"Numerical Techniques in Finance" by Simon Benninga: A Solution and Study Manual

7/22/2019 "Numerical Techniques in Finance" by Simon Benninga: A Solution and Study Manual

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Numerical Techniques in Finance by

Simon Benninga: A Solution and Study

Manual

R.E. Salvino

4329 Thistlewood Terrace

Burtonsville MD 20866

1995 - reformatted: 2 Apr 2013

Abstract

This study guide and solution manual is incomplete and offered in

an as is condition. It was written nearly 20 years ago when I began

a self-study program in finance in anticipation of a career shift. This

career shift never happened and is one of two reasons this guide and

manual was not completed. While writing solutions to the problems,

it occurred to me that the publisher, MIT Press, might be interested

in a solution manual to Professor Benningas book. While MIT Press

was interested in such a solution manual, Professor Benninga felt hisbook was near the end of its life cycle and, consequently, a solution

manual would not b e particularly helpful. This was the second reason

the study guide and solution manual was abandoned.

Keywords: finance, numerical techniques in finance, numerical finance,

financial simulations

Table of Contents

I CORPORATE FINANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Simulating Financial S tatements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Analyzing Mergers and Acquisitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

3 Debt Capacity and Riskless Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Leasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Current address: 9 Thomson Lane, 15-06 Sky@Eleven, Singapore 297726.

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5 Financial Analysis of Leveraged Leases ..............................17

II PORTFOLIO PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

6 Port folio Models - An Int roduct ion ..................................18

7 Calculating the Variance-Covariance Matrix .........................23

8 Calculating Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Estimating Betas and the Security-Market Line . . . . . . . . . . . . . . . . . . . . . 23

10 Entering Portfolio Information - A Macro . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 References for Port folio Models .................................... 25

I I I O P T I O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

12 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

13 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

14 Simulating the Normal and the Lognormal Distributions . . . . . . . . . . . 31

1 5 O p t i o n P r i c i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

16 Portfolio Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

IV DURATION AND IMMUNIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

17 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

18 Immunization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

V THE TECHNICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

19 The Gauss-Seidel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

20 The Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 1 M a t r i c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5

2 2 R a n d o m N u m b e r G e n e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5

23 Lotus Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

24 Macros in Lotus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

25 Data Table Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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PART I: CORPORATE FINANCE

1 Simulating Financial Statements

1.1 Introduction.

We begin with a summary of the main points in the chapter together witha recasting of the discussion into basic algebraic terms. Financial statementmodels may be considered as belonging to one of two main classes:

(a) Models which assume the firm wishes to maintain a given ratio of debt

to equity in its balance sheet (Warren & Shelton, 1971)

(b) Models which assume the firm wishes to maximize its value subject toa set of financing constraints (Myers & Pogue, 1974)

Models of type (b) are preferable to models of type (a), but type (a) modelsare more widespread.

The Warren & Shelton model assumes that most balance sheet items aredirectly or indirectly related to sales and that the firms primary financingconcern is to maintain an appropriate balance between the value of debt andthe value of equity in its balance sheet.

1.2 How Financial Models Work: Theory and Examples.

To set up such a model, we begin by defining a set of ratios of various quan-tities to sales: (1) current assets to sales, (2) current liabilities to sales, (3)net fixed assets to sales, (4) expenses (excluding interest and depreciation)to sales, and (5) debt to equity ratio,

Rca(n) =Ac(n)

S(n)(1.1)

Rcl(n) =Lc(n)

S(n)(1.2)

Rnfa(n) =Anf(n)

S(n)(1.3)

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Rex(n) =Eex

(n)

S(n) (1.4)

RDE(n) =DLT(n)

Eeq(n)(1.5)

where Ac(n) are current assets, S(n) are sales, Lc(n) are current liabilities,Anf(n) are net fixed assets, Eex(n) are expenses excluding interest and de-preciation, DLT(n) is long term debt, Eeq(n) = Es(n) +Er(n) is the equity,Es(n) is the stock component of equity, and Er(n) is the retained earningscomponent of equity, all for the nth year.

We must distinguish between (1) those financial statement items that are

functional relationships of sales and of other financial statement items and(2) those that involve policy decisions. The asset side of the balance sheetis usually assumed to be dependent on functional relationships; current li-abilities may also be taken as functional relationships only; and the majorpolicy decision is the target ratio of long-term debt to equity for each year.

To handle the troublesome modeling of fixed assets and depreciation, weassume (1) a depreciation policy that depreciates all fixed assets over aspecified life span (Td, e.g., 10 years) using straight line depreciation and(2) that new assets are purchased at the end of year (so that items acquiredin year n are first depreciated in year n+ 1). The accumulated depreciationis given by

Dacc(n) = Dacc(n 1) +Afac(n 1)

Td(1.6)

where Dacc(m) is the accumulated depreciation for year m, Aac(m) are theassets at cost for year m, and Td is the average depreciable life span of theassets.

The index valuem = 0 corresponds to the current year; the value for sales forthe current year, S(0), is given; the sales growth rate, GS, the growth of salesper year, is given; finally, the financial statement relations contained in theabove ratios are anticipated, that is, they are assumed given. The financialstatement can then be summarized by the following set of equations:

Assets

Ac(n) = S(n)Rca(n) (1.7)

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Afac(n) = Anf(n) + Dacc(n) (1.8)

Dacc(n) = Dacc(n 1) +Afac(n 1)

Td(1.9)

Anf(n) = S(n)Rnfa(n) (1.10)

ATot(n) = Ac(n) + Anf(n) (1.11)

where Afac(n) are fixed assets at cost and ATot(n) are the total assets foryear n.

Liabilities

Lc(n) = S(n)Rcl(n) (1.12)

DLT(n) = RDE(n)(Es(n) + Er(n)) (1.13)

Es(n) = Es(n 1) + Es(n) (1.14)

Er(n) = Er(n 1) + Er(n) (1.15)

LTot(n) = Lc(n) + DLT(n) + Es(n) + Er(n) (1.16)

where the sales for year n is S(n) = S(0)(1 + Gs)n, the addition to stock

equity is Es(n), the addition to retained earnings is Er(n), and LTot(n)are the total liabilities for year n.

Unknowns

The unknowns in the balance sheet, considered as the primary unknowns,are the increase in stock equity, total new debt, and the new debt (theaddition to total new debt),

Es(n) = ATot(n) Lc(n)DLT(n)Es(n 1) Er(n) (1.17)

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DTN(n) = DLT(n)DLT(0)(1 Pdn) (1.18)

DTN(n) = DTN(n) DTN(n 1) (1.19)DTN(0) = 0 by definition for the case where new debt is raised at thebeginning of the year.

Income Statement

The remaining part of the financial statement, the profit and loss or incomestatement, can be summarized by the following equations:

Eex(n) = Rex(n)S(n) (1.20)

I(n) = DLT(0)(1 Pdn)rc + DTN(n)rnew (1.21)

Edepr(n) = Dacc(n)Dacc(n 1) =Afac(n 1)

Td(1.22)

PBT(n) = S(n)Eex(n) I(n) Edepr(n) (1.23)

PAT(n) = PBT(n)(1 Tr) (1.24)

PDiv(n) = PAT(n)Po (1.25)

Er(n) = PAT(n) PDiv(n) = PAT(1 Po) (1.26)where I(n) is the interest paid in year n, rc is the current interest rate ondebt, rnew is the new interest rate on newly generated debt, Pd is the fractionof old debt to be paid in each year, EDepr(n) is the yearly depreciation,PBT(n) are the profits before taxes, PAT(n) are the after tax profits, Tr, isthe firms tax rate, PDiv(n) are the stockholder dividends, Po is the dividendpayout ratio, and Er(n) is the addition to retained earnings for year n.

Policy decisions are related to stipulated values for ratios such as RDE(n),that is, the values these ratios should have over the periods of interest areconstraints on the model. For example, in the model presented in the chap-ter, the policy decision is related to the behavior of RDE(n) over a 5 yearperiod: it is assumed that RDE(n + 1) = RDE(n) 0.02 for n 5 so thatRDE(0) is reduced to RDE(5) = RDE(0) 0.1.

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1.3 Setting Up and Solving the Model.

This set of equations for the financial statement model may be solved alge-braically by the substitution method. This will give a closed form solutionwhich can be used as a check on purely numerical methods to generate ac-curate solutions. First, in the equation for new stock, we can substitute theexpression for stock and solve for new stock,

Es(n + 1) =ATot(n + 1) Lc(n + 1)

1 + RDE(n + 1)Es(n)Er(n + 1) (1.27)

The only quantity on the right hand side which is unknown at this level ofthe indexing is the retained earnings, Er(n + 1). Since

Er(n + 1) = Er(n) + Er(n + 1) (1.28)

This means that the additional retained earnings part of the statement mustbe found. If we define the temporary variable,

F(n + 1) =ATot(n + 1) Lc(n + 1)

1 + RDE(n + 1) (Es(n) + Er(n)) (1.29)

F(n + 1) =ATot(n + 1) Lc(n + 1)

1 + RDE(n + 1) DLT(n)RDE(n)

(1.30)

then the equation for new stock may be written simply as

Es(n + 1) = F(n + 1) Er(n + 1) (1.31)

Furthermore, if we define the auxiliary variables,

A(n+ 1) (S(n+ 1)Eex(n)Edepr(n+ 1)DLT(0)(1Pd(n+1)) (1.32)

B(n + 1) RDE(n + 1)(Es(n) + Er(n)) (1.33)

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G(n+1) =

(A(n + 1)(rc rnew)B(n + 1)(1 + RDE(n + 1)rnew(1 Tr)(1 Po))

(1Tr)(1Po)) (1.34)

then the equation for the additional retained earnings, obtained from theincome statement, can be written as

Er(n+ 1) = G(n+ 1)RDE(n + 1)rnew(1 Tr)(1 Po)

[1 + RDE(n + 1)rnew(1 Tr)(1 Po)]Es(n+ 1)

(1.35)

These equations for the additional retained earnings and new stock can besolved in closed form to obtain

Es(n + 1) = (1 + RDE(n + 1)rnew(1 Tr)(1 Po))(F(n + 1) G(n + 1)) (1.36)

Er(n + 1) = G(n + 1) RDE(n + 1)rnew(1 Tr)(1 Po)(F(n + 1) G(n + 1)) (1.37)

Now that we have these quantities, everything else in the financial statementcan be obtained. Thus we have a closed form algebraic solution that canbe investigated in detail and used as a benchmark for purely numericalsolutions.

1.4 Where Do We Go From Here?

This is the basic type (a) model. Such a model requires considerable knowl-edge of how the firm works and it may be too simple: these are its majordrawbacks. On the other hand, as more complexity is incorporated, moreguessing is required which generates more opportunities for mistakes, and asimple model will have known faults and inaccuracies that render the simplemodel transparent compared to a more complex model: these are the majoradvantages of the simple model.

There are a number of more realistic elements that we may wish to consideras modifications to the basic algebraic model. Among these are the followingideas:

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(1) If a firm cant raise new equity, all new financing must come from

additional debt, that is, no additional stock is generated Es(n) = 0.This can actually simplify the implementation of the model.

(2) Break down current liabilities into payables (which bear no explicitinterest, although interest is implicit in extension of trade credit) andbank loans (which bear explicit interest charges). The present modelpays no interest on current liabilities, only on long term debt.

(3) Break down current assets into various items: (1) accounts receivablewhich are directly and linearly related to sales (some fixed percent-age of sales), (2) inventories (the optimal inventories increase withthe square root of sales according to operations research models: in-

ventories = 2aQK where a is the fixed cost of reordering, k is theunit carrying cost of inventory, and Q is the level of sales), and (3)cash and market securities which are needed to complete day to daytransactions associated with the business: cash may be a complicatedfunction of sales (see Brealey and Myers) and marketable securites areused to park excess cash and should not be directly related to sales.

(4) Break down fixed assets, detail the classes of assets, their relation tosales, and their depreciation schedules.

(5) Break down current liabilities into component liabilities.

(6) Improve the modeling of fixed asset accounts.

Modeling Fixed Assets. Item (6), however, is a difficult improvement toachieve. The present fixed assets model assumes that assets do not die andmust be removed from the depreciable base. This is troublesome to model.Once accumulated depreciation fully deprciates old assets, the old assetsshould be removed from the books: they are dead. This may be termed theactual fixed assets account, but it should be remembered that depreciationis not commonly accepted as reflecting the actual economic use of assets.However, the actual fixed assets model may be considered a better versionof the present fixed assets model. If the ages of all assets were known anddepreciation schedules were known, the actual fixed assets model could beimplemented into a more accurate model: but this is unrealistic and negatesthe purpose of financial statement modeling, that is, obtaining a simplifiedbut realistic snapshot of a firm at different stages in the future. Thus, theoptions for modeling fixed assets are:

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(a) Stick with the old model as long as it is not stretched too far into the

future and the initial asset base is not too old.

(b) If assets have a constant, depreciable life, in the long run the fixedassets at cost can be modeled by Afac = Anf/(1 1Td ).

1.5 Exercises

1. Generate the model described in the chapter. Assume that current lia-bilities bear no interest.

The policy decisions or anticipations are summarized by the ratiostatements,

Rca(n) = Rca(0) = 0.15 (1.38)

Rcl(n) = Rcl(0) = 0.08 (1.39)

Rnfa(n) = Rnfa(0) = 0.77 (1.40)

Rex(n) = Rex(0) = 0.80 (1.41)

RDE(n) = RDE(0) 0.02n (1.42)where RDE(0) = 0.5 and n 5. We may consider assigned values to anyof these ratios, either to remain fixed in time or to decrease in time by aspecified amount, as policy decisions. Since these ratios are now consideredknown, a method for solving the model can be chosen. In addition, theinitial sales are set at S(0) = 1000, the sales growth is set at GS = 0.10,the initial long term debt is DLT(0) = 280, the current interest rate isrc = 0.105, and this debt is to be repaid in annual equal installments over5 years (Pd = 1/5 = 0.2). The estimated new interest rate on any new debtis rnew = 0.095. The initial common stock balance is Es(0) = 450, and the

initial value of retained earnings is Er(0) = 110. The tax rate for the firmis Tr = 0.47.These equations can not be solved yet until some further information

is obtained: for the current year, the quantitites Afac(0), Dacc(0), andEDepr(0) must be defined or additional previous history of the firm must

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be available in order to calculate these quantities. Actually, since the net

fixed assets Anf(0) is known, only one of Afac(0) or Dacc(0) is needed inaddition to EDepr(0). In the text, it would appear that the following assign-ments were made:

Afac(0) = 1100 (1.43)

EDepr(0) = 110 (1.44)

Consquently, Dacc(0) = 330. The numerical results, rounded to the nearestinteger, are compiled in the table:

Balance Sheet

+ 0 1 2 3 4 5+ Ac(n) 150 165 182 200 220 242+ Afac(n) 1100 1287 1500 1744 2020 2355+ Dacc(n) 330 440 569 719 893 1095+ Anf(n) 770 847 932 1025 1127 1240+ ATot(n) 920 1012 1113 1225 1347 1482+ Lc(n) 80 88 97 106 117 129+ DLT(n) 280 300 320 342 364 387+ Es(n) 450 502 561 628 704 791

+ Er(n) 110 123 136 149 162 175+ LTot(n) 920 1012 1113 1225 1347 1482

Unknowns

+ Es(n) - 52 59 67 76 87+ DTN(n) - 76 77 77 78 79+ DTN(n) 0 76 152 230 308 387

Debt/Equity Ratio

+ RDE(n) 0.50 0.48 0.46 0.44 0.42 0.40

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Profit and Loss Statement+ S(n) 1000 1100 1210 1331 1464 1611+ Eex(n) 800 880 968 1065 1171 1288+ I(n) 29 31 32 34 35 37+ EDepr(n) 110 110 129 150 174 202+ PBT(n) 61 79 81 83 83 83+ PAT(n) 32 42 43 44 44 44+ PDiv(n) 22 29 30 31 31 31+ Er(n) 10 13 13 13 13 13

The calculation for new stock and new debt for the current year (Es(0)

and DTN(0)) either require knowledge for the previous year, since Es(1)and Er(1) are needed for Es(0) and DTN(1) is needed for DTN(0),or they must be assigned a value based on an assumption for the model,such as DTN(0) = 0 for the present model that assumes new debt is raisedat the beginning of the year.

2. Now consider a different problem. The firm feels that it will not be ableto sell any new stock in the next five years.

(a) If it pays no dividends, what will happen to its debt/equity ratio? As-sume the same growth rate of sales as in the previous problem.

(b) What is the maximum annual growth rate of sales so that the firms

debt/equity ratio will not exceed 0.6?

(a) For this problem, the quantity Es(n) is known and equal to 0; thedebt/equity ratio RDE(n) is no longer determined by policy but is now anunknown that takes the place of Es(n) in the algebra. Since Es(n) = 0,

Er(n) = F(n) (1.45)

Er(n) = G(n) (1.46)

where F(n) and G(n) are defined above. Eliminating Er(n) from theseequations determines RDE(n) in terms of known quantities. If we defineauxiliary functions by

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H(n) = [S(n) Eex(n)DLT(0)(1 Pdn)(rc rnew)](1 Tr)(1 Po) (1.47)

H(n) = H(n) rnew(1 Tr)(1 Po)(ATot(n) Lc(n))+Es(0) + Er(n 1) (1.48)

then the solution for the debt/equity ratio can be written as

RDE(n) =ATot(n) Lc(n) Es(0) Er(n 1)H(n)

H(n)(1.49)

This value of the debt/equity ratio is then used in the function F(n) todetermine Er(n),

Er(n) =ATot(n) Lc(n)

1 + RDE(n) (Es(0) + Er(n 1)) (1.50)

The only quantities which change for this problem, relative to the previous

problem, are on the liability side of the balance sheet (DLT(n), Es(n), andEr(n)) and on the profit and loss statement (I(n), PBT(n), PAT(n), PDiv(n),and Er(n)). The numerical values are tabulated below:

Balance Sheet

+ 0 1 2 3 4 5+ Ac(n) 150 165 182 200 220 242+ Afac(n) 1100 1287 1500 1744 2020 2355+ Dacc(n) 330 440 569 719 893 1095+ Anf(n) 770 847 932 1025 1127 1240+ ATot(n) 920 1012 1113 1225 1347 1482

+ Lc(n) 80 88 97 106 117 129+ DLT(n) 280 325 372 442 516 601+ Es(n) 450 450 450 450 450 450+ Er(n) 110 151 191 230 267 301+ LTot(n) 920 1014 1110 1228 1350 1481

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Unknowns+ Es(n) - 0 0 0 0 0+ DTN(n) - 101 103 126 130 141+ DTN(n) 0 101 204 330 460 601

Debt/Equity Ratio

+ RDE(n) 0.50 0.54 0.58 0.65 0.72 0.80

Profit and Loss Statement

+ S(n) 1000 1100 1210 1331 1464 1611+ Eex(n) 800 880 968 1065 1171 1288+ I(n) 29 33 37 43 50 57

+ EDepr(n) 110 110 129 150 174 202+ PBT(n) 61 77 76 73 69 64+ PAT(n) 32 41 40 39 37 34+ PDiv(n) 0 0 0 0 0 0+ Er(n) 32 41 40 39 37 34

There may be some propagation in rounding errors that can be eliminatedwith more care in the computations.

(b) The requirement that the debt/equity ratio go no higher than 0.60,that is, RDE(n) 0.6, places a constraint on the sales growth for a specifiedmaximum value ofn (in this case, taken to be 5). The inequality produces

the condition that

[ATot(n) Lc(n)](1 0.6rnew(1 Tr)(1 Po)) 1.6H(n) 1.6(Es(0) + Er(n 1)) 0 (1.51)

If we define the two auxiliary variables T1(n) and T2(n) by

T1(n) = [Rac(0) + Rnfa(0) Rcl(0)](1 0.6rnew(1 Tr)(1 Po))

1.6(1

Rex(0))(1

Tr)(1

Po) (1.52)

T2(n) = 1.6[DLT(0)(1 Pdn)(rc rnew) + EDepr(n)](1 Tr)(1 Po)+ 1.6(Es(0) + Er(n 1)) (1.53)

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then, upon substitution, eq. (1.51) may be written as

S(n)T1(n) T2(n) 0 (Ex2.9)

Consequently, sales must satisfy the inequality,

S(n) T2(n)T1(n)

(Ex2.10)

Since S(n) = S(0)(1 + Gs)n, this may be taken to be a condition on the

sales growth Gs. However, this requires knowledge ofEr(n1) which is notreadily available unless the calculation is redone with Gs an unknown.

A less computationally intensive approach is to note that a debt/equityratio of 0.6 lies between the results for years 2 and 3, RDE(2) = 0.58 andRDE(3) = 0.65. If we consider sales as a function of the debt/equity ratio,a simple linear interpolation gives S = S(2) + 1729(RDE 0.58) and forRDE = 0.6 this gives S= 1245. Thus, taking this value of sales as that cor-responding to the ratio RDE = 0.6, then S(n) 1245 or Gs (1.245)1/n1since S(0) = 1000. Taking n = 5 as the maximum time into the future, thesales growth must satisfy Gs 0.045 if the debt/equity ratio is to satisfyRDE(n) 0.6 for n 5.

3. The model described in the chapter assumes that all new debt is raised

at the beginning of the year and that repayments od the old debt are madeat the beginning of the year. Somewhat inconsistently, it assumes thatnew assets are acquired only at the end of each year - this is a didacticcompromise. Set out a model in which debt repayment is made at the endof the year and in which new debt is likewise raised at the end of the year.

As stated, the model in the chapter assumes all new debt is raised at thebeginning of the year (n = 1, 2, 3, 4, 5) and repayment of old debt is alsomade at the beginning of the year (n = 1, 2, 3, 4, 5). This can be inferredfrom the fact that the interest payments on new debt and old debt are

Inew(n) = DTN(n)rnew (1.54)

Ic(n) = DLT(0)(1 Pdn)rc (1.55)

and the total long term debt is

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DLT(n) = DLT(0)(1 Pdn) + DTN(n) (1.56)Interest paid on the new debt is calculated for the entire year, showingthe debt for year n was raised at the beginning of year n: year 0 showsno new debt, so new debt for year 1 can not be raised at end of year 0.Similarly, interest paid on the original debt is also calculated for the entireyear, showing the debt repayment for year n was made at the beginningof year n: year 0 shows no principal repayment was made, so principalrepayment applied to year 1 can not be made at end of year 0.

To make debt repayments at the end of the year (n = 1, 2, 3, 4, 5)requires no alteration of DLT(n) since new debt raised at the end of the

year n and repayment of the original debt at the end of the year n are stillapplied to year n. However, interest on new debt raised in year n does notaccrue until year n + 1 and interest on the original debt is not reduced bythe principal repayment at the beginning of the year,

Inew(n) = DTN(n 1)rnew (1.57)

Ic(n) = DLT(0)(1 Pd(n 1))rc (1.58)

Again, the new debt, DTN(m), is raised at the end of year m and therepayment of principal of the original debt, P

dDLT

(0), is also made at theend of the year m. Consequently, the only required model statement changeis

I(n) = DLT(0)(1 Pd(n 1))rc + DTN(n 1)rnew (1.59)

Eq. (1.59) should be used in place of the equation for interest given abovein the summary, eq. (1.21). All other model statements remain the same.

4. Consider the following variation of the model of exercise 2: Let currentassets denote only current assets needed to support sales, and add anothercurrent account called cash. Make the following additional assumptions:

the firm does not repay any debt, the interest on old debt and new debt isthe same (9.5 %), the ratio of net fixed assets to sales is 0.65, the firm putsretained earnings not needed to support sales into its cash accounts, andcash earns 8.7 % interest. Build an appropriate model.

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5. Find a firm whose financial statements are available for the past 10

years. Go back to the last 5 years and try to figure out how the variousfinancial statement items are related to the sales of the firm. Then builda financial statement model for the firm based on what you have learned.Finally, compare the output of your model with the actual results for thelast 5 years.

2 Analyzing Mergers and Acquisitions

3 Debt Capacity and Riskless Cash Flows

4 Leasing

5 Financial Analysis of Leveraged Leases

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PART II: PORTFOLIO PROBLEMS

6 Portfolio Models - An Introduction

6.1 Example.

We begin by defining the relevant quantities: E(Ri) is the expected returnon asset i, Var(Ri) is the variance of asset is return, Cov(Ri, Rj) is thecovariance between the returns of asset i and asset j, and the asset index iruns from 1 to NA where NA is the total number of assets.

The closing price at the end of month t=0,1,2,... is denoted by PA(t)with PA(0) denoting the initial price of the asset. The price return of anasset earned by an investor who bought the asset at the end of month t 1and sold it at the end of month t is defined as RA(t) = (PA(t)PA(t 1) +Div(t))/PA(t 1) where Div(t) is the dividend payment made by the asset.

The heroic assumption that is made is that the average of historic datarepresents the expected monthly return from the investment:

E(RA) =1

Nt

Ntk=1

RA(k) (6.1)

Var(RA) =1

Nt

Ntk=1

(RA(k)E(RA))2 (6.2)

A =

Var(RA) (6.3)

Cov(RA, RB) =1

Nt

Ntk=1

(RA(k)E(RA))(RB(k)E(RB)) (6.4)

A,B =Cov(RA, RB)

AB(6.5)

E(RA) is the expected return, Var(RA) is the variance of the return, A is thestandard deviation of the return, Cov(RA, RB) is the covariance of the returnof asset A with the return of asset B, and A,B is the correlation coefficient.

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The correlation coefficient satisfies the inequalities 1 A,B 1 and

measures the degree of a linear relationship that may exist between thereturns of asset A and asset B. If A,B = 1 then RA(t) = a + bRB(t) whereb > 0; if A,B = 1, the RA(t) = a bRB(t) where b > 0. If RA(t) andRB(t) are independent, then A,B = 0; however, A,B = 0 does not implythat RA(t) and RB(t) are independent (its a necessary condition, but nota sufficient condition).

6.2 Calculating Portfolio Means and Variances.

For a two asset portfolio, the fraction of the portfolio in asset A will bedenoted by and consequently the fraction of the portfolio in asset B is

1 . The expected return, the variance, and the standard deviation of thetwo asset portfolio are

E(Rp) = E(RA) + (1 )E(RB) (6.6)

Var(Rp) = 2Var(RA) + (1 )

2Var(RB) + 2(1 )Cov(RA, RB) (6.7)

2p = 22A + (1 )

22B + 2(1 )A,BAB (6.8)

The standard exercise is to plot E(Rp) as a function of p for rangingfrom zero to one.

6.3 Portfolio Means and Variances - General Case

The general case for mean and variance considers NA assets with i theproportion of asset i in the portfolio so that

NAi=1 i = 1. The asset weights

i are used to construct a column and row vector:

=

1

2...NA

(6.9)

T = (1, 2, . . . , NA) (6.10)

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and corresponding column and row vectors are constructed from the ex-

pected returns of the NA assets

(R) =

E(R1)E(R2)

...E(RNA

(6.11)

T(R) = (E(R1), E(R2), . . . , E (RNA) (6.12)

Consequently, the expected return of the portfolio

E(Rp) =i

iE(Ri) (6.13)

can be written in matrix notation as

E(Rp) = T(R) = T(R) (6.14)

The construction of the portfolio variance in matrix notation requires a littlemore work. First we note that the portfolio variance is

Var(Rp) =

NAi=1

2i Var(Ri) +

NAi=1

NAj=1

ijCov(Ri, Rj) (6.15)

where the double sum excludes the i = j term. Since Var(Ri) = Cov(Ri, Ri),this can be written as

Var(Rp) =

NAi=1

NAj=1


where the double sum includes the i = j term. Another way to write theportfolio variance is by restricting one of the double sum indices:

Var(Rp) =

NAi=1

2i Var(Ri) +

NAi=j+1

NAj=1


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Now we construct a covariance matrix, Sij = Cov(Ri, Rj), Sii = Var(Ri) =

Cov(Ri, Ri) so that the portfolio variance may be written as

Var(Rp) =

NAi=1

NAj=1

ijSij (6.18)

Since the matrix elements for S are

S =

S11 S12 ... S 1NAS21 S22 ... S 2NA.... .... ... ....

SNA1 SNA2 ... S NANA

(6.19)

the portfolio variance can be written very compactly in matrix notation as

Var(Rp) = TS (6.20)

Now, suppose that two portfolios consist of the same assets with the sameexpected returns, variances, etc. but the portfolios are assembled with differ-ent proportions for each asset. In other words, portfolio 1 has asset weightsi while portfolio 2 has asset weights i. Then the covariance between thetwo portfolios is given by

Cov(p1, p2) = ST

= Cov(p2, p1) = ST (6.21)

where is the column vector defined above and is the column vectorformed from the i.

6.4 Efficient Portfolios.

An efficient portfolio is a portfolio of risky assets that gives the lowest vari-ance of return of all portfolios having the same expected return. Alterna-tively, an efficient portfolio is a portfolio of risky assets that gives the highest

expected return of all portfolios having the same variance. Mathematically,an efficient portfolio minimizes the variance Var(Rp) =

i

j ijSij sub-

ject to the constraints

i iE(Ri) = E(Rp) and

i i = 1.

The efficient frontier is the set of all efficient portfolios. It is the locus ofall convex combinations of any two efficient portfolios. This means that if

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and are efficient portfolios, then so is any linear combination +(1)

where is some constant. Thus, we can find the entire efficient frontier ifwe can find any two efficient portfolios.

The minimization process subject to the specified constraints leads tothe following equations:

E(Rk) C =j

Skjzj (6.22)

j =zjk zk

(6.23)

or, in matrix notation,

(R) CI = SZ (6.24)

where I is the identity matrix. Solving this equation for Z gives

Z = S1[(R) CI] (6.25)

Substituting two different values for the constant C gives two different vec-tors and corresponding to two different efficienct portfolios. If corre-sponds to portfolio 1 and corresponds to portfolio 2, then the covariance

between the two efficient portfolios is Cov(p1, p2) = ST

. From these twoefficient portfolios, the entire efficient frontier can be obtained by

+ (1 ) (6.26)

where 0 1. IfR1 is the expected return for portfolio 1 () and R2 isthe expected return for portfolio 2 (), then

R1 = (R) (6.27)

R2 = (R) (6.28)

E(Rp) = R1 + (1 )R2 (6.29)

Var(Rp) = 2Var(R1) + (1 )

2Var(R2) + 2(1 )Cov(p1, p2) (6.30)

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Exercises

7 Calculating the Variance-Covariance Matrix

Exercises

8 Calculating Efficient Portfolios

Exercises

9 Estimating Betas and the Security Market Line

9.1 Introduction.

The Capital Asset Pricing Model (CAPM) is an equilibrium model of capitalmarkets. Its main conclusion is the equilibrium relation between risk andreturn given by the Security Market Line (SML):

E(Ri) = Rf + i[E(Rm) Rf] (9.1)

where E(Ri) is the expected return on asset i, E(Rm) is the expected returnon the market portfolio m, Rf is the return on a risk-free asset, and i isa measure of the riskiness of asset i. Formally, is a linear regressioncoefficient (slope) fitting the expected returns of an asset to the expectedreturns of the market as a whole. Formally,

i =Cov(Ri, Rm)

Var(Rm)(9.2)

An alternative version (Black, 1972) eliminates the assumption of the exis-tence of a risk-free asset with the assumption of the existence of a zero-betaasset:

E(Ri) = E(Rz) + i[E(Rm) E(Rz)] (9.3)

where E(Rz) is the expected return of an asset whose beta is zero, z = 0.It should be noted that the SML should hold when the market portfolio mis replaced by any efficient portfolio.

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Thus, there are really two types of tests relating to CAPM. The first

is a statistical test that determines whether we have an efficient portfolioon which to run regressions. The second is an economic test to determinewhether the portfolio we picked is really the market portfolio. The stepsof the test are: (1) determine a candidate for the market portfolio m, suchas the S&P 500 Index, (2) for each asset in question, determine the assetsbeta, and (3) regress the mean returns of the assets on their respective betasto give the SML.

9.2 Testing the CAPM

The simple test of the CAPM consists of calculating s for a set of assets andthen determining the equation of the SML (this also requires the definitionof the market portfolio, such as the S&P 500). There exists an enormousliterature discussing the statistical and methodological pitfalls of this test:the classic reference is Miller & Scholes (1972); Elton & Gruber (1984) andLevy & Sarnat (1984) provide surveys.

9.3 Setting Up the Spreadsheet

9.4 An Alternative Method for Calculating Beta

As an alternative to the covariance approach for obtaining i, one mayregress the returns of asset i on the returns of the market, however themarket is defined (in this case, the S&P 500 is chosen as the candidate forthe market portfolio m). i is the slope of the linear regression line - thetwo methods had better give the same result. But it seems to me thatthe covariance approach is simply the formula for the linear regression thatyields the slope.

9.5 Estimating the Securities Market Line

Once all the is have been determined, a second regression is then per-formed. Define the

is of the assets as the independent variables and the

mean returns of the assets as the dependent variables. A straight line isthen fitted to these data pairs to yield the SML: Predicted Mean Returnon Asset i is C1 + C2i where the constants C1 and C2 have been obtainedfrom this second regression.

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9.6 Is Our Market Portfolio Merely Efficient?

Take an efficient portfolio as the market portfolio, compute the s andmean returns for each of a number of other efficient portfolios and findexcellent linear behavior, that is, a very good linear regression fit. Thismerely indicates that the market portfolio is an efficient portfolio, and anyother efficient portfolio would work as well (e.g., use the data on page 85and on page 100).

Exercises

10 Entering Portfolio Information - A Macro

11 References for Portfolio Theory

Appendix: SML Minimization

As in statistical mechanics, we define a function that is a linear combina-tion of the function we wish to minimize and constraints with Lagrangemultipliers:

F =i,j

ijSij + 0i

iE(Ri) + 0i

i (A.1)

0 and 0 are arbitrary constants in this minimization process. In particular,0 has nothing to do with the beta of an asset, it is simply a Lagrangemultiplier here. Taking the derivative of F with respect to the portfolioweights k gives

F

k= 2j

jSkj + 0E(Rk) + 0 (A.2)

The symmetry of Sij has been used in obtaining this equation. The second

derivative of F is

2F

kl= 2Skl (A.3)

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The minimization process requires setting the first derivative to zero,

E(Rk) +00

= 2

0

j

Skjj (A.4)

Defining C = 0/0 and 2/0 = 1 gives the equation we seek,

E(Rk) C =j

Skjj (A.5)

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PART III: OPTIONS

12 The Normal Distribution

12.1 Background

The cumulative normal probability or distribution function is the integralof the normal probability or distribution density:

F(x) =

x

f(y)dy (12.1)

f(y) =e(y)2/22

22(12.2)

The random variable x is normally distributed with a mean and variance2 or standard deviation . The standard normal distribution is defined asthe normal distribution with zero mean and unit variance:

N(x) =

x

g(y)dy (12.3)

g(y) =ey

2/2

2(12.4)

This is simply obtained from the normal distribution by setting = 0 and = 1.

A convenient and apparently widely used approximation to the standardcumulative normal distribution is

N(x) = 1 h(x)t4

i=0

biti + (12.5)

h(x) =ex

2/2

2 (12.6)

t =1

1 +px(12.7)

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p = 0.2316419 (12.8)

< 7.5 108 (12.9)

and where the bi are given on page 114 of the text or in Handbook of Math-ematical Functions by Abramowitz and Stegun. This is form of the ap-proximation is valid for x > 0; for x < 0 the properties of the distributionfunction show that N(x) = 1N(x).

12.2 A Long Exercise

12.3 How To Prevent Your Spreadsheet From Thinking

Exercises

13 The Lognormal Distribution

13.1 A Crude Introduction

The price of any risky asset is uncertain. The question should not be whatis its price, but what is the probability distribution of its price. We may listsome reasonable properties: (1) the stock price is uncertain, (2) the stock

price is never zero, that is, we exclude dead companies, (3) the return fromholding a stock tends to increase over time, (4) changes in stock prices arecontinuous: over short periods of time, changes in the stock price are verysmall, tending to zero as t 0 (although it is possible to build occasionaland discontinuous random jumps into the model), and (5) the uncertaintyassociated with the return from holding a stock also tends to increase overtime: given the stock price today, the variance of the stock price tomorrowis small but increases as time goes on.

We will denote the stock price at time t by S(t). We will assume thatthe price S(t + t) is comprised of two parts, one certain or deterministicand the other uncertain or stochastic:

S(t + t) = S(t)e(t+tZ) (13.1)

where is the average rate of growth (assumed constant over time), t isthe elapsed time interval, is the standard deviation of the stock price,

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and Z is a standard random variable, that is, a random number drawn from

a normal distribution with zero mean and unit variance. The t in therandom part guarantees that the variance of the stocks log return is linearin t.

Note that if = 0, then the stock price grows at an exponential ratewith certainty, similar to a riskless bond that bears an interest rate and iscompounded continuously. For > 0, there is still a tendency for the stockprice to increase, but there is a normally distributed uncertain element thatmust be taken into account.

13.2 Some Properties of Equation (13.1)

A simple manipulation of eq. (13.1) yields the log return of the stock:

ln(S(t + t)/S(t)) = t +

tZ (13.2)

If we now take the expected value of this equation, we find that

E(ln(S(t + t)/S(t))) = t (13.3)

since the E(Z) = 0, that is, Z is drawn from a distribution with a zeromean. Calculating the variance yields linear growth in t,

Var(ln(S(t + t)/S(t))) = 2t (13.4)

since Var(Z) = 1, that is, Z is drawn from a distribution with unit vari-ance. In other words, the logarithm of the stock price returns is normallydistributed with mean and variance 2: this is termed the lognormaldistribution.

13.3 Calculating a Lognormal Distribution

We may calculate the density function of a lognormally distributed randomvariable. First, we calculate the densities for the standard normal distri-bution, for instance using the approximation described in Chapter 12. IfFprob(Z0) is the probability density that the standard normal random vari-able Z will have the value Z0, then Fprob(Z0) is the probability density that

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the lognormal distribution will have the value exp(t + 2

tZ0). For

example, if Z0 = 2.975, then Fprob(Z0) = N(Z+ Z0/2) N(Z Z0/2) =0.000239. In other words, our approximation is to attach the probability ofan interval to its midpoint If = 0.2, = 0.5, and t = 1, then the value ofthe lognormal density function is exp(0.2+0.5(2.975)) = 0.275959 occuringwith probability Fprob(Z0) = 0.000239.

13.4 From Price Data to Distributional Parameters

Suppose we are given a set of data in the following form:

+ Month Closing Price 1+R log(1+R)+ 0 25+ 1 24.69219 0.987687 -0.01238+ 2 23.68760 0.959315 -0.04153+ 3 22.88975 0.966317 -0.03426+ 4+ 5+ 6+ 7+ 8+ 9+ 10+ 11

+ 12

The monthly mean is then =

i log(1+ Ri) and the annualized mean issimply 12 = 0.072874. The monthly standard deviation is calculated in theusual way as the square root of the monthly variance, and the annualizedstandard deviation is then

12 = 0.099871. In other words, we write

S(t + t) = S(t)eR (13.5)

which is actually provides the definition of the variable R:

ln R = ln(S(t + t)/S(t)) = t + tZ (13.6)In other words, ln R has mean and standard deviation since the monthlytime increments are t = 1:

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E(ln R) = (13.7)

Var(ln R) = 2 (13.8)

(ln R) = (13.9)

This means that the end of the year price can be written as

S(12) = S(0)e(R1+R2+R3+R4+R5+R6+R7+R8+R9+R10+R11+R12) (13.10)

where S(0) is the beginning of the year price. If the log rates of return Rkare independent, the then annual log return ln(S(12)/S(0)) has an expectedvalue of 12 and expected variance of 122 where and are the monthlymean and standard deviation obtained from the monthly data.

13.5 Exercises

14 Simulating the Normal and Lognormal Distributions

14.1 Introduction

14.2 Simulating the Normal Distribution: A General

Description

The algorithm can be described in the following manner:

(1) Choose two random numbers, r1 and r2, uniformly distributed between-1 and +1

(2) Form the quantity S1 = r21 + r

22. If S1 1 go back to step (1); for

S1 < 1 go to step (3)

(3) From S1 form the quantity S2 = (

2 ln S1/S1)1/2

(4) Finally construct two independent random variables by x1 = r1S1 andx2 = r2S2: these random variables are from a normal distribution withzero mean and unit variance.

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14.3 The Macro Program

14.4 Some Output

14.5 Simulating the Lognormal Distribution

Since S(t + t) = S(t)exp(t + Z

t), it follows that S(t) is givenby S(0) exp(t + Z

t) so that

S(nt) = S(0)en(t+Zt) (14.1)

The quantities that are needed are todays price S(0), the annual log mean

return , the annual log standard deviation , and the time interval ofinterest t. For instance, if the daily price behavior is wanted, then t =1/250 since there are about 250 business days per year. Z is a randomvariable obtained from the standard normal distribution with zero meanand unit variance.

14.6 The Simulation

14.7 Technical Analysis

Security analysts are broadly divided into two groups. The fundamentalists

believe the value of a stock is ultimately determined by underlying economicvariables such as earnings, debt/equity ratio, markets, and so on. The tech-nicians or technical analysts believe that stock prices are determined bypatterns and that they can predict future prices based on the patterns ofpast behavior. The orthodox, ivory tower view of technical analysis is that itis worthless. The efficient market hypothesis, that markets efficiently incor-porate information known about securities that are traded on the markets, isa basic tenet of financial theory. In its weakest form, the weak efficient mar-kets hypothesis states that all information about past prices is incorporatedinto the current price. If this hypothesis is correct, it immediately impliesthat technical analysis can not make predictions of future prices since it is

based solely on past price information which is already incorporated in thecurrent price.

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Exercises

3. Let e(t) = ln X(t) where X(t) is the exchange rate between two cur-rencies. Then e(t) is normally distributed with an expectation value andvariance given by

E(e(t)) = (e(0) E)et + E (14.2)

Var(e(t)) =(e2t 1)2

2(14.3)

where < 0 and E is the log og the purchasing power parity rate, that is, E

is the log of the exchange rate that equilibrates the cost of baskets of goodsin the two countries, E = ln Xeq.

15 Option Pricing

15.1 Introduction

An option on a stock is a security that gives the holder the right to buy orsell shares of the underlying stock on or before a predetermined date fora predetermined price. A call is an option giving the holder the right to buy

shares of the stock. A put is an option giving the holder the right to sellshares of the stock. The exercise price K is the price at which the holder ofthe option can buy or sell the underlying stock. The expiration date T is thedate on or before which the holder can buy or sell the underlying stock. Thestock price S(t) is the price at which the underlying stock is selling at datet. The option price is the price at which the option is bought or sold: C(t)is the price of a call on date t and P(t) is the price of a put on date t. Theactual variable dependence of the options may be written as C(t, S(t), K , T )and P(t, S(t), K , T ).

There are two main types of options: American options and Europeanoptions. American options can be exercised on or before the expiration date

T, while European options can only be exercised on the expiration date T.However, the options sold on both American and European options marketsare usually American options. A theorem relating to calls states thatearly exercise is optimal only if the underlying stock pays dividends beforethe expiration date; even then, early exercise is optimal only at the moment

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before the dividend payment. In other words, most American call options

can be analyzed as if they were European call options.For every buyer of an option there corresponds the writer of the option.

In other words, the purchaser of a call acquires the right to buy sharesof stock for a given price and pays for this right at the time of purchase.Initially, there is a negative cash flow (the price of the option); the futurecash flow is at worst zero (if the option is not worth exercising) and otherwisepositive (if the option is worth exercising). The writer of the call sells theright to buy shares of stock and collects the price of the option in return forthe obligation to deliver the shares of stock in the future at the exercise priceif the purchaser of the call exercises the option. Initially, the cash flow ispositive (the price of the option); the future cash flow is at best zero (if the

option is not exercised) and otherwise negative (if the option is exercised).

For example, GP Sept 50 call option is a security giving the holderthe right to buy shares of GP on or before a specified date in September for$50. In most American markets, the date is the 3rd Friday in the month. Ifthe price of the GP Sept 50 call option today is $4, the buyer pays thisprice for the privilege of purchasing shares of GP at $50 between now andthe specified date in September, no matter what price GP shares are sellingfor at the time the option is exercised.

The purchaser of a put acquires the right to sell shares of stock for agiven price and pays for this right at the time of purchase. Initially, there

is a negative cash flow (the price of the option); the future cash flow is atworst negative (if shares are not owned and the option is exercised), zeroif the option is not exercised, and at best positive (if shares are owned andthe option is exercised). The writer of the put sells the right to sell sharesof stock and collects the price of the option in return for the obligation tobuy the shares of stock in the future for the exercise price if the purchaserof the put option wishes to exercise the option. Initially, the cash flow ispositive (the price of the option); the future cash flow is at best zero (if theoption is not exercised) and otherwise negative (if the option is exercised).In contrast to calls, early exercise of American puts may be optimal even inthe absence of dividend payments.

For example, GP Sept 50 put option is a security giving the holderthe right to sell shares of GP in September (on or before the same day inthe month as the call) for $50. If the price of the GP Sept 50 put optiontoday is $2, the buyer pays this price for the privilege of selling shares ofGP for $ 50 between now and September, no matter what the market price

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of GP shares are selling for at the time the option is exercised. The option

will be exercised only if the market price is less than $50, that is, less thanthe exercise price.

In the payoff patterns at expiration, the word profit is used loosely,since it ignores costs associated with buying an asset: however, this is thestandard and traditional usage. The quantity S(0) denotes the price of stockat date t = 0 and S(T) denotes the price of the stock at date t = T. Theprofit from the stock is simply PS = S(T)S(0) while the profit from a callis

Pcall = Max(0, S(T)K) C(0) (15.1)

where C(0) is the price of the call option and K is the strike price, and theprofit from a put is

Pput = Max(0, K S(T)) P(0) (15.2)

where P(0) is the price of the put option and K is again the strike price.In addition, for the case of the put, it is assumed that the holder of the puthas shares of the stock to sell, that is, that it is a covered put.

For example, suppose S(0) = $50, C(0) = $4, and P(0) = $2. Then thepayoff patterns are given by

PS = S(T) 50 (15.3)

Pcall = 4(50 S(T)) + (S(T) 54)(S(T) 50) (15.4)

Pput = (48 S(T))(50 S(T)) 2(S(T) 50) (15.5)

Pput = (48 2S(T))(50 S(T)) 2(S(T) 50) (15.6)

where (x) is the unit step function ((x) = 1 for x > 0, (x) = 0 forx < 0), the first put payoff is for a covered put, and the second is for anuncovered put. The payoff for a protective put or portfolio insuranceis the payoff from the purchase of the stock at price S(0) and a put on thesame stock with a strike price set at the present price K = S(0),

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Ppp = (S(T) S(0)) + Max(S(0) S(T), 0) P(0) (15.7)where the first component is the stock payoff PS and the last componentis the put payoff Pput for strike price S(0). This is an insurance policyin that it limits the maximum loss to the put price P(0). As an example,suppose that S(0) = K = $50 and P(0) = $2. Then the payoff pattern issimply

Ppp = 2(50 S(T)) + (S(T) 52)(S(T) 50) (15.8)

This is an example of a combination of an option market position with aspot market position. Combinations of option market positions with otheroption market positions are commonly termed spreads.

A bullish spread consists of buying a call with a given exercise price andwriting a call on the same security but with a higher exercise price. Thepayoff pattern is

Pbull = Max(S(T)KL, 0)CL + CHMax(S(T) KH, 0) (15.9)

where the subscripts L and H refer to low and high, respectively. Thefirst component is the payoff for the purchased call (low) and the second

component is the payoff for written call (high). As an example, supposeCL = $4, KL = $50, CH = $2, and KH = $55. Then the payoff is

Pbull = 2(50 S(T)) + (S(T) 52)(S(T) 50)(55 S(T)) + 3(S(T) 55) (15.10)

Another combination is a bearish spread. This consists of buying a call witha given exercise price and writing a call on the same security but with alower exercise price. The payoff pattern is

Pbear = Max(S(T) KH, 0) CH + CL Max(S(T) KL, 0) (15.11)

where,again, the subscripts L and H refer to low and high, respectively. Thefirst component is the payoff for the purchased call (high) and the second

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component is the payoff for the written call (low). For example, if we use

the same numerical values that were used to illustrate the bullish spread,then the payoff is

Pbear = 2(50 S(T)) + (52 S(T))(S(T) 50)(55 S(T)) 3(S(T) 55) (15.12)

A third common combination is known as a butterfly spread. This consists ofbuying two calls each with the same exercise price KB and selling two calls,one with an exercise price K1 and the other with an exercise price K2 suchthat K

1+K

2= 2K

B. For concreteness, we will assume that K

1< K

B< K

2.

The payoff pattern is given by

Pbutter = 2(Max(S(T)KB , 0)CB) + C1 Max(S(T)K1, 0)+C2 Max(S(T)K2, 0) (15.13)

The first component is the payoff for the two purchased calls, the secondcomponent is the payoff for the first written call, and the last componentis the payoff for the second written call. As a final illustration of the typesof spreads that are possible, we consider a straddle. This consists of buying

both a put and a call on the same stock with the same exercise price andthe same expiration date. The payoff pattern is

Pstraddle = Max(S(T) K, 0) C(0) + Max(K S(T), 0) P(0) (15.14)

where the first component is the payoff for the call and the second is thepayoff for the put.

The factors that influence the price of a call option are the exercise priceK, the expiration date T, and the behavior of the stock price S(t) on which

the option is written. The dependence of a call option on these factors canbe described in the following terms: (a) the higher the exercise price, theless worthwhile to exercise the option and thus the lower the value of thecall option; (b) the higher the current price S(t), the more the option isworth, the greater the probability that the option will be exercised; (c) the

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more time left to exercise the option, T t, the greater the probability itwill be worth exercising and thus the higher the value of the option; and (d)as the variability of the stock price S(t) increases, the probability that theoption can be exercised profitably increases: usually we think of variabilityas implying the risk of obtaining lower values, but the call option protectsthe holder against downward movements of the stock price so that the holderonly cares about upward movements.

15.2 The Black-Scholes Equation for Option Pricing

The Black-Scholes equation for option pricing makes two major assumptions:

(1) it assumes the call option is an European option, written on a stock thatpays dividends and (2) it assumes the stocks price is lognormally distributedwith a lognormal standard deviation of . The equation depends upon thestandard deviation of the returns but not on the expected value of thesereturns. If we define S(t) as the stock price at time t, K as the exercise price,T as the expiration date, = T t as the time remaining until expiration, ras the risk-free interest rate, assumed to be continuously compounded andexpressed in terms compatible with (for example, both in annual terms),and C(t) as the value of the call option, then the Black-Scholes pricingequation for call options is

C(t) = S(t)N(d1) KerN(d2) (15.15)

d1 =ln(S(t)/K) + (r + 2/2)

(15.16)

d2 =ln(S(t)/K) + (r 2/2)

(15.17)

where N(x) is the cumulative standard normal distribution function. Notethat d1 and d2 are related by d2 = d2 . As a numerical example,suppose S(t) = $50, k = $50, = 0.35, r = 0.08, and = 0.25 years.

In this case, d1 = 0.20178, d2 = 0.02678, and the cumulative functionsare N(d1) = 0.5800 and N(d2) = 0.5107. Since the exponential factor isexp(r) = 0.9802, the call price is C(t) = $3.9693.

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15.3 Pricing Puts

The put-call parity theorem strictly applies to European options. A Eu-ropean put on a stock with an exercise price of K and expiration date ofT is equivalent to a portfolio in which four asset positions are combined:(1) the purchase of a call on the stock with an exercise price of K andexpiration date of T (an initial cash flow ofC(t) and terminal payoff ofMax(S(T)K, 0); (2) the purchase ofKexp(r) of riskless assets (havingpayoff of K at time T); (3) selling short of shares of the stock (having cashflow of S(t) at time t and cash flow ofS(t) at time T); and (4) writinga European put on the stock (having positive cash flow P(t) at time t andcash flow Max(K S(T), 0) at time T). The terminal cash flow of theentire portfolio (elements (1), (2), (3), and (4)) is zero: (a) if S(T)

K

then C(T) = 0, the riskless asset has value K, the stock has value S(T),and the put has value (K S(T) which totals to zero; (b) if S(T) K,then C(T) = S(T) K, the riskless asset has value K, the stock has valueS(T), and P(T) = 0 which also totals to zero. Arbitrage considerationsrequire that the total initial cash flow of the portfolio must also be zero,C(t)Kexpr+ S(t) + P(t) = 0 which then yields the value of the put,

P(t) = C(t) S(t) + Ker (15.18)

This is known as the put-call parity theorem. If we use the Black-Scholesresult for C(t) and the fact that 1 N(x) = N(x), this may be written as

P(t) = S(t)N(d1) + KerN(d2) (15.19)

where d1 and d2 are defined above.

15.4 Calculating the Implied Variance

Of all the variables in the Black-Scholes option pricing equation and analysis,

the most difficult to estimate is the volatility . A common exercise is toperform the following calculation: given C(t), S(t), , and r, calculate theimplied that makes the pricing equation valid. There is no closed formsolution, and two numerical methods are typically pursued. The problemcan be simply stated in this way: first, define the function f() by

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f() = S(t)N(d1)Ker

N(d2) (15.20)

and find such that f() = C(t).

The first method proceeds in the following way. Note that f() is mono-tonic in , so that (f())min = f(0) where

f(0) = Max(0, S(t) Ker) (15.21)The algorithm is:Step 1: choose a 1 and a 2 such that f(1) < C < f(2)Step 2: compute = (1 + 2)/2. Clearly, 1 < < 2Step 3: if f() < C, let 1 = ; if f() > C, let 2 = Step 4: repeat step 2 until |f() C| is sufficiently small.

The second method uses the Newton-Raphson technique to find the root ofthe equation C f() = 0. This is an iterative procedure that takes theform

(i + 1) = (i) f((i) Cf((i))

(15.22)

where i is the iteration index and the prime refers to the derivative withrespect to . It has been shown (Manaster & Koehler, 1982) that the initialvalue

2(0) =

ln(S(t)/K) + r2 (15.23)

will always lead to convergence. The final quantity needed to complete theiterative procedure is the derivative f() which, after some manipulation,is given by

f() =df()

d= S(t)

N(d1) (15.24)

N(x) =e

x2/2

2 (15.25)

Exercises

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PART IV: DURATION AND IMMUNIZATION

17 Duration

17.1 Introduction

Duration is a measure of how long, on the average, a holder of a bond hasto wait before receiving cash payments. From another point of view, it isa measure of the sensitivity of a bonds price to changes in interest rates.An immunization strategy is an outgrowth of the duration concept. Such astrategy is one in which a portfolio of bonds is managed so that its value isalways as close as possible to the value of another asset, that is, it remainsfixed.

We define the bond payments C(t) for t = 1, 2, 3,...,N as composed ofinterest payments C(t) for t = 1, 2, 3,...,N 1 and the final payment C(N)as the last interest payment plus the repayment of the principal. We canthen construct a time-weighted average of payments

< C >=Nt=1

tC(t)

(1 + r)t(17.1)

The duration is defined as this time-weighted average of bond payments

expressed as a fraction of the bond price

D =< C >

P=

1

P

Nt=1

tC(t)

(1 + r)t(17.2)

P =Nt=1

C(t)

(1 + r)t(17.3)

where r is the current market interest rate at the time the bond is purchased.

17.2 Two Examples

17.3 Using Duration to Measure Price Volatility

If we begin with the expression given above for the bond price,

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P =Nt=1

C(t)(1 + r)t

(17.4)

and take the derivative of the price with respect to the interest rate r, wefind that

dP

dr=

Nt=1

tC(t)

(1 + r)t+1=

DP

(1 + r)(17.5)

It should be remembered that the meaning of dP/dr is the rate of changeof the bond price P with interest rate r. This equation gives rise to twointerpretations of duration: (1) the discount rate elasticity of bond price

D = dP/P

dr/(1 + r)(17.6)

and (2) the measure of price volatility of a bond

dP

P= D

dr

(1 + r)(17.7)

As an approximation, P DPr is typically used, but P DP ln(1+r) and ln P Dln(1+ r) are better approximations over wider rangesof the variables P and r. In the highly exceptional case that D may betreated as independent of r, then the above equation may be integrated togive

ln P = D ln(1 + r) + K (17.8)

P =eK

(1 + r)D(17.9)

where K is a constant of integration. These two equations can be used toprovide the basis of the better approximations given above.

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17.4 Closed-form Formulas for Duration

For the case of a bond with unchanging coupon payments over time, C(t) =C for all t, there are two closed-form formulas for duration. If we denote theface value of the bond by F so that C(N) = C + F, then Chuas formulafor duration is

D =1

P

C

(1 + r)N+1 (1 + r) rN

r2(1 + r)N+

NF

(1 + r)N

(17.10)

If we further define the present value interest factor (PVIF) of an N periodannuity of $ 1 per period, discounted at the interest rate r as

PVIF(r, N) =Nt=1

1

(1 + r)t=

(1 + r)N 1

r(1 + r)N(17.11)

and note that the current yield of the bond is defined as y = C/P, thenBabcocks formula for duration is

D = N

1

y

r

+

y

rPVIF(r, N)

1 + r

(17.12)

There are two so-called insights from Babcocks formula. The first states

that duration is the weighted average of the bond and (1 + r) times thePVIF associated with the bond. The second states that, in many cases, thecurrent yield y of the bond is not greatly different from the yield to maturityr (y r) so that D (1 + r)PVIF(r, N).

17.5 Duration and the Yield to Maturity

Chuas formula for duration can be used to calculate the effect of changesin the yield to maturity r on the duration D. In other words, plot D as afunction of r.

17.6 Calculating the IRR for Uneven Periods

A problem often encountered is to calculate the yield to maturity r of abond when payments are not evenly spaced.

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Exercises

6. A sequence of bonds indexed by i = 1, 2, 3,... all have the same maturity,N, and the same yield to maturity r, but each bond has a different coupon,Ci. Show (using Babcocks formula) that the duration of these bonds maybe written as Di = NKCi where K = N (1 + r)PVIF(r, N). Producea graph of this curve for different values of Ci (see Morrisey and Huang,1987).

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PART V: THE TECHNICAL BACKGROUND

Chapter 19: The Gauss-Seidel MethodExercises

Chapter 20: The Newton-Raphson Method20.1 Introduction20.2 Theory20.3 Finding Minima and Maxima20.4 Using Lotus to Set Up the Newton-Raphson MethodExercises

Chapter 21: Matrices

21.1 Introduction21.2 Matrix Operations21.3 Matrix Inverses21.4 Solving Systems of Simultaneous Linear Equations21.5 Lotus and MatricesExercises

Chapter 22: Random Number Generators22.1 Introduction22.2 Testing the Lotus Random Number GeneratorExercises

Chapter 23: Lotus Functions23.1 Introduction23.2 @NPV23.3 @IRR23.4 @PV23.5 @VLOOKUP23.6 @INT23.7 @MOD(A,B)23.8 @PI23.9 @ABS(x)23.10 @SUM(range), @STD(range), @VAR(range), @COUNT(range)23.11 @NOW

Chapter 24: Macros in Lotus24.1 Introduction24.2 A Simple Example24.3 Some Basic Macro Rules

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24.4 An Advanced Macro

24.5 Macro Keywords24.6 Examples24.7 Minimizing Macro Execution TimeExercises

Chapter 25: Data Table Commands25.1 Introduction25.2 An Example25.3 Setting Up /Data Table 125.4 Using /Data Table 225.5 An Aesthetic Note: Hiding Unnecessary Cells

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"Numerical Techniques in Finance" by Simon Benninga: A Solution and Study Manual