Goal of the Lecture:

Understand how to properly value a series of cash flows.

What is value?

OBJECTIVE VALUATION

EXAMPLE: C.C. Sabbathia’s contract averages 24 mm while Mark Texiera’s averages 23 mm

years: 1 2 3 4 5 6

Sabbathia $20 20 24 24 28 28 Texiera $23 23 23 23 23 23

You can compare the two using "Present Value" which means discounting dollars to be received in the future so that they are equivalent to dollars received now. Maybe adjust for risk if pay depends on uncertain performance. You are probably more familiar with "Future Value" which is the number of dollars you expect to have in the future if you leave a specific number of present dollars in a bank account earning interest at k percent per year. Present value is just the inverse of future value.

Note: Manny Ramirez 2 year contract with L.A. Dodgers for 20 and 25 million made him the “second highest paid” after Alex Rodriguez, but couldn’t agree on deferral.

QUESTION: How do you decide whether to buy a snow blower (lawn mower)?

ANSWER: Estimate present value of future plowing, discount, and compare to snow blower price. Other considerations may include time to shovel, back-ache, plow digs up lawn/pavement and doesn't come on time.

QUESTION: How do you decide how often to change the oil or coolant in your car?

QUESTION: How do you decide whether to replace breaks, tire, clutch etc. in your car?

QUESTION: How do you decide how much to save now for retirement or a major future purchase ?

NEED TO CONSIDER THAT FUTURE CASH FLOWS ARE WORTH LESS THAN PRESENT CASH FLOWS.

Discount - as in "discounting" what a person who exaggerates says.

Compound - how the difficulty increases if you have three finals in one day.

SETTING - MOST PEOPLE USE PRESENT VALUE INTUITIVELY EVERY DAY

Time Value of MoneyTime Value of Money““A Dollar is Worth More Today Than a Dollar Tomorrow”A Dollar is Worth More Today Than a Dollar Tomorrow”

Illustration - Pie Concept

Maximize the Value of a Firm or the Size of the Pie

Three Basic Ideas Affect the Size of the Pie:

1. Timing of Cash Flows - Ingredients in the Pie

2. Valuation of Stocks/Bonds - Size of Pie

3. Risk - Baking Conditions

Visualize as Follows:

Factor First Condition Second Condition

Cash Flows: More Less

Timing of Cash Flows: Sooner Later

Riskiness of Cash Flows: Less More

Produces Larger Pie Produces Smaller Pie

V = S + BV = S + B V= S+ BV= S+ B

Time Value of MoneyTime Value of Money

I. Time Value of Money - A Formula for Every Situation

II. Future Value and Present Value of a Single Cash Flow

III. Present Value is the Inverse of Future Value

IV. Future Value and Present Value of Multiple Cash Flows

a. Equal annual cash flows (Annuities)

b. Infinite Annuities (Perpetuities)

c. Multiple Unequal Cash Flows

V. Solving for an Unknown (Implied) Interest Rate

a. Given Present Value and Future Value, find

IRR (Internal Rate of Return)

Interest RatesInterest Rates““Cost on Borrowed Funds or Rate Received on Money Cost on Borrowed Funds or Rate Received on Money Lent”Lent”

I. Nominal vs. Effective Annual Interest Rates

a. Adjust n for New Number of Periods

b. Adjust k for New (Lower) Rate for Portion of Year

Future ValueFuture Value““Taking a Cash Flow Today and Determining What It will be Taking a Cash Flow Today and Determining What It will be

Worth Sometime in the Future at a Given Interest Rate, k”Worth Sometime in the Future at a Given Interest Rate, k”

I. Future Value

General Formula

FVn = PV0(1+k)n

= PV0(1+k)n = [=FV(k, n, 0, PV)]

Note: FVn = Future Value at the End of n Periods

PV0 = Present Value Now (Time = 0)

k = the Interest (Discount or Compound) Rate

n = the Number of Periods in the Future

[=FV()]= Excel Future Value Function (note that Excel uses I instead of k and N instead of

n)

II. Example: Suppose you have $5,000 and the interest rate is 15% and you wish to invest for 4 years, how much will you have at the end of 4 years?

FV4 = PV0(1 + .15)4

= $5,000(1.15)4

= $5,000(1.75)

= [= FV(0.15, 4, 0, -5000)]

= $8,745

PROBLEM: Suppose your company made $50 million this year and you expect profit to grow by 15% per year for the next 4 years. What will profit be at the end of 4 years?

ANSWER: $50(1 + .15)4 = [= FV(0.15, 4, 0, -50)]

= $87.45 million

Note: This is the same basic problem as the previous problem but with different language.

Present ValuePresent Value““Inverse of Future Value and Gives the Value Today ofInverse of Future Value and Gives the Value Today of

Money Received in the Future”Money Received in the Future”

I. Present Value

General Formula

PV0 =

= FVn = [= PV(k, n, 0, FV)]

[=PV()]= Excel Present Value Function (note that Excel uses I instead of k and N instead of

n)

FV

knn( )1

1

1( ) k n

II. Example: What is the value of $100 to be received at the end of 10 years if the k = 10%?

PV0 = $100/(1+.10)10

= $100/2.54 = [= PV(0.10, 10, 0, -100)]= $38.6

PROBLEM: Suppose you need $10,000 to pay off a loan in 10 years. How much do you need to put in the bank

today at a 10% interest rate to have the $10,000 then?

PV0 = $3,860

Future Value of an Future Value of an OrdinaryOrdinary Annuity Annuity““Future Value of a Series of Equal Cash Flows Received at Future Value of a Series of Equal Cash Flows Received at the the End of Each PeriodEnd of Each Period for a Specified Number of Periods” for a Specified Number of Periods”

I. Future Value of an Ordinary Annuity

General Formula

FVAn

= [= FV(k, n, PMT, PV)]

Note: FVAn = Future Value of an n-Period Annuity

PMT = Constant Annuity Payment

PMTk

k

n( )1 1

II. Example: Suppose you save $100 at the end of each

year for 3 years - k = 10%. How much will you

have in the account after 3 years?

FVA3 = $100 [((1+.10)3 - 1)/.10]

= $100(3.310)

= [= FV(0.10, 3, -100, 0)]

= $331.00

Future Value of an Annuity Future Value of an Annuity DueDue““Future Value of a Series of Equal Cash Flows Received at Future Value of a Series of Equal Cash Flows Received at the the Beginning of Each PeriodBeginning of Each Period for a Specified Number of for a Specified Number of Periods”Periods”

I. Future Value of an Annuity Due

General Formula

FVAn

= [= FV(k, n, PMT, PV, 1)]

Note: Just multiply the ordinary annuity formula by (1 + k).

PMTk

kk

n( )( )

1 11

II. Example:

Suppose you save $100 at the beginning of each

year for 3 years - k = 10%. How much will you

have in the account after 3 years?

FVA3 = $100 [((1+.10)3 - 1)/.10](1 + .10)

= $100(3.310)(1 + .10)

= [= FV(0.10, 3, -100, 0, 1)]

= $364.10

Present Value of an Present Value of an OrdinaryOrdinary Annuity Annuity““Present Value of a Series of Equal Cash Flows Paid at the Present Value of a Series of Equal Cash Flows Paid at the End of Each PeriodEnd of Each Period for a Specified Number of Periods” for a Specified Number of Periods”

I. Present Value of an Ordinary Annuity

General Formula

PVAn = PMT

= [= PV(k, n, PMT, FV)]

Note: PVAn = Present Value of an n-Period Annuity

PMT = Constant Annuity Payment

1 1

1k k k n

( )

II. Example: You just won megabucks for $3.0M to be paid over 20 yearly payments. How much did you reallywin in present value dollars if k = 10%.

PMT = $150,000 (If Taxed at 30%, PMT = $105,000)

PVA20= $150,000[1/.10 - 1/(.1)(1+.1)20]

= $150,000(8.514)

= [= PV(0.10, 20, -150000, 0)]

= $1,277,100 (Less than Half)

If Taxed -> $105,000(8.514) = $894,000

Note: For the Present Value of an Annuity Due, just multiply the above by (1 + k) because the money is earning interest for one additional year. Or use [= PV(k, n, PMT, FV,1)].

For an Infinite Annuity the formula above reduces to

PVA = PMT [1/k] = PMT/k

Example: Redo the Lottery example except assume that you receive payments for ever.

PVA = $150,000/.10 = $1,500,000

This is not much more than a receiving payments for only 20 years because payments after 20 years are not worth much - discounted heavily.

Note: The infinite annuity is often a good approximation to a long annuity and is easy to calculate.

Multiple Unequal Cash FlowsMultiple Unequal Cash Flows““Present and Future Values are Additive so the Value of Present and Future Values are Additive so the Value of Multiple Unequal Cash Flows is Just the Sum of the Multiple Unequal Cash Flows is Just the Sum of the Individual Present or Future Values”Individual Present or Future Values”

Example: Present Value - Multiple Unequal Cash Flows

YEAR CF

1 $200 2 $200 3 $300

PV = $200(1/1.10) + $200(1/1.10)2 + $300(1/1.10)3

= $200(.909) + $200(.826) + $300(.751)

= [= NPV(0.10, 200, 200, 300)]

= $572.5

Alternatively, you could find the value as a $200 annuity for 3 years plus a $100 cash flow in the third year.

Assume in the previous example that you receive 300 in year 3 and in each year forever afterward. The Present Value is

PV = $200 + 300/.10 [1/(1.10)2]

= $200[1.735] + $3000[.826]

= [= PV(0.10, 2, -200, 0)] + [= PV(0.10, 2, 0, -3000)]

= $2825

2)1.01(1.0

1

1.0

1

PROBLEM: Suppose someone offers you any of the following 3 series of cash flows. Which do you want if k = 20%

Year Cash Flows .

1 2 31 $200 0 02 $200 0 03 $200 $1000 $4004 $200 0 $4005 $200 0 $400

ANSWER: Choose 1. At high interest rates ignore 2.

at 20% at 3%

1. PV = 598 PV = 915.94

2. PV = 579 PV = 915.14

3. PV = 585 PV = 1066.5

QUESTION: What would you do if k = 3%? - choose 3.

NOTE: Low Japanese interest rates may explain their long-term view.Remember, behavior is driven by financial incentives.

Note: This is where a calculator or spreadsheat is helpful.

SINGLE CASH FLOWSuppose we have one cash flow and the present value and future value - we get the interest rate by solving for PVk,n.

PV0 = FVn

=

PROBLEM: Suppose you borrow $5,000 Today and agree to payback $10,000 in 7 years. What was the interest rate?

PV0/FV 7 = $5,000/$10,000

= .5 = 1/(1 + k)7

= [=Rate(n, PMT, PV, FV)]

= [=Rate(7, 0, -5000, 10000)]

=> k = 10.4%

Determining Implied Interest RatesDetermining Implied Interest Rates

PV

FVn

0

1

1( ) k n

1

1( ) k n

ANNUITIES - SAME IDEA – but best to use Excel

PROBLEM: Suppose you borrow $5000 and agree to pay $1500 per year for 7 years to pay off the loan. What is the implied interest rate?

[=Rate(7, -1500, 5000, 0)] => k = 23%

QUESTION:In the previous problem you paid a total of $10,000 while here you paid $10,500. Why the large difference in implied interest rate?

ANSWER: Interim payments worth more than one balloon payment in last year.

WHEN WE HAVE AN UNEVEN SERIES OF CASH FLOWS ONE MUST USE IRR FUNCTION.

Example: If you pay $5000 to receive cash flows in the future of (1) 2000, (2) 2000, (3) 3000, you get the interest rate as follows;

Input -5000, 2000, 2000, 3000 into spreadsheet cells A1-A4

Then use [=IRR(A1:A4)] to get k =17.5%

ApplicationsApplicationsGROWTH RATES - EARNINGS OR SALES GROWTH

PROBLEM: A company earned the following amounts in the past 5 years - What was their earnings growth rate?

FIRST: What formula is required?

Earn Year$1000 1$1500 2$1725 3$2250 4$2500 5

$1000 = ($2500)

$1000/$2500 = .40 = 1/(1 + k)4

= [=Rate(4, 0, -1000, 2500)]

=>k = 25.74%

1

1( ) k n

FUTURE SUMSSuppose a company sells $1000 of bonds which mature in 5 years. How much must be put in a sinking fund yearly to pay off the bonds if k = 10%.

FIRST: What formula?

FV = $1000

= PMT(6.105)

= [=PMT(0.10, 5, 0, -1000)]

=> PMT = $163.80

TERM LOANS - CAPITAL RECOVERY / LOAN AMORTIZATION - MORTGAGE

If you take out a $10,000 car loan at 10% interest rate and make 5 yearly payments. What will the payments be?

FIRST: What formula?

PV = $10,000 = PMT

=PMT[3.791]

= [=PMT(0.10, 5, -10000, 0)]

=> PMT = $2637.82

Applications ContinuedApplications Continued

10.0

1)10.01( 5 PMT

5)1.01(1.0

1

1.0

1

Effective vs. Nominal Interest RatesEffective vs. Nominal Interest Rates

““The Effective Rate is the True Rate of Interest per Year The Effective Rate is the True Rate of Interest per Year

and the Nominal Rate is the Quoted Rate per Year”and the Nominal Rate is the Quoted Rate per Year”

I. Effective vs. Nominal

keff = (1+knom/m)m - 1

If interest is compounded “m” times annually, we must adjust k and n in the formulas above, to take account of the additional interest earned during a year.

II. Example:

A nominal rate of 16% is compounded quarterly.

Thus, each quarter we earn 4% on the money in

the account.

k4 = k/m = 16%/4 = 4%

keff = (1+knom/m)m - 1

= (1+.16/4)4-1

= .1698 => 17% (Effective Rate)

Therefore, although the nominal rate is 16%, we effectively earn about 17%.

III. Adjusting formulas for compounding.

With “m” number of compound periods within a year, we can use the earlier PV and FV formulas except that we replace “n” with “mn” and replace “k” with “k/m”.

PROBLEM: Suppose you must make $500 monthly car payments for 2 years. What is the present value (Loan Amount) of the payments if knominal = 24%?

mn = 12 * 2 = 24

k/m = 24%/12 = 2%

PV0 = 500

= 500[18.914]

= [=PV(0.02, 24, -500, 0)]

= 9457

24)02.01(02.0

1

02.0

1