Sum Future Value of a Lump Sum and the Compounding Processand the Compounding ProcessThere are three ways of computing the Future Value:  1. Use the Formula:

Where: R is the discount rate

T is the number of periods to wait for the Cash Flow

2. Use the Future Value Table (Compound Factor) (Table )

3. Use a Financial Calculator

)R+(1 x Flow Cash=Value Future T)R+(1 x Flow Cash=Value Future T

R).Factor(T, Value Future x Flow Cash = Value Future R).Factor(T, Value Future x Flow Cash = Value Future

C x FVF = Value Future TTC x FVF = Value Future TT

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The Relationship Between The Relationship Between Present and Future Value Present and Future Value

PV = FVT ( 1 ) = FVT x (Discount Factor)R,T

( 1 + R)T

FV =PV(1+R)T = PV x (Future Value Factor)R,T

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Example Using TablesExample Using Tables

FV10 = 2159, i = 8%, t=10

DF =

PV10 = 2159 x DF(.08,10) = 2159 x .463

= 1,000

FV10 =1000 x FVF(.08,10) = 1000 x 2.1589

= 2159

2159

1000

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Financial calculators recognize the formulas and relationship above so that they calculate present and future values by balancing the above equations.Typical layout: N I%YR PV PMT FV  Now the idea here is that given N (number of periods) and I%YR the interest rate per period, then the equation 

PV = DF(I, N) X FVN must hold. 

Example Using CalculatorExample Using Calculator

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In the above example perform the following operations: Enter Press calculator shows 10 N 10.000 8 I%YR 8.000 2159 FV 2159.000 PV -1000.035Similarly if you want the future value of 1000 after 5 years at 8% 5 N 5.000 1000 PV 1000.000 FV -1469.328

Financial CalculatorFinancial Calculator

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ExamplesExamples

What is the Present Value of $1 received five years from today if the interest rate is 12%?

 Using the formula:  Using the Discount Factor table: The DF(12%,5) = 0.567 Using the Calculator:5 N 12 I%Y 1 FV PV 0.5674

$0.567 = ).12 + (1 x $1 -5

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ExamplesExamples

What is the Future Value of $1 in five years if the interest rate is 12% ?

Using the formula: Using the future value factor table:

The FVF(12%,5) = 1.762 Using the calculator:

1 PV

FV -1.7623

$1.762 = ).12 + (1 x $1 5

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Future Value of a Lump Sum Future Value of a Lump Sum and the Compounding Processand the Compounding Process

What is the future value of $100 in 3 years if the interest rate is 12% ? Approach 1: Keep track of dollar amount being compounded:Period1 $100.00 + $100.00(0.12) OR $100(1.12) = $112.002 $112.00 + $112.00(0.12) OR $100(1.12)2 = $125.443 $125.44 + $125.44(0.12) OR $100(1.12)3 = $140.49 Approach 2: Keep track of number of times interest is earned:Period1 $100(1.12) $100(1.12) = $112.002 $100(1.12)(1.12) $100(1.12)2 = $125.443 $100(1.12)(1.12)(1.12) $100(1.12)3 = $140.49 Notice that the process earns interest on interest. This is called

compounding. The further out in the future you go the more important is the effect of compounding

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Simple Vs. Compound InterestSimple Vs. Compound Interest

Simple Interest: Is the amount earned on the original principal

Compound Interest: Is the amount earned as interest on interest earned.

 Note: Future Value(R%,2) = Present Value (1+R)2   = Present Value (1 + 2R + R2) Original Principal Simple Interest Compound InterestFuture Value(R%,3) = Present Value (1 + 3R + 3R2 +

R3)

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Valuing Any Financial SecurityValuing Any Financial Security

First: What is a "financial security?“

A Financial Security is a promise by the issuer (usually a firm or government agency, but could be an individual) to make some payments, to the holder of the security, under certain conditions, over some specific period of time in the future.

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ExampleExample

Suppose there is a financial security promising to make specific payments over this coming year. At the "appropriate" interest rate of 10%, you determine that the Present Value of these payments is $110.

Suppose that you can purchase this security at the current price of 100.  

Is this a "good" buy? 

Do you need some additional information? 

Does it depend on the individual's feelings and desires? (i.e. utility function) Efficient Market

The Purchase = -100

Receive = +110 

NPV = 10

c1

c0

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Finding the Present Value of an Finding the Present Value of an Uneven Cash Flow StreamUneven Cash Flow Stream Typically, a security will have an uneven cash flow

stream over time, and the problem is to determine the present value of that cash flow stream.

Suppose we have the following cash flow stream, and that the "interest rate" is 10%:

Time Line:

  0 1 2 3 4 5

800 300 200 200 200

There are several ways of finding this present value:

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Method 1: The Sum of the Present Values of each payment:

0 1 2 3 4 5

└─────┴─────┴─────┴─────┴────┘

800 300 200 200 200

$800 X (.909) = $727.20──┘

$300 X (.826) = 247.80────────┘

$200 X (.751) = 150.20──────────────┘

$200 X (.683) = 136.60────────────────────┘

$200 X (.621) = 124.20─────────────────────────┘

 

Present Value $1,386.00

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Method 2: Recognize that this is a combination of a lump sum and an annuity that begins two periods in the future and lasts for three periods.

That is: 0 1 2 3 4 5 └─────┴─────┴─────┴─────┴────┘ 800 300 200 200 200 

is equivalent to

0 1 2 3 4 5 Plus 0 1 2 3 4 5 └─ ─┴──┴──┴───┴───┘ └─ ─┴──┴──┴───┴───┘ 800 300 0 0 0 0 0 200 200 200

Which in turn is equivalent to:

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Present Values: 0 1 2 3 4 5 CF(t) X DF(10%,t)

└─────┴─────┴─────┴─────┴────┘$800 X (.909) $727.20 800 300 0 0 0$300 x (.826) 247.80

Plus Plus

0 1 2 3 4 5 CF X PVFA(10%,5) └─────┴─────┴─────┴─────┴────┘$200 X (3.791) 758.20 200 200 200 200 200

Minus Minus  CF X PVFA(10%,2) 0 1 2 3 4 5$200 X (1.736) 347.20 └─────┴─────┴─────┴─────┴────┘ TOTAL $1,386.00 200 200 0 0 0

Using Financial Calculator: $1,386.26 (Hewlett Packard)

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Method 3: Treat this as two lump sum payments plus an annuity that begins in period 2.

  0 1 2 3 4 5

└─────┴─────┴─────┴─────┴────┘ 800 300 200 200 200 

Is equivalent to  0 1 2 3 4 5

└─────┴─────┴─────┴─────┴────┘ 800 300 200x PVFA(10%,3)

200 x (2.487) $497.40──┘$800.00 X (.909) = $727.20─────┘ $797.40 X (.826) = $658.65──┘ TOTAL $1,385.85 

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PerpetuitiesPerpetuitiesSome Securities last "forever," and generate the equivalent of a

perpetual cash flow.  Clearly, we cannot evaluate these perpetual cash flows in the

conventional manner. We do however, have formulas which allows us to evaluate these

cash flows. A Perpetuity is a series of equal payments that continues

forever.

0 1 2 3 4 5 .......... 98 99 100......└────┴────┴────┴────┴────┴──..........──┴────┴──

──┘...... 15 15 15 15 15 .......... 15 15 15 .......  The Present Value of a Perpetuity is:How much would you pay for this bond?

Rate Discount

Period per Flow Cash = PV Rate Discount

Period per Flow Cash = PV

r

CF=PV r

CF=PV

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PerpetuitiesPerpetuities

Example: A British Government Bond pays 100,000 pounds a year forever. The market rate of interest is 8%. How much would you pay for this bond?

PV = Cash Flow = 100,000 = 1,250,000of perpetuity r 0.08How much is the bond worth if the first coupon is payable

immediately? PV of Bond = PV Immediate Payment Plus Value of Perpetuity

= 1,250,000 + 100,000  = 1,350,000

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Growing PerpetuityGrowing Perpetuity

If the cash flow grows at a constant rate, then the perpetuity is called a growing perpetuity

 where CF1= Cash flow next year r = Market rate interest g = Constant Growth rate How much would you pay for the previous bond if the cash

flows grow at 5% starting at 105,000 next year PV of growing = 105,000 = 3,500,000

perpetuity 0.08 -0.05  = 105,000 = 3,500,000

0.03

g-rCF=Perpetuity Growing of PV 1

g-rCF=Perpetuity Growing of PV 1

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AnnuityAnnuity

Recall:An annuity provides equal cash flows for a fixed number of periods

C1 C2 C3 .............. CN │__________│___________│________│_________│

0 Notice that the first payment starts next period.The value of an annuity is the difference in the value of two

perpetuities, one that starts now and one that starts N-periods from now. 

Present Value Present Value Present Value of N-period = of Perpetuity - of Perpetuity Annuity That starts now that starts N- Periods from now

What is the Equation for an Annuity?

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A Clarification on Different A Clarification on Different Compounding PeriodsCompounding Periods We have assumed that we are dealing with

compounding only once a year. But what happens when the compounding is done

more than annually? Given the periodic interest rate, you can use the

tables to find the present value of a single payment, the present value of a periodic annuity, as well as the future values.

Example: Suppose you will receive $1,000 per month for 12 months. at an annual (simple) interest rate of 18%, compounded monthly, what is the present value of this cash flow?

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Definition of RatesDefinition of Rates

Periodic Interest Rate: the interest earned inside the compounding period.

 Example: 18% compounded monthly has a periodic rate of 1.5%

 Nominal (Simple) Interest Rate: interest is not compounded. the amount you would earn, annually, if the interest were withdrawn as soon as it is received. (This is the APR (Annual Percentage Rate) you find on credit card and bank statements)

 Example: Invest $1,000 today at 18%, APR paid monthly. you would have $1,180 at the end of one year.

 

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Definition of RatesDefinition of RatesEffective Interest Rate: the annual amount you would have if the interest is allowed to compound. (This is the actual interest earn over the year allowing for compounding) Example: invest $1,000 today at 18%, compounded monthly. Then the periodic interest rate is 1.5% per month. The nominal rate is 18%. Then, allowing for compounding, the effective rate is: .............................. 0┴───1┴───2┴───3┴───4┴─────────────┴───┘  (1+.015)12 - 1 = 19.56% thus if you invested $1,000 at 18% compounded monthly you would have   $1,195.60 To convert from simple rates to effective rates, use the formula: effective rate = (1 + r/m)m – 1 r is the simple rate

m is the number of compounding periods per year.

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Now the present value of monthly cash flow over one year is calculated as the present value of an annuity, received for 12 periods at a periodic rate of 1.5%.

Thus you want to use 12 as N and 1.5% as I%YR in the calculator.  

(Alternatively, if your calculator has an option to set the payments per year you could set it to 12 but this is not recommended. There is a tendency to forget to reset it to annual payments for the next problem, and what to use for N gets confusing)

 

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What if, at the same compounding interval, you received only 2 cash flows of 6,000 each in month 6 and month 12?

6000 6000 │ │

│ ......... │ 0───1───2───3───4───5────6───────--11────12

Finally, what if the compounding of 18% occurs only twice per year? 

effective rate present value

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To pay an Annual Interest Rate r, compounded m times during the year means pay r/m for m-times in a year

Example, to pay 10% compounded quarterly means 2.5% is paid 4 times a year

The Effective Interest Rate is = (1 + 0.025)4 – 1 = 0.1038 or 10.38%

The Effective Interest Rate exceeds 10% since interest is paid on interest.

When the compounding interval approaches zero, we have continuous compounding  (1 + r/m)m - 1 = er - 1 = (2.7183)r - 1.

See Appendix Table 4: Values of ert: Future Value of $1 at a continuously compounded rate r for t years.

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If 1 dollar is continuously compounded at rate r, at the end of the year 1 dollar will grow to er

where e = 2.7183 (e is the base of the natural log) after n years, 1 dollar will grow to = enr

Example: abc bank offers 10.2% compounded quarterly. xyz bank offers 10.1% interest continuously compounded. which is better for you?

in abc bank 1 dollar deposit grows to, after 1 year,= (1 + 0.102/4)4 = 1.1060 in xyz bank 1 dollar deposit after 1 year grows to= e.101 = 1.1063 

therefore, even though xyz only pays 10.1%, continuous compounding makes xyz interest a better deal. notice that both of these offers are better than 10.5% simple