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Summary Investments and Risk Management
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Summary for the course Investments and Risk Management of the Master's degree in Financial Management at the University of Twente.
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Summary IRM – Financial Markets and Corporate StrategyTable of Contents
Summary IRM – Financial Markets and Corporate Strategy ................................................................... 1
Chapter 4 – Portfolio Tools.................................................................................................................. 3
4.1 Portfolio weights ....................................................................................................................... 3
4.2 Portfolio returns ........................................................................................................................ 4
4.3 Expected portfolio returns ........................................................................................................ 4
4.4 Variances and standard deviations ........................................................................................... 4
4.5 Covariances and correlations .................................................................................................... 5
4.6 Variances of portfolios and covariances between portfolios.................................................... 5
4.7 The mean-standard deviation diagram ..................................................................................... 7
Chapter 5 – Mean-variance analysis and the Capital Asset Pricing Model ......................................... 8
5.1 Application of mean-variance analysis and the CAPM in use today ......................................... 8
5.2 The essentials of mean-variance analyses ................................................................................ 8
5.3 The efficient frontier and two-fund separation ........................................................................ 9
5.4 The tangency portfolio and optimal investment ....................................................................... 9
5.6 How useful is mean-variance analysis for finding efficient portfolios? .................................. 10
5.7 The relation between risk and expected return...................................................................... 10
5.8 The capital asset pricing model ............................................................................................... 11
5.9 Estimating betas, risk-free returns, risk premiums and the market portfolio ........................ 12
5.10 Empirical tests of the CAPM .................................................................................................. 13
Chapter 6 – Factor models and the arbitrage pricing theory ............................................................ 15
6.1 The market model: the first factor model ............................................................................... 15
6.2 The principle of diversification ................................................................................................ 16
6.3 Multifactor models .................................................................................................................. 16
6.4 Estimating the factors ............................................................................................................. 17
6.5 Factor betas ............................................................................................................................. 17
6.6 Using factor models to compute covariances and variances .................................................. 18
6.7 Factor models and tracking portfolios .................................................................................... 18
6.9 Tracking and arbitrage............................................................................................................. 19
Chapter 7 – Pricing derivatives .......................................................................................................... 21
7.1 Examples of derivatives ........................................................................................................... 21
7.2 The basics of derivatives pricing .............................................................................................. 23
7.3 Binomial pricing models .......................................................................................................... 24
, 7.4 Multi-period binomial valuation.............................................................................................. 25
7.5 Valuation techniques in the financial services industry .......................................................... 25
7.6 Market frictions and lessons from the fate of amaranth advisors .......................................... 26
Chapter 22 – The practice of hedging ............................................................................................... 27
22.1 Measuring risk exposure ....................................................................................................... 27
22.2 Hedging short-term commitments with maturity-matched forward contracts ................... 28
22.6 Hedging with swaps............................................................................................................... 29
22.7 Hedging with options ............................................................................................................ 29
Chapter 8 – Options .......................................................................................................................... 32
8.1 A description of options and options markets ........................................................................ 32
8.2 Option expiration .................................................................................................................... 32
8.3 Put-call parity .......................................................................................................................... 32
8.7 Estimating volatility ................................................................................................................. 34
Chapter 21 – Risk management and corporate strategy .................................................................. 36
21.1 Risk management and the Modigliani-Miller theorem ......................................................... 36
21.2 Why do firms hedge? ............................................................................................................ 36
21.3 The motivation to hedge affects what is hedged .................................................................. 38
21.4 How should companies organize their hedging activities? ................................................... 38
21.5 Do risk management departments always hedge ................................................................. 38
21.6 How hedging affects the firm’s stakeholders ....................................................................... 39
21.7 The motivation to manage interest rate risk......................................................................... 39
21.8 Foreign exchange risk management ..................................................................................... 40
21.9 Which firms hedge? The empirical evidence ........................................................................ 42
Article: The theory and practice of corporate risk management ...................................................... 43
Article: CEO age, risk incentives, and hedging strategy .................................................................... 50
Article: Three approaches to risk management – and how and why Swedish companies use them54
Article: Is operational hedging a substitute for or a complement to financial hedging? ................. 57
Article: R&D investments and high-tech firms firms’ stock return volatility .................................... 61
Article: The cross-section of expected stock returns ........................................................................ 64
Article: Portfolio concentration and performance of institutional investors worldwide ................. 67
Article: The good news in short interest ........................................................................................... 72
,Chapter 4 – Portfolio Tools
The modern theory of how to invest was developed to help investors form a portfolio that achieves
the highest possible expected return2 for a given level of risk.
Portfolio = a combination of investments.
The theory assumes that investors are mean-variance optimizers.
Mean-variance optimizers = seekers of portfolios with the lowest possible return variance for
any given level of mean (or expected) return.
The variance of an investment return is the appropriate measure of risk.
𝑃1 + 𝐷1 + 𝑃0
𝑅𝑒𝑡𝑢𝑟𝑛 (𝑅) =
𝑃0
Mean-variance analysis = mathematically description about how the risk of individual securities
contributes to the risk and return of portfolios.
Most large corporations contain a number of different investment projects: thus a corporation can be
though of as a portfolio of real assets.
Diversification = the holding of many securities to lessen risk. When more equities are added to the
portfolio, the additional equities diversify the portfolio if they do not covary (that is, move together)
too much with other equities in the portfolio. A portfolio is diversified if it contains companies from a
variety of regions and industries.
4.1 Portfolio weights
The portfolio weight for stock j, denoted xj, is the fraction of a portfolio’s wealth held in stock j.
Money held in stock 𝑗
𝑥𝑗 =
Monetary value of the portfolio
Portfolio weights must sum to 1.
The two-stock portfolio
Short sales and portfolio weights
Sell short = investors can sell investments that they do not currently own. To sell short shares or bonds,
the investor must borrow the securities from someone who owns them (= taking short position in a
security). To close out the short position, the investor buys the investment back and returns it to the
original owners.
→ Example, borrowing from a bank.
Selling short an investment is equivalent to placing a negative portfolio weight on it.
A long position achieved by buying an investment has a positive portfolio weight.
To compute portfolio weights when some investments are sold short, sum the amount invested in each
asset of the portfolio, treating shorted (or borrowed) investments as negative numbers. Then divide
each investment by the sum.
Feasible portfolios
,Feasible portfolios = the set of portfolios that one can invest in.
The many-security portfolio
-
4.2 Portfolio returns
Two equivalent methods to compute portfolio returns:
- Ratio method
- Portfolio-weighted average method
For N securities, indexed from 1 to N, the portfolio return formula becomes:
(4.1) 𝑅̃𝑃 = 𝑥1 𝑟̃1 + 𝑥2 𝑟̃2 + ⋯ + 𝑥𝑁 𝑟̃𝑁 = ∑𝑁
𝑖=1 𝑥𝑖 𝑟̃𝑖
4.3 Expected portfolio returns
Portfolios of two securities
Expected returns have some useful properties:
1. The expected value of a constant times a return is the constant times the expected return: that
is,
(4.2) 𝐸(𝑥𝑟̃ ) = 𝑥𝐸(𝑟̃ )
2. The expected value of the sum or difference of two returns is the sum or difference between
the expected returns themselves: that is,
(4.3) 𝐸(𝑟̃1 + 𝑟̃2 ) = 𝐸(𝑟̃1 ) + 𝐸(𝑟̃2 ) 𝑎𝑛𝑑 𝐸(𝑟̃1 − 𝑟̃2 ) = 𝐸(𝑟̃1 ) − 𝐸(𝑟̃2 )
Combining these two equations implies the following.
3. The expected return of a portfolio is the portfolio-weighted average of the expected returns.
That is, for a portfolio of two securities:
(4.4a) 𝐸(𝑅̃𝑝 ) = 𝐸(𝑥1 𝑟̃1 + 𝑥2 𝑟̃2 ) = 𝑥1 𝐸(𝑟̃1 ) + 𝑥2 𝐸(𝑟̃2 )
Leveraging an investment = selling short an investment with a low expected return (that is, borrowing
at a low rate) and using the proceeds to increase a position in an investment with a higher expected
return results in a larger expected return than can be achieved by investing only in the investment with
the high expected return.
Portfolios of many securities
The expected portfolio return is the portfolio-weighted average of the expected returns of the
individual stocks in the portfolio:
(4.4b) 𝑅̃𝑝 = ∑𝑁
𝑖=1 𝑥𝑖 𝑟̃𝑖
4.4 Variances and standard deviations
Return variances
Risk aversion = the concern investors have for losses.
Mean-variance analysis defines the risk of a portfolio as the variance of its return. The variance of a
constant times a return is the square of that constant times the variances of the return: that is,
(4.5) 𝑣𝑎𝑟(𝑥𝑟̃ ) = 𝑥 2 𝑣𝑎𝑟(𝑟̃ )
,Estimating variances: statistical issues
-
Standard deviation
The standard deviation of a constant times a return is the constant times the standard deviation of the
return: that is,
(4.6) 𝜎(𝑥𝑟̃ ) = 𝑥𝜎(𝑟̃ )
4.5 Covariances and correlations
A positive covariance or correlation means that the two returns tend to move in the same direction. A
negative covariance or correlation means that the returns tend to move in opposite directions.
Covariance
Covariance = a measure of relatedness that depends on the unit of measurement.
The covariance between two returns is the expected product of their demeaned outcomes.
Covariances and joint distributions
Joint distributions of the two returns = the set of probabilities attached to each pair.
To compute a covariance with the forward-looking approach, determine the probability-weighted
average of the product of the two demeaned returns associated with each of the paired outcomes
using the joint distribution.
Estimating covariances with historical data
Typically, covariances are estimated by looking at the average demeaned product of historical returns.
Variance is a special case of the covariance
The variance measures the covariance of a return with itself: that is,
𝑐𝑜𝑣(𝑟̃ , 𝑟̃ ) = 𝑣𝑎𝑟(𝑟̃ )
Translating covariances into correlations
The correlation between two returns, denoted ρ, is the covariance between the two returns divided
by the product of their standard deviations: that is,
𝑐𝑜𝑣(𝑟̃1 ,𝑟̃2 )
(4.7) 𝜌(𝑟̃1 , 𝑟̃2 ) = 𝜎1 𝜎2
A coefficient of +1 is defined as perfect positive correlation and a coefficient of -1 is defined as perfect
negative correlation.
Translating correlations into covariances
(4.8) 𝜎𝑖𝑗 = 𝜌𝑖𝑗 𝜎𝑖 𝜎𝑗
4.6 Variances of portfolios and covariances between portfolios
Variances for two-security portfolios
If the returns of two securities covary positively, a portfolio that has positive weights on both securities
will have a higher return variance than if they covary negatively.
,Computing portfolio variances with given inputs
(4.9a) 𝑣𝑎𝑟(𝑥1 𝑟̃1 + 𝑥2 𝑟̃2 ) = 𝑥12 𝜎12 + 𝑥22 𝜎22 + 2𝑥1 𝑥2 𝜎12
Using historical data to derive the inputs: the backward-looking approach
-
Correlations, diversification and portfolio variances
(4.9b) 𝑣𝑎𝑟(𝑥1 𝑟̃1 + 𝑥2 𝑟̃2 ) = 𝑥12 𝜎12 + 𝑥22 𝜎22 + 2𝑥1 𝑥2 𝜌12 𝜎1 𝜎2
Given positive portfolio weights on two assets, the lower the correlation, the lower the variance of the
portfolio.
Portfolio variances when one of the two investments in the portfolio is riskless
If investment 1 is riskless, σ1 and ρ12 are both zero. In this case, the portfolio variance is 𝑥22 𝜎22 . The
standard deviation is either x2σ2, if x2 is positive or - x2σ2 if x2 is negative.
Portfolio variances when the two investments in the portfolio have perfectly correlated returns
Equation (4.9b) also implies than when two securities are perfectly positively or negatively correlated,
it is possible to create a riskless portfolio form them – that is, one with a variance and standard
deviation of zero.
The standard deviation of either (1) a portfolio of two investments where one of the investments is
riskless, or (2) a portfolio of two investments that are perfectly positively correlated, is the absolute
value of the portfolio-weighted average of the standard deviations of the two investments.
Portfolios of many assets
To compute the variance of a portfolio of many assets, one needs to know:
− The variances of the returns of each asset in the portfolio
− The covariances between the returns of each pair of assets in the portfolio
− The portfolio weights
A portfolio variance formula based on covariances
The formula for the variance of a portfolio return is given by:
(4.9c) 𝜎𝑝2 = ∑𝑁 𝑁
𝑖=1 ∑𝑗=1 𝑥𝑖 𝑥𝑗 𝜎𝑖𝑗
where σij is the covariance between the returns of asset i and j.
A portfolio variance formula based on correlations
-
Covariances between portfolio returns and asset returns
For any asset, indexed by k, the covariance of the return of a portfolio with the return of asset k is the
portfolio-weighted average of the covariances of the returns of the investments in the portfolio with
asset k’s return: that is,
𝑁
𝜎𝑝𝑘 = ∑ 𝑥𝑖 𝜎𝑖𝑘
𝑖=1
,4.7 The mean-standard deviation diagram
Mean-standard deviation diagram = a graph that plots the means (Y-axis) and the standard deviations
(X-axis) of all feasible portfolios in order to develop an understanding of the feasible means and
standard deviations that portfolios generate.
Combining a risk-free asset with a risky asset in the mean-standard deviation diagram
When both portfolio weights are positive
Whenever the portfolio mean and standard deviations are portfolio-weighted averages of the means
and standard deviations of two investments, the portfolio mean-standard deviation outcomes are
graphed as a straight line connecting the two investments in the mean-standard deviation diagram.
(4.10) 𝑅̅𝑝 = 𝑥1 𝑟𝑓 + 𝑥2 𝑟̅2 and 𝜎2 = 𝑥2 𝜎2
When the risky investment has a negative portfolio weight
When investors employ risk-free borrowing (that is, leverage) to increase their holdings in a risky
investment, the risk of the portfolio increases.
Exhibit 4.3 (106, 166)
Portfolios of two perfectly positively correlated or perfectly negatively correlated assets
Perfect positive correlation
Exhibit 4.4 (p108, 168)
Perfect negative correlation
-
The feasible means and standard deviations from portfolios of other pairs of assets
For sufficiently large correlation, the minimum variance portfolio, the portfolio of risky investments
with the lowest variance, requires a long position in one investment and a short position in the other.
, Chapter 5 – Mean-variance analysis and the Capital Asset Pricing Model
The Capital Asset Pricing Model (CAPM) is a model of the relation of risk to expected return.
5.1 Application of mean-variance analysis and the CAPM in use today
Investment applications of mean-variance analysis and the CAPM
-
Corporate applications of mean-variance analysis and the CAPM
One of the lessons of the CAPM is that, while diversifying investments can reduce the variance of a
firm’s share price, it does not reduce the firms cost of capital.
Cost of capital = weighted average of the expected rates of return required by the financial
markets for a firm’s debt and equity financing.
As a result, a corporate diversification strategy can create value for a corporation only if the
diversification increases the expected returns of the real asset investments of the corporation.
5.2 The essentials of mean-variance analyses
The feasible set
Feasible set (of the mean-standard deviation diagram) = the set of mean and standard deviation
outcomes, plotted with mean return on the vertical axis and standard deviation on the horizontal axis,
that are achieved from all feasible portfolios.
Exhibit 5.1
Investors achieve higher means and lower variances by ‘moving to the north-west’, or up and to the
left, while staying within the feasible set.
The assumptions of mean-variance analysis
Two assumptions of mean-variance analysis:
1. In making investment decisions today, investors care only about the means and variances of
the returns of their portfolios over a particular period. Their preference is for higher means
and lower variances.
2. Financial markets are frictionless.
The diagram can help to rule out dominated portfolios, which are plotted as points in the diagram that
lie below and to the right (that is, to the south-east) of some other feasible portfolios. These portfolios
are dominated in the sense that other feasible portfolios have higher mean returns and lower return
variances, and thus are better.
The assumption that investors care only about the means and variances of returns
Investors prefer portfolios that generate the greatest amount of wealth with the lowest risk.
The assumption that financial markets are frictionless
In frictionless markets, all investments are tradable at any price and in any quantity, both positive or
negative (that is, there are no short sales restrictions). In addition, there are no transaction costs,
regulations or tax consequences of asset purchases or sales.
How restrictive are the assumptions of mean-variance analysis?
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