This slide set is a work in progress and is embedded in my Principles of Finance course site (under construction) that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://financefortechies.weebly.com/
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://financefortechies.weebly.com/
The document summarizes key concepts in financial economics, including:
1. Merton Miller identified five "pillars" of finance, including the CAPM, EMH, modern portfolio theory, and options pricing theory.
2. Eugene Fama defined the EMH as a market where prices "fully reflect available information." The EMH implies prices adjust rapidly to new information.
3. Supporters and critics have debated the EMH for decades, with empirical evidence both supporting and contradicting aspects of the hypothesis. The EMH does not claim prices perfectly equal value, but that no trading strategy can consistently beat the market.
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This document introduces concepts related to investment portfolios including mean, variance, covariance, and matrix algebra. It discusses how to calculate the mean, variance, and covariance of random variables and how these concepts apply to portfolio optimization. Specifically, it explains how to calculate the variance and expected return of a portfolio using covariance matrices and weight vectors. The document also provides examples of generating random portfolios and plotting the efficient frontier.
This document outlines the syllabus for a Principles of Finance course. It includes topics such as financial decision making, valuation, risk and return, capital structure, and models. It lists five pillars of modern finance according to Nobel Prize winner Merton Miller. Assignments will involve weekly programming assignments in R markdown and a semester-long project. Grading will be based on weekly assignments, exams, and a project. The course will provide an introduction to key concepts in finance from various perspectives.
This document discusses key financial concepts related to capital, including invested capital, return to capital, profitability ratios, cash flow to capital, capital structure, and cost of capital. It provides examples of balance sheets, income statements, and calculations of various ratios such as return on invested capital (ROIC), return on assets (ROA), return on equity (ROE), interest coverage ratios, and profit margins. The document uses a company called Fairway Corp as an example to demonstrate calculations of invested capital, net fixed assets, net operating assets, net profit after tax, and various profitability ratios.
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://financefortechies.weebly.com/
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://financefortechies.weebly.com/
The document summarizes key concepts in financial economics, including:
1. Merton Miller identified five "pillars" of finance, including the CAPM, EMH, modern portfolio theory, and options pricing theory.
2. Eugene Fama defined the EMH as a market where prices "fully reflect available information." The EMH implies prices adjust rapidly to new information.
3. Supporters and critics have debated the EMH for decades, with empirical evidence both supporting and contradicting aspects of the hypothesis. The EMH does not claim prices perfectly equal value, but that no trading strategy can consistently beat the market.
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This document introduces concepts related to investment portfolios including mean, variance, covariance, and matrix algebra. It discusses how to calculate the mean, variance, and covariance of random variables and how these concepts apply to portfolio optimization. Specifically, it explains how to calculate the variance and expected return of a portfolio using covariance matrices and weight vectors. The document also provides examples of generating random portfolios and plotting the efficient frontier.
This document outlines the syllabus for a Principles of Finance course. It includes topics such as financial decision making, valuation, risk and return, capital structure, and models. It lists five pillars of modern finance according to Nobel Prize winner Merton Miller. Assignments will involve weekly programming assignments in R markdown and a semester-long project. Grading will be based on weekly assignments, exams, and a project. The course will provide an introduction to key concepts in finance from various perspectives.
This document discusses key financial concepts related to capital, including invested capital, return to capital, profitability ratios, cash flow to capital, capital structure, and cost of capital. It provides examples of balance sheets, income statements, and calculations of various ratios such as return on invested capital (ROIC), return on assets (ROA), return on equity (ROE), interest coverage ratios, and profit margins. The document uses a company called Fairway Corp as an example to demonstrate calculations of invested capital, net fixed assets, net operating assets, net profit after tax, and various profitability ratios.
This document discusses key concepts related to capital including invested capital, return on invested capital, cash flow to invested capital, capital structure, cost of capital, and economic profit. It provides definitions and formulas for calculating invested capital, net operating profit after tax (NOPAT), free cash flow (FCF), return on invested capital (ROIC), and economic profit. Free cash flow is defined as NOPAT minus increases in net fixed assets and net working capital. The document also notes that free cash flow should be invested in projects with positive net present values above the cost of capital, and that conflicts can arise when managers invest excess free cash flow in low return projects.
- The document discusses bonds, including types of bonds, bond parameters, bond yields, prices, and applications such as computing bond prices and yields.
- It covers topics like coupon bonds, zero coupon bonds, bond yields, prices, the yield curve, and bond pricing applications.
- Examples are provided to demonstrate bond pricing calculations and how to determine yields and prices for different bonds.
This document discusses various methods for valuing a firm and its equity, including:
1) The present value discounted cash flow (DCF) method and constant free cash flow and dividend growth methods for valuing a firm and its equity.
2) How the price/earnings ratio relates to equity value.
3) Formulas for valuing a firm under constant free cash flow growth and variable growth assumptions using the DCF approach.
This document provides information on calculating free cash flow (FCF) and determining a firm's cost of equity capital using the Capital Asset Pricing Model (CAPM).
It defines FCF as net operating profit after tax (NOPAT) minus changes in net working capital and capital expenditures. It also outlines how to modify FCF calculations to exclude non-operating cash flows. The document then explains how the CAPM model is used to estimate a firm's cost of equity capital (ke) based on the market risk premium and the stock's beta relative to the market. It provides an example of calibrating the CAPM for Microsoft using historical stock return data versus an S&P 500 index fund.
- The startup needs $3M total investment over 5 years, raised in 3 rounds at the start of years 1, 3, and 5
- The rounds are 30%, 40%, and 30% of total needed, so $0.9M, $1.2M, and $0.9M respectively
- The startup expects $3.6M EBIT and 18x P/E ratio at exit, valuing the startup at $64.8M
- Target rates of return are 40%, 30%, 20% for each round
- The ownership percentages and ROI for each round must be calculated to meet
1) Bond prices and yields are impacted by changes in the yield curve. If the yield curve shifts down, bond prices will increase and yields will decrease. If it shifts up, prices will decrease and yields will increase.
2) There is a difference between dirty and clean bond prices. The dirty price includes accrued interest, while the clean price excludes it. To calculate the clean price, accrued interest is subtracted from the dirty price.
3) Bond dealers quote prices as bids and asks relative to par value. The bid price is what the dealer will pay to purchase bonds, while the ask price is what the dealer will sell bonds for.
The document discusses a venture capital investment in LeanTech, a startup software company. LeanTech has no revenue and needs $3.5 million in funding over the next 5 years. A venture capitalist wants to invest and targets a 50% annual return over 5 years. Using LeanTech's forecasted future earnings and typical industry valuation ratios, the expected value of LeanTech after 5 years is $37.5 million. For the venture capitalist to achieve a 50% annual return, they would need to invest $3.5 million and own 70.875% of LeanTech. This values LeanTech at $4.938 million immediately after the investment.
Capital structure and cost of equity pdfDavid Keck
This document discusses capital structure and cost of equity. It begins by outlining learning objectives around basic corporate finance concepts like capital structure, cost of equity, and dividend policy. It then provides assumptions for calculating rates of return, including that free cash flow is a perpetuity. The document uses an example firm to demonstrate calculating unlevered and levered costs of equity and the effects of leverage on firm value under the assumptions of Miller and Modigliani's propositions.
The document discusses key concepts related to cash flow and cost of capital. It provides definitions and formulas for various cash flow terms including:
- Cash flow from operating activities (CFO) which is net profit plus non-cash expenses/revenues and changes in working capital items.
- Cash flow from investing activities (CFI) which is cash flows from investments in or sales of long-term operating assets.
- Cash flow from financing activities (CFF) which is flows from changes in debt, equity or dividends.
- Free cash flow (FCF) which is cash flow from operating activities plus cash flow from investments in operating assets, available to providers of capital.
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://awesome.weebly.com/
The document discusses portfolio theory and diversification from a mathematical perspective. It introduces Harry Markowitz's efficient frontier and how diversifying investments reduces portfolio risk. The expected return of a portfolio is a weighted average of the component returns, while the variance is not a linear combination and depends on the covariance between components. Diversification is most effective at reducing risk when the component returns are negatively correlated. The mathematics of diversification for multiple assets uses a variance-covariance matrix to calculate portfolio risk.
This document summarizes key concepts in building multiple regression models, including:
1) Analyzing nonlinear variables, qualitative variables, and building and evaluating regression models.
2) Transforming variables to improve model fit, including using indicator variables for qualitative data.
3) Common model building techniques like stepwise regression, forward selection, and backward elimination.
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://financefortechies.weebly.com/
1CHAPTER 6Risk, Return, and the Capital Asset Pricing Model.docxhyacinthshackley2629
1
CHAPTER 6
Risk, Return, and the Capital Asset Pricing Model
2
Topics in Chapter
Basic return concepts
Basic risk concepts
Stand-alone risk
Portfolio (market) risk
Risk and return: CAPM/SML
1
Value = + + +
FCF1
FCF2
FCF∞
(1 + WACC)1
(1 + WACC)∞
(1 + WACC)2
Free cash flow
(FCF)
Market interest rates
Firm’s business risk
Market risk aversion
Firm’s debt/equity mix
Cost of debt
Cost of equity
Weighted average
cost of capital
(WACC)
Net operating
profit after taxes
Required investments
in operating capital
−
=
Determinants of Intrinsic Value:
The Cost of Equity
...
For value box in Ch 4 time value FM13.
4
What are investment returns?
Investment returns measure the financial results of an investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
5
An investment costs $1,000 and is sold after 1 year for $1,100.
Dollar return:
Percentage return:
$ Received - $ Invested
$1,100 - $1,000 = $100.
$ Return/$ Invested
$100/$1,000 = 0.10 = 10%.
2
6
What is investment risk?
Typically, investment returns are not known with certainty.
Investment risk pertains to the probability of earning a return less than that expected.
The greater the chance of a return far below the expected return, the greater the risk.
2
7
Probability Distribution: Which stock is riskier? Why?
8
Consider the Following
Investment AlternativesEcon.Prob.T-BillAltaRepoAm F.MPBust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0Avg. 0.40 8.0 20.0 0.0 7.0 15.0Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0Boom 0.10 8.0 50.0 -20.0 30.0 43.0 1.00
9
What is unique about the T-bill return?
The T-bill will return 8% regardless of the state of the economy.
Is the T-bill riskless? Explain.
5
10
Alta Inds. and Repo Men vs. the Economy
Alta Inds. moves with the economy, so it is positively correlated with the economy. This is the typical situation.
Repo Men moves counter to the economy. Such negative correlation is unusual.
7
11
Calculate the expected rate of return on each alternative.
r = expected rate of return.
rAlta = 0.10(-22%) + 0.20(-2%)
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.
^
^
n
∑
r =
^
i=1
riPi.
12
Alta has the highest rate of return. Does that make it best?^rAlta 17.4%Market15.0Am. Foam13.8T-bill 8.0Repo Men 1.7
13
What is the standard deviation
of returns for each alternative?
σ = Standard deviation
σ = √ Variance = √ σ2
n
∑
i=1
= √
(ri – r)2 Pi.
^
14
= [(-22 - 17.4)20.10 + (-2 - 17.4)20.20
+ (20 - 17.4)20.40 + (35 - 17.4)20.20
+ (50 - 17.4)20.10]1/2
= 20.0%.
Standard Deviation of Alta Industries
11
15
T-bills = 0.0%.
Alta = 20.0%.
Repo = 13.4%.
Am Foam = 18.
This document discusses key concepts related to investment returns and risk. It defines return as the financial results of an investment expressed in dollar or percentage terms. Risk is defined as the probability of earning a return lower than expected. Diversification across many stocks can reduce risk, as stock returns are not perfectly correlated. A portfolio's risk is measured by its beta coefficient, which represents the portfolio's volatility relative to the market. The Security Market Line (SML) shows the relationship between risk and required return in the Capital Asset Pricing Model (CAPM).
The document introduces an ARMA-GARCH time series model to explain and predict the behavior of stock return in the financial industry. It analyzes weekly stock price data of 5 major banks from 2005 to 2014. Summary statistics show the portfolio return has higher kurtosis and negative skewness than a normal distribution. The volatility cluster graph indicates that volatility is time-varying rather than constant. Autocorrelation functions imply the return series is stationary. Model selection based on AIC and BIC criteria finds that ARMA(1,1)-GARCH(2,2) fits the data best.
This document discusses risk and return concepts including stand-alone risk, portfolio risk, and the capital asset pricing model (CAPM). It defines investment returns, risk, and probability distributions. It shows how diversification reduces risk and calculates portfolio returns and risk. The security market line (SML) from CAPM is used to calculate required returns based on betas and determine if securities are over or undervalued. Calculating betas using regression analysis on historical returns is also demonstrated.
The document discusses key concepts related to risk and return in investments. It defines return as the motivating force for making investments, while risk refers to uncertainty in future cash flows. There are different types of returns such as historical returns and expected returns. Risk is measured using concepts like variance, standard deviation, and beta. The Capital Asset Pricing Model relates risk and return, specifying that expected return is equal to the risk-free rate plus a risk premium based on systematic risk. Diversification can reduce unsystematic risk but not systematic risk.
The document discusses key concepts related to risk and return in investments. It defines return as the motivating force for making investments, while risk refers to uncertainty in future cash flows. There are different types of returns such as historical returns and expected returns. Risk is measured using concepts like variance, standard deviation, and beta. The Capital Asset Pricing Model relates risk and return, with expected return equal to the risk-free rate plus a risk premium based on systematic risk. Diversification can reduce unsystematic risk but not systematic risk. Understanding the risk-return relationship is important for investment decision making.
This document discusses key concepts related to capital including invested capital, return on invested capital, cash flow to invested capital, capital structure, cost of capital, and economic profit. It provides definitions and formulas for calculating invested capital, net operating profit after tax (NOPAT), free cash flow (FCF), return on invested capital (ROIC), and economic profit. Free cash flow is defined as NOPAT minus increases in net fixed assets and net working capital. The document also notes that free cash flow should be invested in projects with positive net present values above the cost of capital, and that conflicts can arise when managers invest excess free cash flow in low return projects.
- The document discusses bonds, including types of bonds, bond parameters, bond yields, prices, and applications such as computing bond prices and yields.
- It covers topics like coupon bonds, zero coupon bonds, bond yields, prices, the yield curve, and bond pricing applications.
- Examples are provided to demonstrate bond pricing calculations and how to determine yields and prices for different bonds.
This document discusses various methods for valuing a firm and its equity, including:
1) The present value discounted cash flow (DCF) method and constant free cash flow and dividend growth methods for valuing a firm and its equity.
2) How the price/earnings ratio relates to equity value.
3) Formulas for valuing a firm under constant free cash flow growth and variable growth assumptions using the DCF approach.
This document provides information on calculating free cash flow (FCF) and determining a firm's cost of equity capital using the Capital Asset Pricing Model (CAPM).
It defines FCF as net operating profit after tax (NOPAT) minus changes in net working capital and capital expenditures. It also outlines how to modify FCF calculations to exclude non-operating cash flows. The document then explains how the CAPM model is used to estimate a firm's cost of equity capital (ke) based on the market risk premium and the stock's beta relative to the market. It provides an example of calibrating the CAPM for Microsoft using historical stock return data versus an S&P 500 index fund.
- The startup needs $3M total investment over 5 years, raised in 3 rounds at the start of years 1, 3, and 5
- The rounds are 30%, 40%, and 30% of total needed, so $0.9M, $1.2M, and $0.9M respectively
- The startup expects $3.6M EBIT and 18x P/E ratio at exit, valuing the startup at $64.8M
- Target rates of return are 40%, 30%, 20% for each round
- The ownership percentages and ROI for each round must be calculated to meet
1) Bond prices and yields are impacted by changes in the yield curve. If the yield curve shifts down, bond prices will increase and yields will decrease. If it shifts up, prices will decrease and yields will increase.
2) There is a difference between dirty and clean bond prices. The dirty price includes accrued interest, while the clean price excludes it. To calculate the clean price, accrued interest is subtracted from the dirty price.
3) Bond dealers quote prices as bids and asks relative to par value. The bid price is what the dealer will pay to purchase bonds, while the ask price is what the dealer will sell bonds for.
The document discusses a venture capital investment in LeanTech, a startup software company. LeanTech has no revenue and needs $3.5 million in funding over the next 5 years. A venture capitalist wants to invest and targets a 50% annual return over 5 years. Using LeanTech's forecasted future earnings and typical industry valuation ratios, the expected value of LeanTech after 5 years is $37.5 million. For the venture capitalist to achieve a 50% annual return, they would need to invest $3.5 million and own 70.875% of LeanTech. This values LeanTech at $4.938 million immediately after the investment.
Capital structure and cost of equity pdfDavid Keck
This document discusses capital structure and cost of equity. It begins by outlining learning objectives around basic corporate finance concepts like capital structure, cost of equity, and dividend policy. It then provides assumptions for calculating rates of return, including that free cash flow is a perpetuity. The document uses an example firm to demonstrate calculating unlevered and levered costs of equity and the effects of leverage on firm value under the assumptions of Miller and Modigliani's propositions.
The document discusses key concepts related to cash flow and cost of capital. It provides definitions and formulas for various cash flow terms including:
- Cash flow from operating activities (CFO) which is net profit plus non-cash expenses/revenues and changes in working capital items.
- Cash flow from investing activities (CFI) which is cash flows from investments in or sales of long-term operating assets.
- Cash flow from financing activities (CFF) which is flows from changes in debt, equity or dividends.
- Free cash flow (FCF) which is cash flow from operating activities plus cash flow from investments in operating assets, available to providers of capital.
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://awesome.weebly.com/
The document discusses portfolio theory and diversification from a mathematical perspective. It introduces Harry Markowitz's efficient frontier and how diversifying investments reduces portfolio risk. The expected return of a portfolio is a weighted average of the component returns, while the variance is not a linear combination and depends on the covariance between components. Diversification is most effective at reducing risk when the component returns are negatively correlated. The mathematics of diversification for multiple assets uses a variance-covariance matrix to calculate portfolio risk.
This document summarizes key concepts in building multiple regression models, including:
1) Analyzing nonlinear variables, qualitative variables, and building and evaluating regression models.
2) Transforming variables to improve model fit, including using indicator variables for qualitative data.
3) Common model building techniques like stepwise regression, forward selection, and backward elimination.
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://financefortechies.weebly.com/
1CHAPTER 6Risk, Return, and the Capital Asset Pricing Model.docxhyacinthshackley2629
1
CHAPTER 6
Risk, Return, and the Capital Asset Pricing Model
2
Topics in Chapter
Basic return concepts
Basic risk concepts
Stand-alone risk
Portfolio (market) risk
Risk and return: CAPM/SML
1
Value = + + +
FCF1
FCF2
FCF∞
(1 + WACC)1
(1 + WACC)∞
(1 + WACC)2
Free cash flow
(FCF)
Market interest rates
Firm’s business risk
Market risk aversion
Firm’s debt/equity mix
Cost of debt
Cost of equity
Weighted average
cost of capital
(WACC)
Net operating
profit after taxes
Required investments
in operating capital
−
=
Determinants of Intrinsic Value:
The Cost of Equity
...
For value box in Ch 4 time value FM13.
4
What are investment returns?
Investment returns measure the financial results of an investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
5
An investment costs $1,000 and is sold after 1 year for $1,100.
Dollar return:
Percentage return:
$ Received - $ Invested
$1,100 - $1,000 = $100.
$ Return/$ Invested
$100/$1,000 = 0.10 = 10%.
2
6
What is investment risk?
Typically, investment returns are not known with certainty.
Investment risk pertains to the probability of earning a return less than that expected.
The greater the chance of a return far below the expected return, the greater the risk.
2
7
Probability Distribution: Which stock is riskier? Why?
8
Consider the Following
Investment AlternativesEcon.Prob.T-BillAltaRepoAm F.MPBust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0Avg. 0.40 8.0 20.0 0.0 7.0 15.0Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0Boom 0.10 8.0 50.0 -20.0 30.0 43.0 1.00
9
What is unique about the T-bill return?
The T-bill will return 8% regardless of the state of the economy.
Is the T-bill riskless? Explain.
5
10
Alta Inds. and Repo Men vs. the Economy
Alta Inds. moves with the economy, so it is positively correlated with the economy. This is the typical situation.
Repo Men moves counter to the economy. Such negative correlation is unusual.
7
11
Calculate the expected rate of return on each alternative.
r = expected rate of return.
rAlta = 0.10(-22%) + 0.20(-2%)
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.
^
^
n
∑
r =
^
i=1
riPi.
12
Alta has the highest rate of return. Does that make it best?^rAlta 17.4%Market15.0Am. Foam13.8T-bill 8.0Repo Men 1.7
13
What is the standard deviation
of returns for each alternative?
σ = Standard deviation
σ = √ Variance = √ σ2
n
∑
i=1
= √
(ri – r)2 Pi.
^
14
= [(-22 - 17.4)20.10 + (-2 - 17.4)20.20
+ (20 - 17.4)20.40 + (35 - 17.4)20.20
+ (50 - 17.4)20.10]1/2
= 20.0%.
Standard Deviation of Alta Industries
11
15
T-bills = 0.0%.
Alta = 20.0%.
Repo = 13.4%.
Am Foam = 18.
This document discusses key concepts related to investment returns and risk. It defines return as the financial results of an investment expressed in dollar or percentage terms. Risk is defined as the probability of earning a return lower than expected. Diversification across many stocks can reduce risk, as stock returns are not perfectly correlated. A portfolio's risk is measured by its beta coefficient, which represents the portfolio's volatility relative to the market. The Security Market Line (SML) shows the relationship between risk and required return in the Capital Asset Pricing Model (CAPM).
The document introduces an ARMA-GARCH time series model to explain and predict the behavior of stock return in the financial industry. It analyzes weekly stock price data of 5 major banks from 2005 to 2014. Summary statistics show the portfolio return has higher kurtosis and negative skewness than a normal distribution. The volatility cluster graph indicates that volatility is time-varying rather than constant. Autocorrelation functions imply the return series is stationary. Model selection based on AIC and BIC criteria finds that ARMA(1,1)-GARCH(2,2) fits the data best.
This document discusses risk and return concepts including stand-alone risk, portfolio risk, and the capital asset pricing model (CAPM). It defines investment returns, risk, and probability distributions. It shows how diversification reduces risk and calculates portfolio returns and risk. The security market line (SML) from CAPM is used to calculate required returns based on betas and determine if securities are over or undervalued. Calculating betas using regression analysis on historical returns is also demonstrated.
The document discusses key concepts related to risk and return in investments. It defines return as the motivating force for making investments, while risk refers to uncertainty in future cash flows. There are different types of returns such as historical returns and expected returns. Risk is measured using concepts like variance, standard deviation, and beta. The Capital Asset Pricing Model relates risk and return, specifying that expected return is equal to the risk-free rate plus a risk premium based on systematic risk. Diversification can reduce unsystematic risk but not systematic risk.
The document discusses key concepts related to risk and return in investments. It defines return as the motivating force for making investments, while risk refers to uncertainty in future cash flows. There are different types of returns such as historical returns and expected returns. Risk is measured using concepts like variance, standard deviation, and beta. The Capital Asset Pricing Model relates risk and return, with expected return equal to the risk-free rate plus a risk premium based on systematic risk. Diversification can reduce unsystematic risk but not systematic risk. Understanding the risk-return relationship is important for investment decision making.
This document provides an overview of portfolio theory and asset pricing models. It discusses key concepts such as the factors that affect stock prices like cash flows, risk, and timing. It also covers the weighted average cost of capital (WACC) and how it is used to calculate intrinsic value. Other topics include the capital market line, security market line, beta estimation through regression analysis, and tests of the capital asset pricing model (CAPM). The document provides examples of how to calculate portfolio expected returns, standard deviations, and betas. It also discusses the relationship between total, market, and diversifiable risk.
This document provides 14 problems related to risk and return concepts including expected rate of return, standard deviation, portfolio return and risk, and the Capital Asset Pricing Model. The problems cover calculating expected returns and standard deviations for single assets and portfolios. Other problems involve using the CAPM formula to calculate expected returns given risk free rates, market returns and betas. The solutions show the step-by-step workings for each problem.
This document discusses the concepts of risk and return as they relate to investment portfolio diversification. It provides an example to illustrate how combining investments with imperfect correlations can reduce overall portfolio risk compared to holding individual assets. Specifically:
- It analyzes potential returns and risks of investing in bus and taxi companies under different economic scenarios.
- It then shows how a 50% allocation to each lowers total risk compared to the individual investments, from 10.68% portfolio standard deviation versus 26.42% and 34.13% for each individually.
- This demonstrates how diversification across imperfectly correlated assets reduces risk for the same expected return compared to holding single assets.
This chapter discusses key concepts related to risk and return in investments. It defines investment returns as the financial results of an investment expressed in dollar or percentage terms. Investment risk is the probability that the actual return will be lower than expected. Diversifying investments across many uncorrelated assets reduces overall portfolio risk. The Security Market Line and Capital Asset Pricing Model relate the expected return of an asset to its beta, a measure of how volatile the asset is relative to the market.
The document presents a methodology for removing serial correlation from hedge fund return time series data in order to determine the "true" underlying returns. It describes the Okunev White model, which can eliminate autocorrelation of any order from a time series. The document then applies this model to various hedge fund indices, finding significant reductions in autocorrelation and changes to the distributions and risk measures of the returns. Key impacts included a right shift of negatively skewed distributions and reduced kurtosis, as well as lower values for risk ratios like the Sharpe ratio.
This chapter discusses key concepts related to investment risk and return. It defines investment returns as the financial results of an investment expressed in dollar or percentage terms. Investment risk is the probability of earning a return lower than expected and is impacted by how widely returns can vary from expectations. Diversifying a portfolio by holding many stocks from different industries can significantly reduce risk without lowering expected returns. The Security Market Line shows the expected return required based on a security's beta, which measures its non-diversifiable risk relative to the overall market.
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Volatility
This document discusses efficient numerical PDE methods to solve calibration and pricing problems in local stochastic volatility models. It begins with an overview of volatility modelling, including local stochastic volatility models that combine local volatility, jumps, and stochastic volatility. It then discusses calibrating both parametric and non-parametric local volatility models using PDE methods. The document provides examples of modelling stochastic volatility factors using implied volatility data and estimating jump parameters from historical returns. It also discusses calibrating local volatility models to vanilla option prices while including jumps and stochastic volatility.
The document discusses various methods of measuring and evaluating risk:
1. Standard deviation is a statistical tool used to measure risk by quantifying the variation of returns around the mean. It allows comparison of risk across stocks.
2. Covariance and correlation measure how returns of two securities move together, allowing analysis of portfolio risk. Diversification across negatively correlated securities can reduce overall risk.
3. Beta indicates how sensitive a stock's returns are to market returns. It is estimated using regression analysis and implies the level of systematic risk. Stocks with beta greater than 1 are riskier than the market.
A Study on the Short Run Relationship b/w Major Economic Indicators of US Eco...aurkoiitk
The objective of this study
was to develop an economic indicator system for the US
economy that will help to forecast the turning points in the
aggregate level of economic activity. Our primary concern
is to study the short run relationship between the major
economic indicators of US economy (eg: GDP, Money
Supply, Unemployment Rate, Inflation rate, Federal Fund
Rate, Exchange Rate, Government Expenditure &
Receipt, Crude Oil Price, Net Import & Export).
Ciaran Phillip Directed Research Thesis DefenseCiaran Phillip
This document describes a study that used physical and chemical analysis to develop a multivariate forensic signature for classifying and determining the brand of matchsticks. Stereomicroscopy, FTIR, and ICP-MS were used to analyze samples from different matchstick brands. Multivariate statistical analysis including PCA, ANOSIM, and NMDS showed that the combination of techniques can distinguish between matchstick classes and determine commercial brands. The results provide an excellent method for forensic analysis and classification of matchsticks.
This document outlines a proposed data and models sequence for undergraduate students. The sequence consists of three courses - RAIK 270 taken in the fall sophomore year covering fundamentals of data analysis, RAIK 370 taken in the spring sophomore year covering fundamentals of data science, and RAIK 371 taken in the spring junior year covering fundamentals of management science. These courses aim to teach traditional topics in probability, statistics, data analytics, machine learning, and optimization applied across disciplines like engineering, science, and business with an emphasis on critical thinking, model thinking, cross-disciplinarity, and working with uncertainty.
This document provides a list of topics related to data science, machine learning, social modeling, and decision making. Some of the key topics included are neural networks, random processes, linear and nonlinear regression models, emergence, growth models, heuristics, Markov processes, game theory, and prediction. The list touches on concepts from mathematics, computer science, economics, and other domains relevant to analyzing and modeling data.
This document discusses model thinking and different types of models. It states that models are approximations of reality and while all models are wrong, some can be useful. It discusses using models to think more clearly, understand data, and make decisions. Model thinking involves using data to create data models to gain insights and make predictions. A single model can apply to multiple problems and multiple models can be used to study one problem. The document lists different types of models used across individual decision making, organizations, biological systems, physical systems, and more.
This document discusses three levels of cross-disciplinarity: multidisciplinarity, interdisciplinarity, and transdisciplinarity. Multidisciplinarity involves several disciplines addressing a topic without integrating their perspectives. Interdisciplinarity aims to integrate knowledge from different disciplines by developing new methodologies and approaches. Transdisciplinarity seeks to understand the world as a whole by going beyond individual disciplines and integrating them at a higher level of abstraction.
This document discusses three levels of cross-disciplinarity: multidisciplinarity, interdisciplinarity, and transdisciplinarity. Multidisciplinarity involves several disciplines addressing a topic without integrating their perspectives. Interdisciplinarity aims to integrate knowledge from different disciplines by developing new methodologies and approaches. Transdisciplinarity seeks to understand the world as a whole by going beyond individual disciplines and integrating them at a higher level of abstraction.
This document discusses concepts related to certainty, complexity, chaos, randomness, and uncertainty. It notes that certainty involves order and complexity can involve emergence, self-organization, learning and adapting, self-organized criticality, interdependency, and nonlinear and dynamic systems with heterogeneous elements. Uncertainty comes from ignorance, randomness, natural variation, measurement error, and nonlinear dynamics. The document asks whether elements in a system follow rules, behave randomly, or adapt.
The document discusses key concepts relevant to innovation including opportunity, uncertainty, interdependency, and wicked problems. It notes that innovation emerges from a space characterized by these factors. The document also outlines what is needed for innovation, including foundational knowledge, cross-disciplinarity, critical thinking, and design/model thinking. It discusses how uncertainty can be reduced through data, analytics, learning, and gaining information to make better decisions.
The document discusses the history of innovation and entrepreneurship from ancient times to the modern era. It traces key concepts from early thinkers like Richard Cantillon and Adam Smith, through economists like Thomas Malthus, Jean-Baptiste Say, and John Stuart Mill. Joseph Schumpeter is discussed as a major theorist who defined innovation as new combinations and the entrepreneur as the agent of innovation. Later sections cover innovation diffusion theory, endogenous growth theory, and frameworks for human-centered design thinking. The document aims to provide context around innovation concepts over time.
This document introduces concepts related to investment portfolios including mean, variance, covariance, and matrix algebra. It defines formulas to calculate the mean, variance, and covariance of random variables and how these concepts extend to portfolios containing multiple assets. The document also describes how to model a portfolio of multiple assets using vectors and matrices and defines the formulas to calculate the variance and expected return of the portfolio based on the individual asset expected returns, variances, and covariances. It concludes by describing how to optimize a portfolio by minimizing variance for a given expected return level, known as the efficient frontier.
This document discusses various criteria for evaluating investment decisions: net present value, internal rate of return, discounted payback period, payback period, and break-even analysis. It provides definitions and formulas for calculating each, such as using net present value to determine if the NPV of an investment is greater than $0, or using internal rate of return to find the rate that results in an NPV of $0. Homework assignments 7A through 7E are also listed.
The document provides an introduction to the Capital Asset Pricing Model (CAPM) for calculating the cost of equity for publicly traded companies. It discusses key concepts such as the value of a company being equal to the value of its debt plus the value of its equity. It also covers the equity return rate being a function of the risk-free rate plus a risk premium related to the market return rate. The CAPM model sets the expected return on equity equal to the risk-free rate plus the product of the market risk premium and the stock's beta coefficient.
The document discusses stock prices and rates of return using Intel Corporation stock as an example. It provides adjusted closing prices for Intel stock from August 2006 to August 2016 sampled monthly. It defines simple and natural log rates of return calculated from the stock prices and discusses how dividends and stock splits are handled. The document also discusses how random errors in stock prices accumulate and how rates of return are distributed, noting that additive errors accumulate normally while multiplicative errors accumulate lognormally. Various R commands for analyzing distributions and rates of return are also provided.
This document discusses interest rates, including simple interest rates, compound interest rates with annual and periodic compounding, and continuous compounding. It also covers topics like present and future values, inflation adjustments, and uses of interest rates for things like bonds, mortgages, and consumer rates. Worked examples are provided for calculations involving simple interest, annual compounding, periodic compounding, and continuous compounding.
- LeanTech, a software startup, needed to raise $3.5 million in capital to fund costs over the next 5 years as it had no revenue.
- A venture capitalist was interested in investing and targeted a 50% annual return over 5 years.
- To achieve this return, the venture capitalist would receive a portion of LeanTech's equity and become a shareholder and board member, with the intent to exit via IPO or acquisition after 5 years.
- The key terms of the venture capital deal, including LeanTech's valuation pre- and post-investment, the venture capitalist's expected return and ownership stake, and LeanTech's expected future value were calculated using standard venture capital models.
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This slide set is under serious development!
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
Economic Risk Factor Update: June 2024 [SlideShare]Commonwealth
May’s reports showed signs of continued economic growth, said Sam Millette, director, fixed income, in his latest Economic Risk Factor Update.
For more market updates, subscribe to The Independent Market Observer at https://blog.commonwealth.com/independent-market-observer.
Independent Study - College of Wooster Research (2023-2024) FDI, Culture, Glo...AntoniaOwensDetwiler
"Does Foreign Direct Investment Negatively Affect Preservation of Culture in the Global South? Case Studies in Thailand and Cambodia."
Do elements of globalization, such as Foreign Direct Investment (FDI), negatively affect the ability of countries in the Global South to preserve their culture? This research aims to answer this question by employing a cross-sectional comparative case study analysis utilizing methods of difference. Thailand and Cambodia are compared as they are in the same region and have a similar culture. The metric of difference between Thailand and Cambodia is their ability to preserve their culture. This ability is operationalized by their respective attitudes towards FDI; Thailand imposes stringent regulations and limitations on FDI while Cambodia does not hesitate to accept most FDI and imposes fewer limitations. The evidence from this study suggests that FDI from globally influential countries with high gross domestic products (GDPs) (e.g. China, U.S.) challenges the ability of countries with lower GDPs (e.g. Cambodia) to protect their culture. Furthermore, the ability, or lack thereof, of the receiving countries to protect their culture is amplified by the existence and implementation of restrictive FDI policies imposed by their governments.
My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.
Every business, big or small, deals with outgoing payments. Whether it’s to suppliers for inventory, to employees for salaries, or to vendors for services rendered, keeping track of these expenses is crucial. This is where payment vouchers come in – the unsung heroes of the accounting world.
[4:55 p.m.] Bryan Oates
OJPs are becoming a critical resource for policy-makers and researchers who study the labour market. LMIC continues to work with Vicinity Jobs’ data on OJPs, which can be explored in our Canadian Job Trends Dashboard. Valuable insights have been gained through our analysis of OJP data, including LMIC research lead
Suzanne Spiteri’s recent report on improving the quality and accessibility of job postings to reduce employment barriers for neurodivergent people.
Decoding job postings: Improving accessibility for neurodivergent job seekers
Improving the quality and accessibility of job postings is one way to reduce employment barriers for neurodivergent people.
OJP data from firms like Vicinity Jobs have emerged as a complement to traditional sources of labour demand data, such as the Job Vacancy and Wages Survey (JVWS). Ibrahim Abuallail, PhD Candidate, University of Ottawa, presented research relating to bias in OJPs and a proposed approach to effectively adjust OJP data to complement existing official data (such as from the JVWS) and improve the measurement of labour demand.
Vicinity Jobs’ data includes more than three million 2023 OJPs and thousands of skills. Most skills appear in less than 0.02% of job postings, so most postings rely on a small subset of commonly used terms, like teamwork.
Laura Adkins-Hackett, Economist, LMIC, and Sukriti Trehan, Data Scientist, LMIC, presented their research exploring trends in the skills listed in OJPs to develop a deeper understanding of in-demand skills. This research project uses pointwise mutual information and other methods to extract more information about common skills from the relationships between skills, occupations and regions.
Dr. Alyce Su Cover Story - China's Investment Leadermsthrill
In World Expo 2010 Shanghai – the most visited Expo in the World History
https://www.britannica.com/event/Expo-Shanghai-2010
China’s official organizer of the Expo, CCPIT (China Council for the Promotion of International Trade https://en.ccpit.org/) has chosen Dr. Alyce Su as the Cover Person with Cover Story, in the Expo’s official magazine distributed throughout the Expo, showcasing China’s New Generation of Leaders to the World.
2. The
Five
Pillars
2
Nobel
Prize
winner
and
former
Univ.
of
Chicago
professor,
Merton
Miller,
published
a
paper
called
the
“The
History
of
Finance”
Miller
idenBfied
five
“pillars
on
which
the
field
of
finance
rests”
These
include
1. Miller-‐Modigliani
ProposiBons
• Merton
Miller
1990
and
Franco
Modigliani
1985
2. Capital
Asset
Pricing
Model
• William
Sharpe
1990
3. Efficient
Market
Hypothesis
• (Eugene
Fama,
Paul
Samuelson,
…)
4. Modern
Por+olio
Theory
• Harry
Markowitz
1990
5. OpBons
• Myron
Scholes
and
Robert
Merton
1997
3. Learning
ObjecBves
¨ Build
a
por[olio
an
opBmal
por[olio
of
securiBes
consistent
with
your
expected
risk
and
return
requirements
¤ DiversificaBon
is
key
¤ Single,
not
mulBperiod,
investment
horizons
n So
can
use
r
&
d
or
α
and
δ
for
simple
¨ Understand
¤ Random
variables
with
cross
correlaBons
¤ matrix
algebra
and
¤ quadraBc
opBmizaBon
¨ Note
¤ r
and
σ
are
used
as
generic
symbols
to
represent
expected
(mean)
return
rate
and
standard
deviaBon
over
the
planning
period
n Can
be
conBnuously
or
discretely
compounded,
but
must
be
consistent
3
4. 4
Por[olio
of
M
Risky
Assets
¨ Each asset has returns expected to be normally distributed
¨ The portfolio’s expected returns are also normally
distributed
¨ A stock’s expected return might come from the CAPM model
¨ A bond’s expected return come from a similar model
¤ bexpected = rforecast + ( bhistorical - rhistorical )
Mi1
)σ,(r ii ≤≤
)σ,(r PP
)rr(rr FMFE −⋅β+=
5. 5
Por[olio
of
M
Risky
Assets
¨ Expected
variance
for
an
asset
is
o`en
assumed
to
be
the
historical
variance
¨ In
this
topic
we
will
also
assume
that
the
expected
return
is
the
long
term
historical
average
return
¨ What
is
the
proper
length
of
the
historical
record
and
the
sampling
frequency?
6. 6
A
Por[olio
With
Two
Risky
Assets
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Std
Dev
Return
(rA,σA)
(rB,σB)
7. 7
A
Por[olio
With
Two
Risky
Assets
¨ rP
=
wA·∙rA
+
wB·∙rB
¤ wA
+
wB
=1
n requires
that
the
por[olio
is
fully
invested
in
the
2
assets
A
and
B
¤ wA ≥ 0,
wB ≥ 0
n prohibits
short
selling
or
borrowing
an
asset
¤ 1 ≥ wA,
1 ≥ wB
n Restricts
buying
an
asset
on
margin
ABBABA
2
B
2
B
2
A
2
A
2
p
ABBA
2
B
2
B
2
A
2
A
2
p
ABBABB
2
BAA
2
A
2
p
ρσσw2wσwσwσ
σw2wσwσwσ
σw2wσwσwσ
++=
++=
++=
AAAA
2
A σσσσ ≡≡
8. 8
Por[olios
With
Two
Risky
Assets
¨ σA= 8.3%
¨ σB= 16.3%
¨ σAB = .004
¨ rA =0.9%
¨ rB = 2.3%
¨ ρAB = .28
A
AVBV
AB
2
B
2
A
AB
2
B
AV
w-‐1w
2σσσ
)σ(σ
w
=
−+
−
=
ABBABA
2
B
2
B
2
A
2
A
2
p ρσσw2wσwσwσ ++=
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Std
Dev
Return
A
B
Minimum
variance
portfolio
11. 11
Two
Risky
and
One
Risk
Free
Asset
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
1.6%
1.8%
2.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18%
Std
Dev
Return
Asset
B
Min
Variance
Portfolio
V
risk
free
asset
F
Tangent
Portfolio
T
Asset
A
ABA TT
ABFBFA
2
AFA
2
BFA
ABFB
2
BFA
T w-‐1w
σ)]r(r)r[(rσ)r(rσ)r(r
σ)r(rσ)r(r
w =
⋅−+−−⋅−+⋅−
⋅−−⋅−
=
12. 12
Now
Determine
Your
OpBmal
Por[olio
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Std
Dev
Return
Indifference
curves
A=2
,
4,
7
T:
OpBmal
Risky
Por[olio
F
P:
Your
opBmal
por[olio
A
B
V
13. 13
Por[olio
with
2
Risky
Assets
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Std
Dev
Return
Indifference
curves
A=4
T:
OpBmal
Risky
Por[olio
F
P:
Your
opBmal
por[olio
A
B
V
rCE
14. 14
Now
Consider
M
>
2
Risky
Assets
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17%
Extected
Std
Dev
%/mo.
Expeced
Return
%/mo
Now
where
is
the
opBmal
risky
por[olios
?
Symbol ri
σ
i
IBM 1.07% 9.03%
TM 0.92% 7.82%
XOM 1.21% 5.25%
BRK-‐B 1.06% 5.94%
GE 0.79% 6.42%
WMT 0.99% 7.30%
C 0.96% 8.35%
ORCL 2.36% 16.07%
15. 15
Compute
rP
and
σP
with
M
risky
assets
1w0 i
≤≤
1w
M
1i
i
=∑=
i
M
1i
iP rwr ∑=
⋅= ij
M
1j
ji
M
1i
2
P σwwσ ⋅⋅= ∑∑ ==
∑∑∑
≠
===
⋅⋅+⋅=
M
ij
1j
ijji
M
1i
M
1i
2
i
2
i
2
P σwwσwσ
16. 16
Now
Use
Array
NotaBon
For
rP
and
σP
⎣ ⎦[ ]{ }jiji
T2
P wσwwCwσ =⋅⋅=
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
2
MM2M1
2M
2
221
1M12
2
1
MMM2M1
2M2221
1M1211
σσσ
σσσ
σσσ
σσσ
σσσ
σσσ
C
{ }i
σ=σ
⎣ ⎦i
T
σ=σ
{ }i
r
r =
⎣ ⎦i
T
rr =
ij
M
1j
ji
M
1i
2
P σwwσ ⋅⋅= ∑∑ ==
17. 17
Compute
Covariance
–
Variance
Matrix
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
NMN2N1
2M2221
1M1211
rrr
rrr
rrr
R
stocks
1
to
M
returns
1
to
N
N
AA
C
T
=
ji
ij
ij
σσ
σ
ρ
⋅
=
N
r
r
N
1k
ki
i
∑=
=
N
)r(r
σ
N
1k
2
iki
2
i
∑=
−
=
N
)r)(rr(r
σ
N
1k
jkjiki
ij
∑=
−−
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−−
−−−
−−−
=
MNM2N21N1
M2M222121
M1M212111
rrrrrr
rrrrrr
rrrrrr
A
18. Compute
Por[olio
Return
18
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
=
NMMN22N11
2MM222211
1MM122111
rwrwrw
rwrwrw
rwrwrw
R
∑ ∑
∑ ∑
∑ ∑
∑
∑∑
= =
= =
= =
=
==
⋅=
⋅=
⋅=
⋅=
=⋅
⋅
=
M
1i
N
1k
kii
M
1i
N
1k
kii
M
1i
N
1k
kiiP
M
1i
iiP
N
1k
ki
N
1k
kii
i
i
rw
N
1
r
N
1
w
r
N
1
wr
rwr
r
N
1
rw
wN
1
r
19. 19
Example
Matrices
Covariance
matrix
CorrelaBon
matrix
Visualize
IBM TM XOM BRK-‐B GE WMT C ORCL
IBM 0.00815 0.00162 0.00149 0.00046 0.00226 0.00150 0.00394 0.00483
TM 0.00162 0.00612 0.00054 0.00084 0.00224 0.00146 0.00205 0.00341
XOM 0.00149 0.00054 0.00276 0.00053 0.00056 0.00010 0.00111 0.00052
BRK-‐B 0.00046 0.00084 0.00053 0.00353 0.00139 0.00151 0.00174 -‐0.00066
GE 0.00226 0.00224 0.00056 0.00139 0.00412 0.00185 0.00237 0.00416
WMT 0.00150 0.00146 0.00010 0.00151 0.00185 0.00533 0.00270 0.00299
C 0.00394 0.00205 0.00111 0.00174 0.00237 0.00270 0.00697 0.00231
ORCL 0.00483 0.00341 0.00052 -‐0.00066 0.00416 0.00299 0.00231 0.02582
IBM TM XOM BRK-‐B GE WMT C ORCL
IBM 1.00 0.23 0.31 0.09 0.39 0.23 0.52 0.33
TM 0.23 1.00 0.13 0.18 0.45 0.26 0.31 0.27
XOM 0.31 0.13 1.00 0.17 0.17 0.03 0.25 0.06
BRK-‐B 0.09 0.18 0.17 1.00 0.37 0.35 0.35 -‐0.07
GE 0.39 0.45 0.17 0.37 1.00 0.39 0.44 0.40
WMT 0.23 0.26 0.03 0.35 0.39 1.00 0.44 0.26
C 0.52 0.31 0.25 0.35 0.44 0.44 1.00 0.17
ORCL 0.33 0.27 0.06 -‐0.07 0.40 0.26 0.17 1.00
ji
ij
ij
σσ
σ
ρ
⋅
=
22. 22
CorrelaBon
Between
Por[olios
A
&
B
wT
=
⎣
wIBM
wTM
wXOM
wBRK-‐B
wGE
wWMT
wC
wORCL
⎦
rT
=
⎣
rIBM
rTM
rXOM
rBRK-‐B
rGE
rWMT
rC
rORCL
⎦
Example:
Por[olio
A
has
weight
vector
a
and
is
half
TM
and
half
GE
aT
=
⎣
.0
.5
.0
.0
.5
.0
.0
.0
⎦
ij
M
1j
ji
M
1i
AB σbaσ ⋅⋅= ∑∑ ==
i
M
1i
iA
rar ⋅= ∑=
i
M
1i
iB
rbr ⋅= ∑=
23. 23
Diversifiable
Risk
2
σ
σ
ρ
ρσ2
⋅
is
the
avg
var
of
the
M
assets
is
the
avg
std
dev
of
the
M
assets
is
the
avg
corr
between
the
M
assets
is
the
avg
cov
between
the
M
assets
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25 30 35 40
M
M
1
M
1M−
∑∑∑
≠
===
⋅⋅+⋅=
M
ij
1j
ijji
M
1i
M
1i
2
i
2
i
2
P σwwσwσ
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⋅+⋅= ∑∑∑
≠
===
M
ij
1j
2
ij
M
1i
M
1i
2
i2
P
M
σ
M
1
M
σ
M
1
σ
ρσ
M
1)(M
σ
M
1
σ 222
P ⋅⋅
−
+⋅=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−⋅
⋅
−
+⋅= ∑∑∑
≠
===
M
ij
1j
2
ij
M
1i
M
1i
2
i2
P
1)(MM
σ
M
1M
M
σ
M
1
σ
24. 0%
5%
10%
15%
20%
25%
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99
Number
of
Assets,
M
Por[olio
Std
Dev
-‐0.50
-‐0.25
0.00
0.25
0.5
0.75
1.00
Avg
Std
Dev
=
20%
24
Diversifiable
Risk
10%1%σ
1%.25.2.2σ
ρσσ
P
2
P
22
P
=⇒
=⋅⋅⇒
⋅⇒
ρσσσ
M
ρσ
M
1)(M
σ
M
1
σ
222
P
222
P
⋅⋅+⋅⇒
∞→
⋅⋅
−
+⋅=
10
Diversifiable
risk
for
ρ=0.25
Non-‐diversifiable
risk
for
ρ=0.25
ρ
25. 25
OpBmal
Por[olios
of
M
Risky
Assets
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17%
Expected
Std
Dev
%/mo.
Expeced
Return
%/mo
IBM TM XOM BRK-‐B GE WMT C ORCL
26. 26
Find
the
Minimum
Risk
Por[olio
via
QuadraBc
OpBmizaBon
¨ Minimize
this
quadraBc
objecBve
funcBon
¨ Subject
to
these
linear
constraints
¨ Solve
Using
Excel
Solver
1w
0
1w
i
M
1i
i
≥≥
=∑=
ij
M
1j
ji
M
1i
2
V σwwσ ⋅⋅= ∑∑ ==
Symbol r σ
Equal 1.17% 5.04%
Min
Risk 1.09% 3.81%
SPX 0.38% 4.25%
IBM TM XOM BRK-‐B GE WMT C ORCL
1.4% 9.4% 43.1% 23.3% 8.6% 13.3% 0.0% 0.9%
i
M
1i
iV rwr ⋅= ∑=
27. 27
Find
the
Minimum
Risk
Por[olio
via
QuadraBc
OpBmizaBon
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17%
Expected
Std
Dev
%
Expected
Return
%.
IBM TM XOM BRK-‐B GE WMT
C ORCL Equal Min
Risk SPX
V
28. 28
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17%
Expected
Return
%.
Expected
Std
Dev
%
IBM TM XOM BRK-‐B GE WMT
C ORCL Equal Min
Risk SPX
¨ Determine
the
other
por[olios
¨ Minimize
¨ Subject
to
these
constraints
Find
the
other
opBmal
risky
por[olios
%36.2r
1.09%
*
P <<
ij
M
1j
ji
M
1i
2
P σwwσ ⋅⋅= ∑∑ ==
1w
0
1w
i
M
1i
i
≥≥
=∑=
i
M
1i
i
*
P rwr ⋅= ∑=
29. 29
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17%
Expected
Return
%.
Expected
Std
Dev
%
IBM TM XOM BRK-‐B GE WMT
C ORCL Equal Min
Risk SPX
¨ Determine
the
other
por[olios
¨ Minimize
¨ Subject
to
these
constraints
Por[olio
with
more
than
2
risky
assets
ij
M
1j
ji
M
1i
2
P σwwσ ⋅⋅= ∑∑ ==
1w
0
1w
i
M
1i
i
≥≥
=∑=
i
M
1i
i
*
P rwr ⋅= ∑=
30. 30
Find
the
other
opBmal
risky
por[olios
Port
Mean
Port
Std
Dev
IBM TM XOM BRK-‐B GE WMT C ORCL
1.09% 3.81% 1.4% 9.4% 43.1% 23.3% 8.6% 13.3% 0.0% 0.9%
1.24% 4.00% 0.0% 4.4% 48.1% 29.1% 0.0% 8.9% 0.0% 9.4%
1.44% 5.00% 0.0% 0.0% 49.7% 27.1% 0.0% 0.0% 0.0% 23.1%
1.55% 6.00% 0.0% 0.0% 48.7% 19.4% 0.0% 0.0% 0.0% 31.9%
1.64% 7.00% 0.0% 0.0% 16.8% 0.0% 0.0% 0.0% 83.2% 0.0%
1.73% 8.00% 0.0% 0.0% 46.7% 6.8% 0.0% 0.0% 0.0% 46.4%
1.82% 9.00% 0.0% 0.0% 45.8% 1.1% 0.0% 0.0% 0.0% 53.1%
0.25%
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17%
Expected
Return
%.
Expected
Std
Dev
%
IBM TM XOM BRK-‐B GE WMT
C ORCL Equal Min
Risk SPX
31. 31
One
Risk
Free
Asset
&
M
Risky
Assets
¨ The
tangency
por[olio
is
the
opBmal
risky
por[olio
(asset).
¨ The
opBmal
risky
asset
is
dependent
on
the
return
of
the
risk
free
asset,
but
is
independent
of
the
investor’s
risk
preference
¨ The
slope
of
the
CAL
line
is
the
called
the
“Sharpe
raBo”
and
has
the
steepest
slope
of
any
line
connecBng
the
risk
free
asset
and
a
tangency
por[olio
on
the
efficient
fronBer
¨ A
por[olio
containing
the
risk
free
asset
and
the
opBmal
risky
asset
is
opBmal
for
the
investor
¨ The
allocaBon
of
investor
funds
between
the
risk
free
and
risky
asset
depends
on
the
investor’s
astude
towards
risk.
¨ Extension
of
the
CAL
beyond
the
opBmal
risky
asset
requires
the
investor
to
short
or
borrow
the
risk
free
asset.
¤ In
this
case
the
risk
free
asset
weight
will
be
negaBve
and
the
weight
for
the
opBmal
risky
asset
will
be
greater
than
1.
¤ For
the
CAL
to
be
straight
beyond
the
opBmal
risky
asset,
the
borrowing
rate
must
equal
the
risk
free
rate.
32. 32
EssenBal
Concepts
¨ Asset
and
por[olio
returns
other
than
the
risk
free
asset
are
modeled
as
normally
distributed
random
variables
¨ This
topic
uses
historical
staBsBcs
as
expected
staBsBcs
for
simplicity;
however,
this
is
not
always
a
good
assumpBon.
¤ However,
historical
variances
and
covariances
are
quite
stable
unless
a
firm
undergoes
significant
changes
to
its
business
or
financial
model.
¨ Lack
of
correlaBon
between
asset
returns
reduces
por[olio
risk.
¨ In
the
case
of
more
than
two
risky
assets,
opBmal
por[olios
lie
along
a
curve
called
the
efficient
fronBer
(of
opBmal
risky
por[olios)
¨ When
M
is
large,
covariance
terms
dominate
the
calculaBon
of
por[olio
variance
and
thus
consBtute
non-‐diversifiable
risk
¨ Por[olio
risk
can
be
reduced
by
diversificaBon
i.e.,
by
including
non-‐correlated
assets
¨ The
efficient
fronBer
is
computed
by
sequenBal
applicaBon
of
quadraBc
programming