BKM Chapter 6 Flashcards
Speculation
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assumption of considerable investment risk to obtain commensurate gain
Risk premium (aka excess return)
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excess returns = expected return - risk-free rate
Gamble
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bet or wager on an uncertain outcome
aka a fair game
Key difference between a speculation and a gamble
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a gamble is undertaken for the enjoyment of risk
speculation is undertaken in spite of risk because of a perceived favorable risk-return tradeoff
Utility score
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U = E[r] - .5 * A * sigma^2
Risk aversion index
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quantification of an investor’s risk preferences
high A = more risk-averse
Variance of risk-free portfolios
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variance = 0
utility = expected return
Risk aversion index for risk-averse, risk-lovers, and risk-neutral investors
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risk-averse: A > 0
risk-neutral: A = 0
risk-lover: A < 0 (does not reject fair games/gambles)
Indifference curves
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plot expected returns against risk (x-axis)
all points on the curve represent portfolios with the same utility score (e.g. investor is indifferent between them)
Mean-variance criterion
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one portfolio dominates another if:
- it has the same/better expected return AND
- it has the same/lower risk/volatility
(with one inequality being strict)
Relationship between the risk aversion index (A) and indifference curves and explanation
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more risk-averse investors (higher A) will have a steeper indifference curve (b/c they require a greater increase in return for an increase in risk)
Capital allocation decision
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split of investments between risky and risk-free portfolios
Expected return of the complete portfolio (risky & risk-free portfolio) + alternative formula
E[r-sub C]
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E[r-sub C] = y * expected return for the risky portfolio + (1 - y) * risk-free rate
y = % invested in the risky portfolio
Alternative:
E[r-sub C] = risk-free rate + sharpe ratio * std. deviation of the complete portfolio
Standard deviation of the complete portfolio
sigma-C
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sigma-C = y * sigma-P
b/c the std. dev. of the risk-free portfolio is 0
sigma-P = std. dev. of the risky portfolio
Simplest way to reduce risk
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shift funds away from the risky asset to the risk-free asset
Capital allocation line (CAL)
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linear combination of all risk-return combinations available to investors with varying proportions y invested in the risky asset
plots expected return against risk (x-axis)
Sharpe ratio (S, aka reward-to-volatility ratio) & interpretation
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S = expected excess return / standard deviation
interpretation: average % excess return for every 1% increase in standard deviation
A borrowing or levered position, calculation of y, and relative risk
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a short position where y > 1
in this case, y = total funds available / initial investment budget and the amount invested in the risk-free portfolio is still = 1 - y
generally will have higher risk
CAL with borrowing
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line will be “kinked” at point p (y = 1) because investors usually cannot borrow at the risk-free rate
(to determine the slope, replace the risk-free rate with the borrowing rate)
Optimal risky position (y*) definition and graphical representation
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the proportion invested in the risky asset that maximizes utility (U)
graphically: point on the highest possible indifference curve that is tangential to the CAL
Factors that determine the optimal risky position (2)
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- risk aversion (A) - impacts the slope of the indifference curve
- Sharpe ratio (S) - impacts the slope of the CAL (opportunity set)
Reasonability of standard deviation as a measure of risk and alternative
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only appropriate with normality
use VaR or expected shortfall with non-normality
Capital market line (CML)
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a CAL using 1-month T-bills as the risk-free portfolio and a well-diversified portfolio of common stocks that represents the risky portfolio
also = the opportunity set represented by a passive strategy
Passive strategy
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a portfolio decision avoiding direct or indirect security analysis
Reasons to pursue a passive strategy (2)
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- passive strategies are less expensive than active strategies (both time & cost of information/market knowledge)
- free-rider benefit: with many active knowledgeable investors most assets in the market will be fairly priced (because investors buy and drive up the price of underpriced stocks and sell/drive down the price of overpriced stocks)
Certainty equivalent rate of return
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expected return with std. dev = 0 and same utility as the risky portfolio
(return received with certainty that the risk-free investment would need to provide to achieve the same utility as the risky portfolio)
When to borrow funds to invest in the risky asset
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borrow funds when expected return of risky portfolio > risk-free borrowing rate
