15.401 Finance Theory I - MITweb.mit.edu/astomper/www/univie/pof/Chapter 2.pdf · 2009. 12. 29. · 15.401 Lecture 2: Present value Visualizing cash flows (assets): Example. Drug

15.401

15.401 Finance Theory I15.401 Finance Theory I

Alex Alex StomperStomperMIT Sloan School of ManagementMIT Sloan School of Management

Institute for Advanced Studies, ViennaInstitute for Advanced Studies, Vienna

Lecture Lecture 22: Present Value: Present Value

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15.401 Lecture 2: Present value

_ Present value

_ Future value

_ Special cash flows

_ Compounding

_ Nominal versus real cash flows and discount rates

_ Extensions

Readings:

_ Brealey, Myers, and Allen, Chapters 2 ‒ 3

2

AgendaAgenda

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_ Objective of a financial manager: maximize firm’s market value

_ The value of a firm/an asset/a project/a portfolio depends onfeatures of the cash flow (CF) that the firm/asset/project/portfoliogenerates.

_ Two important characteristics of CFs: timing and riskiness.

_ Financial market prices can be used to value CFs.

_ Cost of capital: expected return on equivalent investments infinancial markets.

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Review of last classReview of last class

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Visualizing cash flows (assets):

Example. Drug company has developed a flu vaccine:

_ Strategy A: To bring to market in 1 year, invest $1B (billion) now andreturns $500M (million), $400M and $300M in years 1, 2 and 3,respectively

_ Strategy B: To bring to market in 2 years, invest $200M in years 0 and 1,and returns $300M in years 2 and 3

How to value/compare CFs?4

Valuing cash flowsValuing cash flows

Invest CF0 CF3

Earn CF1 CF2 CF4 ...


Example. How much is a sure cash flow of $1,100 in one year worthnow?

Market: Traded safe assets offer 5% annual return

A potential buyer of the sure CF also expects 5% return. Let theprice she is willing to pay be X. Then X (1+5%)=$1,100.

Thus, X=$1,100/1.05=$1,048 which is the CF's present value, i.e., itscurrent market value.

Observation: Present value properly adjusts for time

5

Present value (PV)Present value (PV)

X (1+ 0:05) = 1;100


Example. How much is a risky cash flow in one year with aforecasted value of $1,100 worth now?

Market: Traded assets of similar risk offer 20% annual return

A potential buyer of the risky CF also expects 20% return. Let theprice be X. Then

Thus, the present value of the risky CF is X=$1,100/1.20=$917

Observation: Present value properly adjusts for risk

An asset’s present value equals its expected cash flow discounted atthe appropriate cost of capital (discount rate).

6

Present Present vvalue alue (PV)(PV)

X (1+20%)=$1,100


7

Expected returnsExpected returns

32.917.4Small stocks

20.212.3Large stocks

8.56.2Long-term corp debt

9.25.8Long-term gov bond

-3.8Short-term gov bond

Standard deviationMeanAsset

Average annual returns on different assets (1926-2005, nominal)


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Example. (1) $10M in 5 years or (2) $15M in 15 years. Which isbetter if r = 5%?

PVA = 101:055 = 7:84; PVB = 15

1:0515 = 7:22

PV =CFT(1+ r)T

PVA = 101.055 = 7.84

PVB = 151.0515 = 7.22


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$0.0

$0.2

$0.4

$0.6

$0.8

$1.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Year when $1 is received

r = 0.04 r = 0.08 r = 0.12

PV of $1 Received In Year t


Solution to Example. Flu Vaccine. Assume that r = 5%.

Strategy A:

10


Time 0 1 2 3Cash Flow -1,000 500.0 400.0 300.0Present Value -1,000 476.2 362.8 259.2

Total PV 98.2

Time 0 1 2 3Cash Flow -200 -200.0 300.0 300.0Present Value -200 -190.5 272.1 259.2

Total PV 140.8

PV =CF11+ r

+CF2(1+ r)2

+ ...+ CFT(1+ r)T

=CFt(1+ r)tt=0

T

∑

98.2Total259.2362.8476.2-1,000PV300400500-1,000CF3210Time


Solution to Example. Flu Vaccine. Assume that r = 5%.

Strategy B:

Firm should choose strategy B. Firm value would increase by $140.8 M

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Time 0 1 2 3Cash Flow -1,000 500.0 400.0 300.0Present Value -1,000 476.2 362.8 259.2

Total PV 98.2

Time 0 1 2 3Cash Flow -200 -200.0 300.0 300.0Present Value -200 -190.5 272.1 259.2

Total PV 140.8

PV =CF11+ r

+CF2(1+ r)2

+ ...+ CFT(1+ r)T

=CFt(1+ r)tt=0

T

∑

140.8Total259.2272.1-190.5-200PV300300-200-200CF3210Time


How much will $1 today be worth in one year?

Current interest rate is, say, 4%

_ $1 investable at a rate of return

_ FV in 1 year: $(1+4%)

_ FV in T years: $(1+4%) (1+4%) … (1+4%)=$(1+4%)T

Example. Bank pays an annual interest of 4% on 2-year CDs and youdeposit $10,000. What is your balance two years later?

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Future value (FV)Future value (FV)

r = 4%

FV = 1+ r = $1:04

FV = $1£ (1+ r)£ ¢¢¢£ (1+ r)= (1+ r)T

FV = 10;000£ (1+ 0:04)2 = $10;816

FV = $10,000(1+ 0.04)2 = $10,816


Annuity: A constant cash flow for T periods (starting in period 1)

Today is t=0 and cash flows start at t=1.

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Special cash flowsSpecial cash flows

-t = 0 1 2 T time

6 6 6A A A

¢¢¢

PV =A1+ r

+A

(1+ r)2+ ...+ A

(1+ r)T= A 1

r1− 1

(1+ r)T⎛⎝⎜

⎞⎠⎟

FV = PV (1+ r)T


Example. An insurance company sells an annuity of $10,000 per yearfor 20 years. Suppose r = 5%. What should the company sell itfor?

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PV = 10;000£ 10:05£

µ1¡ 1

1:0520¶

= 10;000£ 12:46

= 124;622:1

PV = $10,000 10.05

1− 11.0520

⎛⎝⎜

⎞⎠⎟= $124,622.1


15


Annuity with constant growth rate g

-t = 0 1 2 T time

6 66

A A(1+ g) A(1+ g)T ¡ 1

¢¢¢

PV = A 11+ r

+1+ g

(1+ r)2 + ...+ (1+ g)T −1

(1+ r)T⎛⎝⎜

⎞⎠⎟= A

1r − g

1− 1+ g1+ r

⎛⎝⎜

⎞⎠⎟T⎛

⎝⎜⎞

⎠⎟ if r ≠ g

T1+ r

if r = g

⎧

⎨⎪⎪

⎩⎪⎪


Example. Saving for retirement - Suppose that you are now 30 andneed $2 million at age 65 for your retirement. You can save eachyear an amount that grows by 5% each year. How much shouldyou start saving now, assuming that r = 8%?

16


PV = A 10.08 − 0.05

1− 1.051.08

⎛⎝⎜

⎞⎠⎟

35⎛

⎝⎜⎞

⎠⎟⎛

⎝⎜

⎞

⎠⎟

20.898

FV = PV 1.0835 = A 308.978A = 2,000,000 / 308.978


Example. You just won the lottery and it pays $100,000 a year for 20years. Are you a millionaire? Suppose that r = 10%.

_ What if the payments last for 50 years?

_ How about forever - a perpetuity?

17


PV = 100;000£ 10:10

µ1¡ 1

1:1050¶

= 100;000£ 9:915

= 991;481

PV = $100,000 10.1

1− 11.120

⎛⎝⎜

⎞⎠⎟= $851,356

PV = $100,000 10.1

1− 11.1∞

⎛⎝⎜

⎞⎠⎟=$100,0000.1

= $1,000,000

PV = $100,000 10.1

1− 11.150

⎛⎝⎜

⎞⎠⎟= $991.481


Perpetuity with constant grow g

Example. Super Growth Inc. will pay an annual dividend next year of$3. The dividend is expected to grow 5% per year forever. Forcompanies of this risk class, the expected return is 10%. Whatshould be Super Growth's price per share?

18


-t = 0 1 2 3 time

6 6 6A A(1+ g) A(1+ g)2

¢¢¢

PV = 31:10+ 3(1+ 0:05)

1:102 + 3(1+ 0:05)21:103 + ¢¢¢= 3

0:10¡ 0:05= 60

PV =A

r − g, for r > g

PV =$3

0.10 − 0.05= $60


Interest may be credited/charged more often than annuallyBank accounts: dailyLoans and leases: monthlyBonds: semi-annually

For the same quoted interest rate, the effective annual rate may differ

Why?

Typical quote convention:

Annual Percentage Rate (APR)

k periods of compounding

Interest per period is APR/k

Actual annual rate differs from APR

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CompoundingCompounding

10% Compounded Annually, Semi-Annually, Quarterly, and Monthly


Example. Bank of America's one-year CD offers 5% APR, with semi-annual compounding. If you invest $10,000, how much money doyou have at the end of one year? What is the actual annual rateof interest you earn?

Quoted APR of 5% is not the actual annual rate

It is only used to compute the 6-month interest rate:

(5%)(1/2) = 2.5%

Investing $10,000, at the end of one year you have:

10,000(1+0.025)(1+0.025) = 10,506.25

In the second 6-month period, you earn interest on interest

The actual annual rate, the Effective Annual Rate (EAR), is

20


rE AR = (1+ 0:025)2 ¡ 1= 5:0625%

rEAR = 1.0252 −1 = 5.0625%


Let rAPR be the APR and k be the number of compounding intervalsper year. In one year, one dollar invested today yields:

Effective annual rate, rEAR is given by:

or

Example. Suppose rAPR = 5%.Here, e ≈ 2.71828

21


≥1+ rA P R

k´k

(1+ rE A R ) =≥1+ rA P R

k´k

1+ rAPRk

⎛⎝⎜

⎞⎠⎟k

rEAR = 1+ rAPRk

⎛⎝⎜

⎞⎠⎟k

−1


Nominal vs. real CFs

Example. Inflation is 4% per year. You expect to receive $1.04 inone year, what is this CF really worth next year?

The inflation adjusted or real value of $1.04 in a year is

_ Nominal cash flows ⇒ expressed in actual-dollar cash flows_ Real cash flows ⇒ expressed in constant purchasing power

At an annual inflation rate of i , we have

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Nominal Nominal vsvs. real . real CFsCFs/rates/rates

Real CF = Nominal CF1+ in° ation = 1:04

1+ 0:04 = $1:00

RCF =NCF1+ i

=$1.041+ 0.04

= $1

RCFt =NCFt(1+ i)t


23


EMU = Euro-zone (MUICP), EU = EICP, AT = Austria, BE = Belgium, BG = Bulgaria, CY = Cyprus, CZ = Czech Republic, DE = Germany, DK =Denmark, EE = Estonia, EL = Greece, ES = Spain, FI = Finland, FR = France, HU = Hungary, IE = Ireland, IT = Italy, LT = Lithuania, LU =Luxembourg, LV = Latvia, MT = Malta, NL = Netherlands, PL = Poland, PT = Portugal, RO = Romania, SE = Sweden, SI = Slovenia, SK =Slovakia, UK = United Kingdom.

EU = EICP includes 15 member countries up to April 2004, 25 member countries from May 2004 and 27 member countries from Jan 2007. EMU =MUICP includes 13 member countries.

Inflation in Europe (2008)


Nominal vs. Real Rates_ Nominal rates of return ⇒ prevailing market rates_ Real rates of return ⇒ inflation adjusted rates

Example. $1.00 invested at a 6% interest rate grows to $1.06 nextyear. If inflation is 4% per year, then the real value is

$1.06/1.04 = 1.019

The real return is 1.9%.

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rreal =1+ rnom1+ i

−1 ≈ rnom − i


Example. Sales is $1M this year and is expected to have a realgrowth of 2% next year. Inflation is expected to be 4%. Theappropriate nominal discount rate is 5%. What is the presentvalue of next year's sales revenue?

_ Next year’s nominal sales forecast: (1)(1.02)(1.04) = 1.0608

_ Next year’s real sales forecast: (1)(1.02) = 1.02

For PV calculations, treat inflation constantly_ Discount nominal cash flows using nominal discount rates_ Discount real cash flows using real discount rates

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Nominal Nominal vsvs. real . real CFs CFs and ratesand rates

PV =1.06081.05

= 1.0103

rreal =1.051.04

−1 = 0.9615%

PV =1.02

1.009615= 1.0103


Example. Fixed rate mortgage calculation in the U.S._ 20% down payment, and borrow the rest from bank using property as collateral_ Pay a fixed monthly payment for the life of the mortgage_ Have the option to prepay

Suppose that you bought a house for $500,000 with $100,000 down paymentand financed the rest with a thirty-year fixed rate mortgage at 8.5% APRcompounded monthly

_ The monthly payment M is determined by

_ Effective annual interest rate (EAR):

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Mortgage exampleMortgage example

400;000 =360X

t= 1

M[1+ (0:085=12)]t

= M(0:085=12)

n1¡ 1

[1+ (0:085=12)]360o

= M £ (0:9212)(0:085=12)

M = 3075:65

[1+ (0:085=12)]12 ¡ 1= 1:08839¡ 1= 8:839%

$400,000 = M

1+ 0.08512( )t=

M0.085

121− 1

1+ 0.08512( )360⎛

⎝

⎜⎜

⎞

⎠

⎟⎟t=1

360

∑

M = $3075.65

1+ 0.08512( )12 −1 = 8.839%


Monthly payments

_ Total monthly payment is the same for each month_ The percentage of principal payment increases over time_ The percentage of interest payment decreases over time

27

Mortgage exampleMortgage example

t (month) P rincipal I nterest Sum Remaining P.1 242.37 2833.33 3075.7 399,757.632 244.08 2831.62 3075.7 399,513.553 245.81 2829.89 3075.7 399,267.74...

..

....

..

....

120 561.29 2514.42 3075.7 354,415.49121 565.26 2510.44 3075.7 353,850.23

..

....

..

....

..

.240 1309.27 1766.43 3075.7 248,068.95241 1318.54 1757.16 3075.7 246,750.41

..

....

..

....

..

.359 3032.60 43.10 3075.7 3,054.07360 3054.07 21.63 3075.7 0.00


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ExtensionsExtensions

_ Taxes

_ Currencies

_ Term structure of interest rates

_ Forecasting cash flows

_ Choosing the right discount rate (risk adjustments)

PV =E[CF1]1+ r1

+E[CF2 ](1+ r2 )

2 + ...


29

SummarySummary

_ Present value

_ Future value

_ Special cash flows

_ Compounding

_ Nominal versus real cash flows and discount rates

_ Extensions

Documents

15.401 Finance Theory I - MITweb.mit.edu/astomper/www/univie/pof/Chapter 2.pdf · 2009. 12. 29. · 15.401 Lecture 2: Present value Visualizing cash flows (assets): Example. Drug