Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier"

Transcription

1 Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Prof. Massimo Guidolin Portfolio Management Spring 2016

2 Overview The two-asset case o Perfectly correlated assets o Perfectly negatively correlated assets o The case of -1 < < +1 (and = 0, uncorrelated assets) The shape of the mean-variance frontier The efficient frontier in the general N-asset case The efficient frontier with unrestricted borrowing and lending and the riskless rate The tangency portfolio and the separation theorem One practical issue in the construction of the efficient frontier: parameter uncertainty 2

3 The two-asset case Call X A the fraction of a portfolio held in asset A and X B the fraction held in asset B o However these may as well be portfolio and not individual assets We require the investor to be fully invested, X A + X B = 1 and X B = 1 X A, so that: Such a simple, weighted way of combining is not necessarily true of the risk (standard deviation of the return) of the portfolio: Using the equation for means returns and, we obtain In order to learn more about this relationship, we study specific cases involving different degrees of co-movement btw. securities Case 1: Perfect Positive Correlation ( = +1), the securities move in unison 3

4 The two-asset case: = +1 When = +1, there is a linear relationship btw. expected ptf. returns and ptf. standard deviation o This derives from the presence of a perfect square inside brackets o C and S refer to an example (see above) o The expected return on the portfolio is o Thus with = + l, risk and return are linear combinations of the risk and return of each security and because X C = (E[R P ] - E[R S ])/(E[R C ] - E[R S ]), we have When = +1, there is a linear relationship btw. expected ptf. returns and std. dev. 4

5 The two-asset case: = -1 o E.g., with our inputs for Colonel Motors and Separated Edison, substituting this expression for X C into the equation for E[R P ] and rearranging yields: In the case of perfectly correlated assets, there is no reduction in risk from purchasing both assets o In the plot, any combinations of the two assets lie along a straight line connecting the two assets Case 2: Perfect Negative Correlation ( = -1), the securities move perfectly together but in exactly opposite directions In this case the standard deviation of the portfolio is: The term in the brackets is equivalent to either of the following: or Ptf. volatility is (*)or (**) o Since we took the square root and the square root of a negative number is imaginary, the equations hold only when right-hand side >0 5

6 The two-asset case: = -1 If two securities are perfectly negatively correlated, it is always possible to find some combination that has zero risk o Since one is always positive when the other is negative (except when both equations equal zero), each also plots as a straight line when expected return is plotted against volatility o In fact, the value of P in this case is always smaller than the value of P for the case where = + 1, for all values of X C between 0 and 1 We can go one step further: If two securities are perfectly negatively correlated, it should always be possible to find some combination that has zero risk By setting either (*) or (**) equal to 0, we find that a portfolio with will have zero risk 6

7 The two-asset case: -1 < < +1 o Because S + C > C this implies that 0 < X C < 1 the zero risk ptf. will always involve positive investment in both securities o In our example, zero risk obtains for a simple 1/3-2/3 portfolio because 3/(3 + 6) = 1/3 o Two equations relating mean and standard deviation, and for each selection of X C the appropriate one is the one that guarantees P 0 Case 3: Uncorrelated assets ( = 0), in this case shows for any value for X C between 0 and 1 the lower the correlation the lower is the standard deviation of the ptf. Ptf. standard deviation reaches its lowest value for = -1 (curve SBC) and its highest value for = + 1 (curve SAC) These two curves represent the limits within which all portfolios of these two securities must lie for intermediate values of 7

8 The two-asset case: = 0 GMVP o E.g., when = 0, noting that the covariance term drops out, the expression for standard deviation becomes There is one point on this figure that is worth special attention: the portfolio that has minimum risk, the global minimum variance ptf. 8

9 The two-asset case: -1 < < +1 In the case -1 < < +1, the global minimum variance ptf. is the set of weights that minimizes the resulting ptf.risk o This portfolio can be found by looking at the equation for risk and minimizing by taking the FOC w.r.t. X C and solving: (***) = 0 o Continuing with our example, the value of X C that minimizes risk is The correlation between any two actual stocks is almost always greater than 0 and considerably less than 1 o E.g., in the case of = 0.5, the ptf. risk equation becomes o In our example, minimum risk is obtained at a value of X C = 0 or 100% in Separated Edison 9

10 The critical coefficient in the two-asset case There is some critical value of such that ptf. risk cannot be made less than the risk of the least risky asset without selling short o Analytically, In this example (i.e., = 0.5) there is no combination of the two securities that is less risky than the least risky asset by itself, though combinations are still less risky than under = +1 The particular value of the correlation coefficient for which no combination of two securities is less risky than the least risky security depends on the characteristics of the assets in question For all assets there is some critical value of such that the risk on the portfolio can no longer be made less than the risk of the least risky asset in the portfolio without selling short Setting X C equal to zero in (***) above and solving for * gives * = S / C so that when is equal to or higher than *, the least risky combination involves short selling the most risky security and may be impossible 10

11 The shape of the mean-variance frontier Note that the portion of the portfolio possibility curve (aka meanvariance frontier) that lies above the MVP is concave while that which lies below the minimum variance portfolio is convex This is a general characteristic of all portfolio problems. o The three figures represent three hypothesized shapes for combinations of Colonel Motors and the MV portfolio Shape (b) is impossible because combinations of assets cannot have more risk than on a straight line connecting two assets 11

12 The shape of the mean-variance frontier The (efficient) segment of the mean-variance frontier above the GMVP must have a concave shape In (c) all portfolios have less risk than the straight line connecting Colonel Motors and the MVP, but this shape is impossible o Examine the portfolios labeled U and V, combinations of the minimum variance portfolio and Colonel Motors o Since U and V are portfolios, all combinations of U and V must lie either on a straight line connecting U and V or above such a line The only legitimate shape is that shown in (a), which is concave o Analogous reasoning can be used to show that if we consider combinations of the MVP and a security or portfolio with higher variance and lower return, the curve must be convex What if the number of assets is some general N >> 2? In theory we could plot all conceivable risky assets and their combinations in a diagram in return standard deviation space In theory," not because there is a problem, but because there are an infinite number of possibilities that must be considered 12

13 The efficient frontier in the general N-asset case Mean-variance dominance criteria simplify the opportunity set If we were to plot all possibilities in risk-return space, we would get Examine the diagram and see if we can eliminate any part of it from consideration by the investor A rational investor would prefer a higher mean return to less and would prefer less risk to more Thus, if we can find a set of ptfs that (i) offered a bigger mean return for the same risk, or (ii) offered a lower risk for the same mean return, we have the choice set o E.g., ptf. B would be preferred by all investors to ptf. A because it offers a higher return with the same level of risk o Ptf. C would be preferable to portfolio A because it offers less risk for the same level of return 13

14 The efficient frontier in the general N-asset case However, we can find no portfolio that dominates portfolio C or portfolio B For this reason, an efficient set of ptfs. cannot include interior ptfs. Moreover, for any point in riskreturn space we want to move as far as possible in the direction of increasing mean return and as far as possible in the direction of decreasing risk o Therefore we can eliminate D since portfolio E exists, which has higher mean return for the same risk o This is true for every other portfolio as we move up the outer shell from D to point C o Point C cannot be eliminated because it is the GMVP o Ptf. F is on the outer shell, but E has less risk for the same return o As we move up the outer shell from point F, all ptfs are dominated 14

15 The efficient frontier in the general N-asset case The efficient frontier consists of the envelope curve of all portfolios that lie between the global MVP and the maximum return portfolio o This until we come to B that cannot be eliminated for there is no ptf. that has same return and less risk or the same risk and more return o Point B represents that ptf. (usually a single security) that offers the highest expected return of all ptfs. The efficient set consists of the envelope curve of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio See a graph of the efficient frontier Based on our earlier proof, it is a concave function Only linear segments may exist 15

16 The efficient frontier with short selling The portfolio problem, then, is to find all portfolios along this frontier, which we shall examine later So far, one could only combine long positions in existing assets In many capital markets, an investor can often sell a security that he or she does not own, a process called short selling o In practice, this amounts to borrowing an asset under the promise to the lender that she will be no worse off lending it and with a commitment to return it at same date (say, end-of day) o This requires re-funding any cash flows (e.g., dividends or coupons that the asset may pay out over time) Short selling allows us to leverage up the return of best performing securities but also increases risks With short sales, ptfs. exist that give infinite expected returns 16

17 Unrestricted Borrowing and Lending Up to this point we have dealt with portfolios of risky assets only The introduction of a riskless asset, that yields R F, into the investment opportunity set considerably simplifies the analysis o Because the return is certain, the standard deviation of the return on the riskless asset must be zero Of course such a step requires assuming that a risk-free asset exists Borrowing can be considered as selling such a security short, so that also borrowing can take place at the riskless rate The investor is interested in placing part of the funds in some portfolio A and either lending or borrowing Call X the fraction of original funds that the investor places in ptf A o X may exceed 1 because we are assuming that investors can borrow at the riskless rate and invest more than his initial funds in ptf. A The expected return on the combination of riskless asset and risky portfolio is given by 17

18 Unrestricted Borrowing and Lending The risk on the combination is (C stands for combination) Since we have already argued that σ F is zero, Solving this expression for X yields X = σ C /σ A and substituting this expression into the expression for expected return and rearranging, yields This is the equation of a straight line with slope = Sharpe ratio of ptf. A: The line passes through point (σ A, E[R A ]) To the left of point A we have combinations of lending and portfolio A, whereas to the right of point A we have combinations of borrowing and ptf. A Problem: ptf. A we selected has no special properties; we could have combined portfolio B with riskless lending and borrowing 18

19 The tangency portfolio All investors facing the same efficient frontier ABGH will select the same tangency portfolio G Combinations along the ray R F B are superior to combinations along R F A since they offer greater expected return for the same risk We would like to rotate the straight line passing through R F as far as we can in a counterclockwise direction The furthest we can rotate it is through G Point G is the tangency point between the efficient frontier and a ray through R F Cannot rotate the ray further because by definition there are no ptfs lying above the line passing through R F and G All investors who believed they faced the efficient frontier and riskless lending and borrowing rates shown in the figure would hold the same ptf. of risky asset: G 19

20 The tangency portfolio and the separation theorem According to the separation theorem, all investors facing the same efficient frontier select the tangency ptf. of risky assets regardless of their preferences towards risk o Investors who are very risk-averse select a ptf along the segment R F G and place some money in a riskless asset and some in risky ptf G o Others who were much more tolerant of risk would hold portfolios along the segment G-H, borrowing funds and placing their original capital plus the borrowed funds in portfolio G Yet all of these investors would hold the tangency portfolio G. Thus, for the case of riskless lending and borrowing, identification of portfolio G constitutes a solution to the problem The ability to determine the optimal ptf. of risky assets without knowing anything about an investor is the separation theorem Our our assumptions realistic? While there is no question about the ability of investors to lend at the risk-free rate (buy government securities), they could possibly not borrow at this rate 20

21 The effects of frictions on the separation theorem Whe borrowing at the riskless rate is impossible, the efficient frontier becomes R F GH Certain investors will hold portfolios of risky assets located between G and H However, any investor who held some riskless asset would place all remaining funds in the tangency portfolio G The separation theorem fails: different investors may select different risky ptfs A possibility is that investors can lend at one rate (R F ) but must pay a different and higher rate to borrow (R F ) The efficient frontier becomew R F GHI There is a small range of risky ptfs that is optional for investors to hold and two different tangency ptfs, G and H 21

22 One practical issue in portfolio choice Also in this case the separation theorem fails Reliable inputs on means, variances, and covariances are crucial to the proper use of mean-variance optimization Common to use historical risk, return, and correlation as a starting point in obtaining inputs for calculating the efficient frontier If return characteristics do not change through time, then the longer the data are available the more accurate are the estimates o E.g., the formula for the standard error of the mean of a sequence of independent random variables is σ 2 /N where N is the sample size o This effect may be first-order: imagine an investor choosing between two investments, each with identical sample means and variances o The standard approach would view the two investments as equivalent o If you consider the additional information that the first sample mean was based on 1 year of data and the second on 10 years of data, common sense would suggest that the second alternative is less risky: 22

23 One practical issue in portfolio choice There is a trade-off between using a long time frame to improve the estimates and having potentially inaccurate estimates from the longer time period because the characteristics have changed o The first part of the expression captures the inherent risk in the return; the second term captures the uncertainty that comes from lack of knowledge about the true mean return Characteristics of asset returns usually change over time There is a trade-off between using a long time frame to improve the estimates and having potentially inaccurate estimates from the longer time period because the characteristics have changed Because of this, most analysts modify historical estimates to reflect beliefs about how current conditions differ from past conditions The choice of the time period is more complicated when a relatively new asset class is added to the mix, and the available data for the new asset is much less than for other assets For example, consider the case of CDS or CDOs as asset classes 23

24 One practical issue in portfolio choice o An analyst who wishes to use historical data could use all available data or use shorter data only from the common period of observation Consider the IFC emerging markets index example in the table o Differences may be substantial: e.g., statistics over the longer term are consistent with an equilibrium in which a higher investor risk is compensated by higher investor expected return o Statistics over the period of common observation, beginning in 1985, are inconsistent with this argument Also correlations are very different and will affect the frontier 24

25 The stock-bond choice again (shorting allowed) Consider again the allocation between equity and debt The estimated historical inputs are: The minimum variance portfolio is given by Unsurprisingly, it implies selling the index short (write futures?) The associated st. dev. is 4.75%, which is slightly less than the one associated with 100% in bonds, so were slightly below the critical ρ This is the efficient frontier with short sales allowed (it continues to the right) (tangency) 25

26 The stock-bond choice again (no short sales) The tangency portfolio is T and we will see how it is calculated soon Assuming a 5% T-bill rate, we have E[R T ] = 13.54% and T = 16.95% so that the slope of the line connecting the tangency portfolio and the efficient frontier is (13.54% - 5%)/16.95% = 0.5 The equation of the efficient frontier with riskless lending and borrowing is Knowing the expected return of T we can easily determine its composition: The efficient frontier with no short sales is to the right In this case the GMVP is 100% in bonds 26

27 Hints to techniques to calculate the efficient frontier We distinguish among 4 cases: o Short sales are allowed and riskless lending and borrowing is possible o Short sales are allowed but riskless lending or borrowing is not o Short sales are disallowed but riskless lending and borrowing exists o Neither short sales nor riskless lending and borrowing are allowed The derivation of the efficient set when short sales are allowed and there is a riskless lending and borrowing rate is the simplest case In this case, the efficient frontier is the entire length of the ray extending through R F and B An equivalent way of identifying the ray R F -B is to recognize that it is the ray with the greatest slope, θ The efficient set is determined by finding the ptf with the greatest θ (Sharpe) ratio that satisfies the weight sum constraint 27

28 Hints to techniques to calculate the efficient frontier o This is a constrained maximization problem for which there are standard solution techniques o For example, it can be solved by the method of Lagrangian multipliers There is an alternative: the constraint could be substituted into the objective function and the objective function maximized as in an unconstrained problem Making this substitution in the objective function and stating the expected return and standard deviation of return in the general form, one maximizes This problem can be solved in standard ways by imposing and solving first order conditions 28

29 Hints to techniques to calculate the efficient frontier o In this case the FOCs are also sufficient because the objective function is concave (we are dividing by a positive quadratic form) We can prove that Because each X k is multiplied by a constant, define a new variable Z k = X k and substituting Z k for the X k simplifies the formulation: We have one equation like this for each value of i Now solve the system of simultaneous equations: 29

30 Hints to techniques to calculate the efficient frontier To determine the optimum amount to invest, we first solve the equations for the Zs and this is generally possible because there are N equations (one for each security) and N unknowns (the Z k ) Then the optimum proportions to invest in stock k is: When short sales are allowed but there is no riskless lending and borrowing rate, the solution above must be modified o Assume a riskless lending and borrowing rate and find the optimum o Assume that a different riskless lending and borrowing rate exists and find the optimum that corresponds to this rate o Continue changing the riskless rate until the full efficient frontier is deterrnined One can show that the optimal proportion to invest in any security is simply a linear function of R F 30

31 Hints to techniques to calculate the efficient frontier Furthermore, the entire efficient frontier can be constructed as a combination of any two portfolios that lie along it Therefore the identification of the characteristics of the optimal portfolio for any two arbitrary values of R F is sufficient to trace out the total efficient frontier When short sales are not allowed but there is riskless lending and borrowing, the solution comes from solving This is a nonlinear mathematical programming problem because of the inequality restriction on the weights o Equations involving squared terms and cross-product terms are called quadratic equation There are computer packages for solving quadratic programming problems subject to constraints In our case, the Excel solver will do 31

32 Hints to techniques to calculate the efficient frontier The imposition of short sales constraints has complicated the solution technique, forcing us to use quadratic programming Once we resort to this technique, it is a simple matter to impose other requirements on the solution Any set of requirements that can be formulated as linear functions of the investment weights can be imposed on the solution o For example, some managers wish to select optimum ptfs given that the dividend yield on the portfolios is at least some number, D o If one wants no short sale constraints, these can be imposed: o Perhaps, the most frequent constraints are those that place an upper limit on the fraction of the portfolio that can be invested in any asset o Upper limits on the amount that can invested in any one stock are often part of the charter of mutual funds o It is possible to build in constraints on the amount of turnover in a portfolio and to allow the consideration of transaction costs 32

33 Summary and conclusions Provided that < 1 diversification offers a costless payoff: risk reduction without any costs in terms of lower expected return Such a risk-reduction is maximal when = -1, when a special ptf. can be found that implies zero risk but positive expected return When = 0, in the limit risk can also be removed by increasing the total number of assets (N) in the portfolio When > 0, even though N, the total amount of risk does not level off to zero, but converges to the average covariance across all pairs of assets in the economy The efficient mean-variance set (frontier) is the subset of the opportunity set that lies above the global minimum variance portfolio and has a concave shape The tangency portfolio is unique across all investors that perceive the same efficient set The separation theorem states that all investors will demand the same risky portfolio irrespective of their risk aversion 33

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

More information

Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*:

Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*: Problem 1. Consider a risky asset. Suppose the expected rate of return on the risky asset is 15%, the standard deviation of the asset return is 22%, and the risk-free rate is 6%. What is your optimal position

More information

CHAPTER 9: THE CAPITAL ASSET PRICING MODEL

CHAPTER 9: THE CAPITAL ASSET PRICING MODEL CHAPTER 9: THE CAPITAL ASSET PRICING MODEL PROBLEM SETS 1. E(r P ) = r f + β P [E(r M ) r f ] 18 = 6 + β P(14 6) β P = 12/8 = 1.5 2. If the security s correlation coefficient with the market portfolio

More information

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS CHAPTER 7: OPTIMAL RIKY PORTFOLIO PROLEM ET 1. (a) and (e).. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash and real estate.

More information

Lecture 1: Asset Allocation

Lecture 1: Asset Allocation Lecture 1: Asset Allocation Investments FIN460-Papanikolaou Asset Allocation I 1/ 62 Overview 1. Introduction 2. Investor s Risk Tolerance 3. Allocating Capital Between a Risky and riskless asset 4. Allocating

More information

Lesson 5. Risky assets

Lesson 5. Risky assets Lesson 5. Risky assets Prof. Beatriz de Blas May 2006 5. Risky assets 2 Introduction How stock markets serve to allocate risk. Plan of the lesson: 8 >< >: 1. Risk and risk aversion 2. Portfolio risk 3.

More information

Review for Exam 2. Instructions: Please read carefully

Review for Exam 2. Instructions: Please read carefully Review for Exam 2 Instructions: Please read carefully The exam will have 25 multiple choice questions and 5 work problems You are not responsible for any topics that are not covered in the lecture note

More information

1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises.

1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises. 1. Solutions to PS 1: 1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises. 7. The bill has a maturity of one-half year, and an annualized

More information

Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS

Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS 5-1 a. Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by holding only one asset. Risk is

More information

Capital Allocation Between The Risky And The Risk- Free Asset. Chapter 7

Capital Allocation Between The Risky And The Risk- Free Asset. Chapter 7 Capital Allocation Between The Risky And The Risk- Free Asset Chapter 7 Investment Decisions capital allocation decision = choice of proportion to be invested in risk-free versus risky assets asset allocation

More information

15.401 Finance Theory

15.401 Finance Theory Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lecture 13 14 14: : Risk Analytics and Critical Concepts Motivation Measuring Risk and Reward Mean-Variance

More information

CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

More information

2. Mean-variance portfolio theory

2. Mean-variance portfolio theory 2. Mean-variance portfolio theory (2.1) Markowitz s mean-variance formulation (2.2) Two-fund theorem (2.3) Inclusion of the riskfree asset 1 2.1 Markowitz mean-variance formulation Suppose there are N

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

SAMPLE MID-TERM QUESTIONS

SAMPLE MID-TERM QUESTIONS SAMPLE MID-TERM QUESTIONS William L. Silber HOW TO PREPARE FOR THE MID- TERM: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below,

More information

1 Portfolio mean and variance

1 Portfolio mean and variance Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

More information

CFA Examination PORTFOLIO MANAGEMENT Page 1 of 6

CFA Examination PORTFOLIO MANAGEMENT Page 1 of 6 PORTFOLIO MANAGEMENT A. INTRODUCTION RETURN AS A RANDOM VARIABLE E(R) = the return around which the probability distribution is centered: the expected value or mean of the probability distribution of possible

More information

Chapter 13 Composition of the Market Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z,

Chapter 13 Composition of the Market Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z, Chapter 13 Composition of the arket Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z, and a riskless government security. Evaluated at current prices in

More information

CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS

CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e). (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional

More information

Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?

Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10? Return, Risk, and Risk Aversion Holding Period Return Ending Price - Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at $40 last month.

More information

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder

More information

Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing Models ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing Models ANSWERS TO END-OF-CHAPTER QUESTIONS Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing odels ANSWERS TO END-OF-CHAPTER QUESTIONS 7-1 a. A portfolio is made up of a group of individual assets held in combination. An asset that

More information

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

More information

CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM)

CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) Answers to Concepts Review and Critical Thinking Questions 1. Some of the risk in holding any asset is unique to the asset in question.

More information

The Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) Prof. Alex Shapiro Lecture Notes 9 The Capital Asset Pricing Model (CAPM) I. Readings and Suggested Practice Problems II. III. IV. Introduction: from Assumptions to Implications The Market Portfolio Assumptions

More information

Models of Risk and Return

Models of Risk and Return Models of Risk and Return Aswath Damodaran Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for

More information

The Tangent or Efficient Portfolio

The Tangent or Efficient Portfolio The Tangent or Efficient Portfolio 1 2 Identifying the Tangent Portfolio Sharpe Ratio: Measures the ratio of reward-to-volatility provided by a portfolio Sharpe Ratio Portfolio Excess Return E[ RP ] r

More information

CHAPTER 10. Capital Markets and the Pricing of Risk. Chapter Synopsis

CHAPTER 10. Capital Markets and the Pricing of Risk. Chapter Synopsis CHAPE 0 Capital Markets and the Pricing of isk Chapter Synopsis 0. A First Look at isk and eturn Historically there has been a large difference in the returns and variability from investing in different

More information

1 Capital Asset Pricing Model (CAPM)

1 Capital Asset Pricing Model (CAPM) Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available

More information

The CAPM (Capital Asset Pricing Model) NPV Dependent on Discount Rate Schedule

The CAPM (Capital Asset Pricing Model) NPV Dependent on Discount Rate Schedule The CAPM (Capital Asset Pricing Model) Massachusetts Institute of Technology CAPM Slide 1 of NPV Dependent on Discount Rate Schedule Discussed NPV and time value of money Choice of discount rate influences

More information

AFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang

AFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang AFM 472 Midterm Examination Monday Oct. 24, 2011 A. Huang Name: Answer Key Student Number: Section (circle one): 10:00am 1:00pm 2:30pm Instructions: 1. Answer all questions in the space provided. If space

More information

Review for Exam 2. Instructions: Please read carefully

Review for Exam 2. Instructions: Please read carefully Review for Exam Instructions: Please read carefully The exam will have 1 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation questions.

More information

Mean Variance Analysis

Mean Variance Analysis Mean Variance Analysis Karl B. Diether Fisher College of Business Karl B. Diether (Fisher College of Business) Mean Variance Analysis 1 / 36 A Portfolio of Three Risky Assets Not a two risky asset world

More information

MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS

MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS EIGHTH EDITION INTERNATIONAL STUDENT VERSION EDWIN J. ELTON Leonard N. Stern School of Business New York University MARTIN J. GRUBER Leonard N. Stern School

More information

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all. 1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

More information

Mean-Variance Portfolio Analysis and the Capital Asset Pricing Model

Mean-Variance Portfolio Analysis and the Capital Asset Pricing Model Mean-Variance Portfolio Analysis and the Capital Asset Pricing Model 1 Introduction In this handout we develop a model that can be used to determine how a risk-averse investor can choose an optimal asset

More information

Chapter 1. Introduction to Portfolio Theory. 1.1 Portfolios of Two Risky Assets

Chapter 1. Introduction to Portfolio Theory. 1.1 Portfolios of Two Risky Assets Chapter 1 Introduction to Portfolio Theory Updated: August 9, 2013. This chapter introduces modern portfolio theory in a simplified setting where there are only two risky assets and a single risk-free

More information

1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset

1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset Department of Economics Financial Economics University of California, Berkeley Economics 136 November 9, 2003 Fall 2006 Economics 136: Financial Economics Section Notes for Week 11 1 Capital Allocation

More information

Chapter 2 Portfolio Management and the Capital Asset Pricing Model

Chapter 2 Portfolio Management and the Capital Asset Pricing Model Chapter 2 Portfolio Management and the Capital Asset Pricing Model In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that

More information

The application of linear programming to management accounting

The application of linear programming to management accounting The application of linear programming to management accounting Solutions to Chapter 26 questions Question 26.16 (a) M F Contribution per unit 96 110 Litres of material P required 8 10 Contribution per

More information

Chapter 21: The Discounted Utility Model

Chapter 21: The Discounted Utility Model Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal

More information

How to Win the Stock Market Game

How to Win the Stock Market Game How to Win the Stock Market Game 1 Developing Short-Term Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.

More information

Equity Risk Premiums: Looking backwards and forwards

Equity Risk Premiums: Looking backwards and forwards Equity Risk Premiums: Looking backwards and forwards Aswath Damodaran Aswath Damodaran 1 What is the Equity Risk Premium? Intuitively, the equity risk premium measures what investors demand over and above

More information

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

More information

CHAPTER 11: ARBITRAGE PRICING THEORY

CHAPTER 11: ARBITRAGE PRICING THEORY CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times

More information

Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans

Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined

More information

CHAPTER 4 Consumer Choice

CHAPTER 4 Consumer Choice CHAPTER 4 Consumer Choice CHAPTER OUTLINE 4.1 Preferences Properties of Consumer Preferences Preference Maps 4.2 Utility Utility Function Ordinal Preference Utility and Indifference Curves Utility and

More information

Practice Set #4 and Solutions.

Practice Set #4 and Solutions. FIN-469 Investments Analysis Professor Michel A. Robe Practice Set #4 and Solutions. What to do with this practice set? To help students prepare for the assignment and the exams, practice sets with solutions

More information

CHAPTER 6 RISK AND RISK AVERSION

CHAPTER 6 RISK AND RISK AVERSION CHAPTER 6 RISK AND RISK AVERSION RISK AND RISK AVERSION Risk with Simple Prospects Risk, Speculation, and Gambling Risk Aversion and Utility Values Risk with Simple Prospects The presence of risk means

More information

Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).

Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r). Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.

More information

The Market-Clearing Model

The Market-Clearing Model Chapter 5 The Market-Clearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that

More information

6. Budget Deficits and Fiscal Policy

6. Budget Deficits and Fiscal Policy Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 2012 6. Budget Deficits and Fiscal Policy Introduction Ricardian equivalence Distorting taxes Debt crises Introduction (1) Ricardian equivalence

More information

1. Portfolio Returns and Portfolio Risk

1. Portfolio Returns and Portfolio Risk Chapter 8 Risk and Return: Capital Market Theory Chapter 8 Contents Learning Objectives 1. Portfolio Returns and Portfolio Risk 1. Calculate the expected rate of return and volatility for a portfolio of

More information

Estimating Risk free Rates. Aswath Damodaran. Stern School of Business. 44 West Fourth Street. New York, NY 10012. Adamodar@stern.nyu.

Estimating Risk free Rates. Aswath Damodaran. Stern School of Business. 44 West Fourth Street. New York, NY 10012. Adamodar@stern.nyu. Estimating Risk free Rates Aswath Damodaran Stern School of Business 44 West Fourth Street New York, NY 10012 Adamodar@stern.nyu.edu Estimating Risk free Rates Models of risk and return in finance start

More information

Chapter 7 Risk, Return, and the Capital Asset Pricing Model

Chapter 7 Risk, Return, and the Capital Asset Pricing Model Chapter 7 Risk, Return, and the Capital Asset Pricing Model MULTIPLE CHOICE 1. Suppose Sarah can borrow and lend at the risk free-rate of 3%. Which of the following four risky portfolios should she hold

More information

Portfolio Performance Measures

Portfolio Performance Measures Portfolio Performance Measures Objective: Evaluation of active portfolio management. A performance measure is useful, for example, in ranking the performance of mutual funds. Active portfolio managers

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

More information

On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information

On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information Finance 400 A. Penati - G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information

More information

FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver

FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver Question: How do you create a diversified stock portfolio? Advice given by most financial advisors

More information

A Detailed Price Discrimination Example

A Detailed Price Discrimination Example A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include

More information

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

More information

Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM)

Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM) Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier Slide 05-1 Overview Simple CAPM with quadratic utility functions (derived from state-price beta model)

More information

Using the Solver add-in in MS Excel 2007

Using the Solver add-in in MS Excel 2007 First version: April 21, 2008 Last revision: February 22, 2011 ANDREI JIRNYI, KELLOGG OFFICE OF RESEARCH Using the Solver add-in in MS Excel 2007 The Excel Solver add-in is a tool that allows you to perform

More information

Wel Dlp Portfolio And Risk Management

Wel Dlp Portfolio And Risk Management 1. In case of perfect diversification, the systematic risk is nil. Wel Dlp Portfolio And Risk Management 2. The objectives of investors while putting money in various avenues are:- (a) Safety (b) Capital

More information

ANALYSIS AND MANAGEMENT

ANALYSIS AND MANAGEMENT ANALYSIS AND MANAGEMENT T H 1RD CANADIAN EDITION W. SEAN CLEARY Queen's University CHARLES P. JONES North Carolina State University JOHN WILEY & SONS CANADA, LTD. CONTENTS PART ONE Background CHAPTER 1

More information

This paper is not to be removed from the Examination Halls

This paper is not to be removed from the Examination Halls ~~FN3023 ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON FN3023 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,

More information

Chapter 5. Conditional CAPM. 5.1 Conditional CAPM: Theory. 5.1.1 Risk According to the CAPM. The CAPM is not a perfect model of expected returns.

Chapter 5. Conditional CAPM. 5.1 Conditional CAPM: Theory. 5.1.1 Risk According to the CAPM. The CAPM is not a perfect model of expected returns. Chapter 5 Conditional CAPM 5.1 Conditional CAPM: Theory 5.1.1 Risk According to the CAPM The CAPM is not a perfect model of expected returns. In the 40+ years of its history, many systematic deviations

More information

Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance)

Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance) Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance) Mr. Eric Y.W. Leung, CUHK Business School, The Chinese University of Hong Kong In PBE Paper II, students

More information

Chapter 6 - Practice Questions

Chapter 6 - Practice Questions Chapter 6 - Practice Questions 1. If a T-bill pays 5 percent, which of the following investments would not be chosen by a risk-averse investor? A) An asset that pays 10 percent with a probability of 0.60

More information

Risk Decomposition of Investment Portfolios. Dan dibartolomeo Northfield Webinar January 2014

Risk Decomposition of Investment Portfolios. Dan dibartolomeo Northfield Webinar January 2014 Risk Decomposition of Investment Portfolios Dan dibartolomeo Northfield Webinar January 2014 Main Concepts for Today Investment practitioners rely on a decomposition of portfolio risk into factors to guide

More information

An introduction to Value-at-Risk Learning Curve September 2003

An introduction to Value-at-Risk Learning Curve September 2003 An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk

More information

Financial Risk Assessment Part I.1

Financial Risk Assessment Part I.1 Financial Risk Assessment Part I.1 Dimitrios V. Lyridis Assoc. Prof. National Technical University of Athens DEFINITION of 'Risk The chance that an investment's actual return will be different than expected.

More information

One Period Binomial Model

One Period Binomial Model FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

More information

Basic Utility Theory for Portfolio Selection

Basic Utility Theory for Portfolio Selection Basic Utility Theory for Portfolio Selection In economics and finance, the most popular approach to the problem of choice under uncertainty is the expected utility (EU) hypothesis. The reason for this

More information

Expected Utility Asset Allocation

Expected Utility Asset Allocation Expected Utility Asset Allocation William F. Sharpe 1 September, 2006, Revised June 2007 Asset Allocation Many institutional investors periodically adopt an asset allocation policy that specifies target

More information

Shares Mutual funds Structured bonds Bonds Cash money, deposits

Shares Mutual funds Structured bonds Bonds Cash money, deposits FINANCIAL INSTRUMENTS AND RELATED RISKS This description of investment risks is intended for you. The professionals of AB bank Finasta have strived to understandably introduce you the main financial instruments

More information

The Capital Asset Pricing Model

The Capital Asset Pricing Model Journal of Economic Perspectives Volume 18, Number 3 Summer 2004 Pages 3 24 The Capital Asset Pricing Model André F. Perold A fundamental question in finance is how the risk of an investment should affect

More information

FIN 432 Investment Analysis and Management Review Notes for Midterm Exam

FIN 432 Investment Analysis and Management Review Notes for Midterm Exam FIN 432 Investment Analysis and Management Review Notes for Midterm Exam Chapter 1 1. Investment vs. investments 2. Real assets vs. financial assets 3. Investment process Investment policy, asset allocation,

More information

Capital Structure. Itay Goldstein. Wharton School, University of Pennsylvania

Capital Structure. Itay Goldstein. Wharton School, University of Pennsylvania Capital Structure Itay Goldstein Wharton School, University of Pennsylvania 1 Debt and Equity There are two main types of financing: debt and equity. Consider a two-period world with dates 0 and 1. At

More information

Risk and Return Models: Equity and Debt. Aswath Damodaran 1

Risk and Return Models: Equity and Debt. Aswath Damodaran 1 Risk and Return Models: Equity and Debt Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for

More information

Optimal Risky Portfolios Chapter 7 Investments Bodie, Kane and Marcus

Optimal Risky Portfolios Chapter 7 Investments Bodie, Kane and Marcus Optimal Risky ortfolios Section escription 7.0 Introduction 7.1 iversification and ortfolio Risk 7. ortfolios of Two Risky Assets 7.3 Asset Allocation with Stocks, Bonds and Bills 7.4 The Markowitz ortfolio

More information

Midterm Exam:Answer Sheet

Midterm Exam:Answer Sheet Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a risk-free and risky asset, with returns given by r f and E(r p ), respectively. Let y be the

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Answers to Concepts in Review

Answers to Concepts in Review Answers to Concepts in Review 1. A portfolio is simply a collection of investments assembled to meet a common investment goal. An efficient portfolio is a portfolio offering the highest expected return

More information

BUS303. Study guide 2. Chapter 14

BUS303. Study guide 2. Chapter 14 BUS303 Study guide 2 Chapter 14 1. An efficient capital market is one in which: A. all securities that investors want are offered. B. all transactions are closed within 2 days. C. current prices reflect

More information

Econ 422 Summer 2006 Final Exam Solutions

Econ 422 Summer 2006 Final Exam Solutions Econ 422 Summer 2006 Final Exam Solutions This is a closed book exam. However, you are allowed one page of notes (double-sided). Answer all questions. For the numerical problems, if you make a computational

More information

Choice under Uncertainty

Choice under Uncertainty Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory

More information

Chapter 5. Risk and Return. Learning Goals. Learning Goals (cont.)

Chapter 5. Risk and Return. Learning Goals. Learning Goals (cont.) Chapter 5 Risk and Return Learning Goals 1. Understand the meaning and fundamentals of risk, return, and risk aversion. 2. Describe procedures for assessing and measuring the risk of a single asset. 3.

More information

Lecture 3: Put Options and Distribution-Free Results

Lecture 3: Put Options and Distribution-Free Results OPTIONS and FUTURES Lecture 3: Put Options and Distribution-Free Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distribution-free results? option

More information

Portfolio Optimization Part 1 Unconstrained Portfolios

Portfolio Optimization Part 1 Unconstrained Portfolios Portfolio Optimization Part 1 Unconstrained Portfolios John Norstad j-norstad@northwestern.edu http://www.norstad.org September 11, 2002 Updated: November 3, 2011 Abstract We recapitulate the single-period

More information

Note: There are fewer problems in the actual Final Exam!

Note: There are fewer problems in the actual Final Exam! HEC Paris Practice Final Exam Questions Version with Solutions Financial Markets Fall 2013 Note: There are fewer problems in the actual Final Exam! Problem 1. Are the following statements True, False or

More information

Chapter 5 Uncertainty and Consumer Behavior

Chapter 5 Uncertainty and Consumer Behavior Chapter 5 Uncertainty and Consumer Behavior Questions for Review 1. What does it mean to say that a person is risk averse? Why are some people likely to be risk averse while others are risk lovers? A risk-averse

More information

Using Microsoft Excel to build Efficient Frontiers via the Mean Variance Optimization Method

Using Microsoft Excel to build Efficient Frontiers via the Mean Variance Optimization Method Using Microsoft Excel to build Efficient Frontiers via the Mean Variance Optimization Method Submitted by John Alexander McNair ID #: 0061216 Date: April 14, 2003 The Optimal Portfolio Problem Consider

More information

Chapter 12: Cost Curves

Chapter 12: Cost Curves Chapter 12: Cost Curves 12.1: Introduction In chapter 11 we found how to minimise the cost of producing any given level of output. This enables us to find the cheapest cost of producing any given level

More information

CHAPTER 12 RISK, COST OF CAPITAL, AND CAPITAL BUDGETING

CHAPTER 12 RISK, COST OF CAPITAL, AND CAPITAL BUDGETING CHAPTER 12 RISK, COST OF CAPITAL, AND CAPITAL BUDGETING Answers to Concepts Review and Critical Thinking Questions 1. No. The cost of capital depends on the risk of the project, not the source of the money.

More information

ATHENS UNIVERSITY OF ECONOMICS AND BUSINESS

ATHENS UNIVERSITY OF ECONOMICS AND BUSINESS ATHENS UNIVERSITY OF ECONOMICS AND BUSINESS Masters in Business Administration (MBA) Offered by the Departments of: Business Administration & Marketing and Communication PORTFOLIO ANALYSIS AND MANAGEMENT

More information

Enhancing the Teaching of Statistics: Portfolio Theory, an Application of Statistics in Finance

Enhancing the Teaching of Statistics: Portfolio Theory, an Application of Statistics in Finance Page 1 of 11 Enhancing the Teaching of Statistics: Portfolio Theory, an Application of Statistics in Finance Nicolas Christou University of California, Los Angeles Journal of Statistics Education Volume

More information

Financial-Institutions Management

Financial-Institutions Management Solutions 3 Chapter 11: Credit Risk Loan Pricing and Terms 9. County Bank offers one-year loans with a stated rate of 9 percent but requires a compensating balance of 10 percent. What is the true cost

More information