Chapter 1-2

EMSE 6450: Investment Science - Leunberger

EMSE 6450 : Investment Science - Leunberger Page 1Chapter 1 : Introduction Notes by J.Rene van DorpAdapted from notes by: Süleyman Özekici, Department of Industrial Engineering, Koç University www.seas.gwu.edu/~dorpjr

Chapters 1 & 2 : Introduction and Basic Theory of Interest

Lecture Notes by: J. René van Dorp"

Faculty Page: www.seas.gwu.edu/~dorpjr

1Department of Engineering Management and Systems Engineering,School of Engineering and Applied Science,

The George Washington University, 1776 G Street, N.W. Suite 135,ß

Washington D.C. 20052. E-mail: [email protected], [email protected]

Cash Flow and Streams


• Investment: in order to achieveCurrent commitment of resourceslater benefits.

• Broader viewpoint: An investment is a cash flow of expendituresand receipts at different time points over a period of time.

• Investment Engineering: to beThe design of cash flow patterns"as desirable" as possible.

• Cash Flow ( )GJ À

B œ 55 net receipt at the beginning of time period

• Cash Flow Stream ÐGJWÑ À

(B B B ß á ß B ß B Ñ!ß "ß # 8" 8



• Fixed Income Security À Can be represented by a deterministic cashflow stream À

(1.00,0.10,0.10, 1.10Ñ

which represent an investment of $1.00 in a bank account paying %"!interest annually for three years.

Time (years)

1.00

0.10 0.10

1.10

0

1 2 3



• How much should I be willing to pay for a given stream?

• Which of two cash flow streams is more preferable?

• Is sum of two streams worth more than sum of their individual values?

• If I can purchase a share of a stream how much should I purchase?

• Given a portfolio of available cash flow streams what is the mostfavorable combination of them?

• What if timings of of amounts can be controlled? What is the> B5 5

optimal timing?

• What if the future cash flows are uncertain? Still considered a CFS, butdecision analysis becomes more complex.

The Comparison Principle


• Wide Application field: Personal, business and government.

• Investment Analysis (IA): A process of examining alternatives andrecommending the most preferable one.

• Decision Analysis (DA): much of investment analysis relies on thesame general DA tools.

• Contrast DA and IA: Investments are carried out within the contextof a financial market. This provides alternatives not found in otherdecision situations, .making IA unique and powerful

• Comparison Principle:

( preferred to (if for all B B B ß á ß B ß B Ñ C C C ß á ß C ß C Ñ

B C ß 3!ß "ß # 8" 8 !ß "ß # 8" 8

3 3 (what assumption is implicit here?)

Issues and Assumptions


• Arbitrage: Suppose one bank offers % and another bank offers"!"#% interest. This means that without investing money of your ownyou can make 2% on any large amount.arbitrary

• No Arbitrage Assumption: Above scenario is not possible Ê (1) Pricing relations are linear, (2) stock prices satisfy certain relations,

(3) Pricing of derivatives securities, such as options and futures, can bedetermined analytically.

• Dynamics: The value of an asset is not a single number but changeswith time and subject to uncertainty Hence, it is .Þ a stochastic process

With one can structure investmentsspecificied dynamic characteristicsto take advantage of this dynamic nature and increase portfolio value.

Issues and Assumptions


• Risk Aversion Principle:

ÐB ß B Ñ ÐB ß\ Ñß IÒ\ Ó œ B ß Z Ò\ Ó !! " ! " " " "is preferred over

if the expectedOne only accepts additional risk (uncertainty)payoff is larger.

• Mean-Variance Analysis: Faced with the same expected pay-off's butdifferent variances, one selects the one with the smallest variance (bythe risk aversion principle).

• Mean-Variance Method: The basis for the most well-known methodof quantitative portfolio analysis. (who wonPioneered by Markowitzthe Nobel prize in economics for his work).

Typical Investment Problems


• Pricing: What is the of the cash flow streamvalue/price

Ð\ ß\ ß\ ßá ß\ Ñß \! " # 8 3 where random.

Usually solved using . Clever argumentsthe comparison principleallow complex investments to be separated into parts that can becompared with .other investments for which the prices are known

• Hedging: How can one invest to eliminate or reduce financialrisks due to uncertainty e.g. fire, theft or adverse price changes?

• Portfolio Selection: What is the portfolio that meets theoptimalrequirements of an investor?

Where do we go from here?


• Part 1: Deterministic cash flow streams: Streams of this type, eitherwith one or several periods are analyzed with various interest rateconcepts. Requires a discussion/review of interest rate theory.

• Part 2: Single period random cash flow streams: Single period withony a , but with beginning and ending flow the second one beinguncertain. Requires a description of uncertain returns (using probabilitytheory) and how an individual assesses returns. Starts with the mean-variance analysis assessment method.

• Part 3: Derivative assets: Random flows at each time points, butwhere the asset producing this stream is to anotherfunctionally relatedasset .whose price characteristics are known

A stock option to buy shares of stock in the future. TheExample:option derives its value from the stock and its value changes over time.

The Basic Theory of Interest

EMSE 6450 : Investment Science - Leunberger Page 10Chapter 2 : The Basic Theory of Interest Notes by J.Rene van DorpAdapted from notes by: Süleyman Özekici, Department of Industrial Engineering, Koç University www.seas.gwu.edu/~dorpjr

• Let denote the principal investment. Let denote the future value.E Z

• Simple annual interest for years:< 8

Z œ Ð" <8ÑE

• Simple annual interest for units of time (in years):< >

Z œ Ð" <>ÑEß i.e. value function is linear in >

• Yearly compounding with interest for years:< 8

Z œ Ð" <Ñ E8 , i.e. value function is geometric in 8

• Compounding times annualy, annual interest for periods:7 < 5

Z œ Ð" Ñ E<

75

Interest: The Time Value of Money


• Effective annual interest rate in latter case satisfies:<w

< œ Ð" Ñ " < ´<

7' , annual interest rate (called nominal rate)7

• Continuous compounding after periods:5 ¸ 7>

lim lim7 Ä ∞ 7 Ä ∞

Ð" Ñ œ Ð" Ñ œ /< <

7 77> 7 <>

> Effective annual interest rate œ / "<

Table 2.1. Continuous Compounding

Nominal 1.00% 5.00% 10.00% 20.00% 30.00% 50.00% 75.00% 100.00%

Effective 1.01% 5.13% 10.52% 22.14% 34.99% 64.87% 111.70% 171.83%

Interest rate (%)

Interest: The Time Value of Money


$0

$200

$400

$600

$800

$1,000

$1,200

0 2 4 6 8 10 12 14 16 18 20 22 24

Valu

e

YearsExponential Growth Compound Twice AnnualyCompound Annualy Simple

Money Markets


Table 2.2. Market Interest RatesInterest rates (August 9, 1995)

U.S. Treasury bills and notes3-month bill 5.396-month bill 5.391-year bill 5.363-year note (% yield) 6.0510-year note (% yield) 6.4930-year note (% yield) 6.92

Fed funds rate 5.6875Discount rate 5.26Prime rate 8.75Commercial paper 5.84Certificate of deposit

1 month 5.172 months 5.241 year 5.28

Banker's acceptances (30 days) 5.68London late Eurodollars (1 month) 5.75London Interbank offer rate (1 month) 5.88Federal Home Loan Mortgage Corp. (Freddie Mae) (30 years) 7.94Many different rates apply on any given day. This is a sampling.

Present Value


• Present Value: Money invested today leads, with interest, to anincreased value in the future. This thinking tocan also be reversedconvert .future dollars to present value

1. $100 today at % is worth $ dollars one year later "! ""! Ê 2. $110 in one year given 10% interest is worth $100 now.

• Discount Factor: Future value may be converted to present values viathe discount factor.. If annual interest was % and compounding<occurs times per year, the discount factor for future values after 7 5periods is:

Z œ Ð" Ñ E Í E œ Z Ê . œ< " "

7 Ò" Ð<Î7ÑÓ Ò" Ð<Î7ÑÓ5

5 55

Present and Future Values of Streams


• The Ideal Bank: An ideal bank applies the same rate of interest toboth deposits and loans, regardless of the amount, and it has noservice charges or fees. Interest rates .may differ per length of a term

• The Constant Ideal Bank: If has an interest rate thatan ideal bankis for which it applies, independent of the length of term and interestis compounded a constant ideal bank, it is said to be .

• Reference Point: The concept is used as aconstant ideal bankreference point via to evaluate outsidethe comparison principle financial markets.

• Cash Flow Stream ( )GJW À

(B B B ß á ß B ß B Ñ!ß "ß # 8" 8

Present and Future Values of Streams


• Future value of CFS after periods8 À

JZ œ B Ð" <Ñ B Ð" <Ñ á B Ð" <Ñ B! " 8" 88 8"+

• Present value of CFS À

TZ œ B á B B B

Ð" <Ñ Ð" <Ñ Ð" <Ñ!

" 8" 8

8" 8+

Example 2.1: GJW œ Ð #ß "ß "ß "Ñ < œ "! and %

JZ œ ‚ Ð"Þ"Ñ " ‚ Ð"Þ"Ñ " ‚ "Þ" " œ !Þ'%)2 + $ #

Example 2.2: GJW œ Ð #ß "ß "ß "Ñ < œ "! and %

TZ œ œ !Þ%)(" " "

"Þ" Ð"Þ"Ñ Ð"Þ"Ñ2 +

# $

Frequent and Continuous Compounding


• Frequent Compounding: For ( withGJW B B B ß á ß B ß B Ñ!ß "ß # 8" 8

annual interest rate and compounding over periods annually< 7 À

TZ œ 5œ! 5œ!

8 8B

Ò" Ð<Î7ÑÓœ . ‚ B ß . œ "ÎÒ" Ð<Î7ÑÓ

5

5 5 5 55

• Continuous Compounding: Denoting as the cash flow at timeBÐ> Ñ5> > > > ß á ß > ß > Ñ5 !ß "ß # 8" 8, we have for a cash flow stream at times ( withnominal interest an continuous compounding:<

TZ œ 5œ!

8

/ ‚ B > Ñ<>5

5 (

• Two deterministic cash flow streams are equivalent if they havethe same PV (via the comparison principle).

Internal Rate of Return (IRR)


• Consider a cash flow stream ( with interest rateB B B ß á ß B ß B Ñ!ß "ß # 8" 8

< then

TZ œ 5œ!

8 B

Ð" <Ñ5

5.

If the CFS above corresponds to a series of deposits and withdrawlsfrom PV .an Constant Ideal Bank Ê œ !

• We can turn this around. Given a cash flow stream (B B B ß!ß "ß #

á ß B ß B Ñ8" 8 , for what value of the interest rate could this have<been a CFS from an constant ideal bank?. The answer is thesolution to the equation.

! œ B B - B - á B - ß - œ "ÎÐ" <Ñ! " # 8# 8 where



• < œ Ð"Î-Ñ " is called the Internal Rate of Return (IRR)

• Internal Rate of Return of ( is thatGJW B B B ß á ß B ß B Ñ!ß "ß # 8" 8

interest rate for which the of this equals < TZ GJW !Þ

Example 2.3 : + For the IRR is found by solvingÐ #ß "ß "ß "Ñ

! œ # - - - Ê - œ !Þ)" Ê MVV œ Ð"Î-Ñ " œ !Þ#$# $

Example 2.3b: For the IRR is found by solvingÐ#ß "ß "ß "Ñ

! œ # - - - Ê - œ !Þ)" Ê MVV œ Ð"Î-Ñ " œ !Þ#$# $

What is the difference between the IRR's in Examples 2.3a and 2.3b?



• To interpret the one needs to look at the initial outlay of theMVV B!

GJW ( .B B B ß á ß B ß B Ñ!ß "ß # 8" 8

If the cash flow stream involves and oneB !! an initial investmentinterprets The investment isthe as interest that is received.MVV B!

worthwhile if the is MVV larger than the prevailing market interestrate.

If the cash flow stream involves and one interprets B !! a loan theMVV as interest that is paid. The loan is worthwhile if the B MVV!

is .less than the prevailing market interest rate

• Main IRR Theorem: A ( withGJW œ B B B ß á ß B ß B Ñ!ß "ß # 8" 8

B ! B ! 5 œ "ß #ß $ß á ß 8! 5and for and at least one term being



strictly positive has a unique IRR , where is the root ofœ Ð"Î-Ñ " -the equation:

0Ð-Ñ œ !ß 0Ð-Ñ œ B B - B - á B - ß ! " # 8# 8

and implies .5œ!

8

5B ! MVV !

Proof:1. ,0Ð!Ñ œ B !!

#Þ 0Ð-Ñ B !ß b B ! is a strictly increasing function since ,5 5

$Þ 0Ð-Ñ œ ∞- Ä ∞

lim

"ß # Ê 0Ð-Ñ œ ! - − Ð!ß∞Ñand 3 has a unique solution .!



4. .5œ!

8

5 ! !B ! Ê 0Ð"Ñ ! Ê ! - " Ê MVV œ Ð"Î- Ñ " !

Evaluation Criteria


• Evaluation Criteria: The essence of investment analysis involves theselection from a number of CFS's. Net Present Value (NPV) and theIRR criterion are the most commonly used.

• Net Present Value (NPV): The same as PV calculation of CFS, butNPV notation is used to emphasize that all cash flows (both positiveand negative) Choose that alternative with need to be included. highest NPV.

• Internal Rate of Return (IRR): Investment would only be worthwhile if IRR is larger than the prevailing interest rate. It would appearthat the higher the internal rate of return on an investment (i.e.B !Ñ! , the more desirable it is. This is not always consistent with theNPV criterion, however.

Evaluation Criteria


Example 2.4: Suppose that you have the opportunity to plant trees and cutthem later for profit. There are 2 investment strategies with :< œ !Þ"!(a) Cut Early or (b) Cut Later, specifically:

(a) : Harvest after 1 year with cash flow stream ,Ð "ß #Ñ

(b) : Harvest after 2 years with cash flow stream .Ð "ß !ß $Ñ

(a) : RTZ œ " #Î"Þ" œ !Þ)#Þ

(b) :RTZ œ " $ÎÐ"Þ"Ñ œ "Þ%)Þ#

Hence, using NPV criterion it is best to cut later!

Evaluation Criteria


Example 2.5: Suppose that you have the opportunity to plant trees and cutthem later for profit. There are 2 investment strategies where .< œ !Þ"!There are two alternatives: (a) Cut Early, (b) Cut Later, i.e.:

(a) : Harvest after 1 year with cash flow stream ,Ð "ß #Ñ(b) : Harvest after 2 years with cash flow stream .Ð "ß !ß $Ñ

(a) : " #- œ ! Ê - œ "Î# Ê MVV œ " œ "Þ"-

(b) : " $- œ ! Ê - œ "Î$ Ê MVV œ $ " ¸ !Þ(# Hence, using IRR criterion it is best to cut earlier!

Comparison of NPV and IRR


• NPV is appropriate for a single period CFS. That is the investmentinvolves that is terminated at the end.a one-time project

• IRR is more appropriate for CFS when the investment involves aproject that may be .repeated or cycled

• Conclusion will be the same if this is dealt with properly!

Example 2.4 & 2.5 continued: Cycle the 's alternativesGJWusing 1 unit of profits for reinvestment to obtain .equal time horizon

Table: Repeat Cutting Early after 4 YearsCombined

Time 1 2 3 4 5 CFS0 -1 -11 2 -1 12 2 -1 13 2 -1 14 2 -1 1

Iteration



Table: Repeat Cutting Late after 4 YearsCombined

Time 1 2 3 CFS0 -1 -11 0 02 3 -1 23 0 04 3 -1 2

Iteration

Alternative (a): while retaining 1 for cyclingÐ "ß "ß "ß "ß "Ñ

Alternative (b): ( while retaining 1 for cycling "ß !ß #ß !ß #Ñ

RTZ Ð+Ñ œ #Þ"( MVVÐ+Ñ œ *$, %RTZ Ð,Ñ œ #Þ!# MVVÐ,Ñ œ '&, %

Hence, it is best to cut earlier according to both criteria!



• Caution: An IRR CFS comparison does not take the scales of thevalues into account. An IRR analysis on two CFS's ought to beB5

conducted on the incremental cash flows (See, Homework 2.8).

Example: Consider the following two cash flow streams. Assume theinterest rate equals %. Please observe that both IRR 's are larger than< "#"#%. What if not both could be executed? Based on NPV and the IRR ofthe CFS of one would choose project EF EÞ

Year A B A-B0 -9000 -9000 01 480 5800 -53202 3700 3250 4503 6550 2000 45504 3780 1561 2219

IRR 18% 20% 15%NPV $1,442.61 $1,185.06 $257.55

Applications and Extensions


Example 2.6 (Simple Gold Mine): You are considering leasing a goldmine for 10 years. Gold can be extracted at a rate of ounces per"!ß !!!year at a cost of $ per ounce. Currently, the market price of gold is $#!! %!!per ounce. . Assuming the price of gold, theThe interest rate is %"!operating cost, and the interest rate remaining constant how much is thelease worth to you?

1. Operating at full capacity the profit per year equals:"!ß !!! ‚ Ð %!! #!! œ$ $ ) $2 Million per year

2. which isAssuming cash flows occur at the end of the year, conservative, the NPV of the 10 year cash flow of profits equals

RTZ œ œ # ‚ œ # ‚ Ò" Ð"Î"Þ"ÑÓ ‚ "! œ "#Þ#*# "

Ð"Þ"Ñ "Þ" 5œ" 5œ"

"! "!

5

5 "!



Year Cash flow PV1 2 1.818

2 2 1.653

3 2 1.503

4 2 1.366

5 2 1.242

6 2 1.129

7 2 1.026

8 2 0.933

9 2 0.848

10 2 0.771

Total NPV 12.289

3. The lease is worth $ to youConclusion: "#Þ#*Q Þ



Extra Homework: Show that À

3œ"

83

8

"ÎC œ" Ð"ÎCÑ

C "

• Cycle Problems: When using interest rate theory to evaluate ongoing(repeatable activities) it is that essential alternatives be comparedover the same time horizon. (Recall also the tree cutting example).

Example 2.7 (Automobile purchase): You are considering purchasing acar and have narrowed it down to two cars A and B.

Car A cost $ , Maintenance of $ per year (payable at beginning#!ß !!! "!!!of each year after first year) and has a useful life of 4 years.



Car B cost $ , Maintenance of $ per year (payable at beginning$!ß !!! #!!!of each year after first year) and has useful life of years.'

Neither car has a salvage value. Interest is constant at %. Which car< œ "!should you buy?

Car A: One cycle (4 years):

TZ œ #!ß !!! "!!! œ ##ß %)("

Ð"Þ"ÑE

5œ"

$

5 $

Three Cycles (12 years):

TZ œ TZ " œ %)ß $$'" "

Ð"Þ"Ñ Ð"Þ"ÑE$ E % )

$



Car A

r 10.00%Salvage Value after 4 years $0.00

Time Car A PVA

0 $20,000 $20,0001 $1,000 $9092 $1,000 $8263 $1,000 $7514 $0 $0 No Salvage Value!

$22,487

Cycle Time CF PV0 0 $22,487 $22,486.851 4 $22,487 $15,358.822 8 $22,487 $10,490.283 12 $0 $0.00 No Salvage Value!

$48,335.96



Car Br 10.00%

Salvage Value after 6 years $0.00

Time Car B PVB

0 $30,000 $30,0001 $2,000 $1,8182 $2,000 $1,6533 $2,000 $1,5034 $2,000 $1,3665 $2,000 $1,2426 $0 $0 No Salvage Value!

$37,582

Cycle Time CF PV0 0 $37,582 $37,581.571 6 $37,582 $21,213.822 12 $0.00 $0.00 No Salvage Value!

$58,795.39



Car : One cycle 6 years):F Ð

TZ œ $!ß !!! #!!! œ $(ß &)#"

Ð"Þ"ÑF

5œ"

&

5 $

Two Cycles (12 years):

TZ œ TZ " œ &)ß (*&"

Ð"Þ"ÑF# F '

$

Hence, choose Car A because NPV is evaluated here in costx

• What would the salvage value of Car B have to be for one to beindifferent if Car A has no salvage value?

Answer evaluated using GoalSeek in Excel: $ 1)(!$Þ'



Example 2.8 (Machine Replacement): An essential machine costs$ and has $ operating cost in the first year which increases by"!ß !!! #ß !!!$ each year. Assume operating cost occur at the end of each year."ß !!!The interest rate is %. How long should the machine be kept until it is"!replaced by a new one. Assume no salvage value.

Replacement after Year PV Cycle Purchase

CostYear 1 Maint.

Year 2 Maint.

Year 3 Maint.

Year 4 Maint.

Year 5 Maint.

Year 6 Maint.

1 $11,818 $10,000 $2,0002 $14,298 $10,000 $2,000 $3,0003 $17,303 $10,000 $2,000 $3,000 $4,0004 $20,718 $10,000 $2,000 $3,000 $4,000 $5,0005 $24,443 $10,000 $2,000 $3,000 $4,000 $5,000 $6,0006 $28,395 $10,000 $2,000 $3,000 $4,000 $5,000 $6,000 $7,000

If machine is replaced every year than the cash flow is (in thousands):

Ð "!ß #Ñ Ð!ß "!ß #Ñ !ß !ß "!ß #Ñand then and then ( , etc.



The of this cash flow has to satisfy the following equationTZ À

TZ œ "! Í TZ œ "$! Ð# TZ

"Þ" "Þ"in thousands $)

In case of a -year cycle we have5 À

TZ œ TZ Í TZ œ ‚ TZTZ Ð"Þ"Ñ

Ð"Þ"Ñ Ð"Þ"Ñ ">9>+6 " -C-6/ >9>+6 " -C-6/

>9>+6

5 5

5

Replacement after Year PV Total Factor

1 $130,000 11.002 $82,381 5.763 $69,577 4.024 $65,359 3.155 $64,481 2.646 $65,196 2.30

Replace Machine every 5 years!



• Taxes: A tax induced distortion of cash flows frequentlyaccompanies the treatment of . Depreciation isproperty depreciationtreated as negative cash-flow by the government, but the timing ofthese flows, as reported for tax purposes rarely coincides with theactual cash outlays.

Example 2.9 (Depreciation): Suppose a firm purchases a machine for$ that has a useful life of years and generates a cash flow of"!ß !!! %$ each year. It has a salvage value of $ at the end of 4 years.$ß !!! #ß !!!

The government does not allows to be reported asthe full machine costan expense the first year. Suppose each year one-fourth of the netmachine cost may be written off as an expense. Suppose the combinedfederal and state tax rate equals % on the taxable income. Determine%$before and after tax profitability.



Analysis results are provided in the table below.

Table: Example 2.9 before and after tax cash flow analysis

YearBefore-Tax Cash Flow Depreciation

Taxable Income Tax

After-Tax Cash Flow

1 -$10,000 -$10,0002 $3,000 $2,000 $1,000 $430 $2,5703 $3,000 $2,000 $1,000 $430 $2,5704 $3,000 $2,000 $1,000 $430 $2,5705 $5,000 $2,000 $1,000 $430 $4,570

NPV $876 -$487

Conclusion: Taxes here turned an otherwiseprofitable enterprise into a non-profitable one



• Inflation: Inflation is another factor that .often causes confusionInflation is characterized by an increase of prices over time. Supposethat the and the , theninflation rate is nominal interest rate is 0 <the follows from:real interest rate <!

" < œ Í < œ ¸ < 0ß 0" < < 0

" 0 " 0! ! when very small

Note that for small levels of inflation the real interest rate isapproximately equal to the nominal interest rate minus the inflationrate.

Example 2.10 (Inflation):Suppose the inflation is 4%, the nominal interest rate is % and we have a"!cash flow of real (or constant) dollars as shown in the second column.



Year Real Cash Flow

Inflation Factor

Nominal Cash Flow

Discount Factor

PV Nominal CFS @ 10%

PV Real CFS @ 5.77%

0 -$10,000 1.00 -$10,000 1.00 -$10,000 -$10,0001 $5,000 1.04 $5,200 0.91 $4,727 $4,7272 $5,000 1.08 $5,408 0.83 $4,469 $4,4693 $5,000 1.12 $5,624 0.75 $4,226 $4,2264 $3,000 1.17 $3,510 0.68 $2,397 $2,397

NPV $5,819 $5,819

It is common to estimate cash flow in constant or real dollars (i.e. currentbuying power). However, this means that prior to discounting at %, one"!should first inflate constant or real dollars to future values. This is referredto as in the third column below.the nominal cash flow

Alternatively, one could have discounted the real dollars at the real interestrate %< œ Ð!Þ"! !Þ!%ÑÎ"Þ!% œ &Þ((!

Homework Set 1


Due Date: Next Week

Problem 2.4Problem 2.6Problem 2.8Problem 2.10

Date post:	17-Feb-2016
Category:	Documents
Upload:	iroyme
View:	215 times
Download:	0 times

Chapter 1-2

Documents