3 decision i riskq - james r. garven
TRANSCRIPT
Finance 4335 (Spring 2020) Course Overview
I. Part 1: January 14 - February 11
A. The overarching Finance 4335 concept in the first part of this course centers
around the notion that people vary in terms of their preferences for bearing
risk. Although we focused most of our attention upon modeling risk averse
behavior, we also considered examples of risk neutrality (where you only
care about expected wealth and are indi↵erent about riskiness of wealth)
and risk loving (where you actually prefer to bear risk and are willing to
pay money for the opportunity to do so).
B. Related to point A: irrespective of whether one is risk averse, risk neutral,
or risk loving, the foundation for decision-making under conditions of risk
and uncertainty is expected utility. Given a choice amongst various risky
alternatives, one selects the choice that has the highest utility ranking.
1. If you are risk averse, then E (W ) > WCE and the di↵erence between
E (W ) andWCE is equal to the risk premium �. Some practical applica-
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3parts i Parti Decisionmaking under
riskq.eu uty
tions – if you are risk averse, then you are okay with buying “expensive”
insurance at a price that exceeds the expected value of payment provided
by the insurer, since (other things equal) you would prefer to transfer
risk to someone else if it is not too expensive to do so. On the other
hand, you are not willing to pay more than the certainty equivalent of
wealth for a bet on a sporting event or a game of chance such as rolling
dice or tossing a coin.
2. If you are risk neutral, then E (W ) = WCE and � = 0; risk is incon-
sequential and all you care about is maximizing the expected value of
wealth.
3. If you are risk loving, then E (W ) < WCE and � < 0; i.e., you are quite
happy to pay for the opportunity to (on average) lose money.
C. We also discussed a couple of di↵erent methods for calculating �.
1. The “exact” method involves calculating expected utility (E(U(W ))),
equating expected utility with certainty-equivalent of wealth (E(U(W )) =
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U(WCE)), and solving for WCE directly; e.g., if E(U(W )) = U(WCE)
= 10 and U(WCE) =pWCE, then WCE = 100; if E (W ) = $110, then
the risk premium � = E(W )�WCE= $10.
2. The “approximate” method involves solving directly for � by evaluating
the Arrow-Pratt coe�cient at the expected value of wealth and multi-
plying it by half of the variance of wealth; i.e., � ⇠= .5�2WRA(E(W )).
This method provides important intuitive insights into the determinants
of risk premiums. Specifically, we find that risk premiums depend upon
two factors: 1) the magnitude of the risk itself (as indicated by vari-
ance), and 2) the degree to which the decision-maker is risk averse (as
indicated by the Arrow-Pratt coe�cient).
D. We talked about “special cases” of expected utility – specifically, the mean-
variance and stochastic dominance models. If we impose various restrictive
assumptions upon expected utility, then these models emerge as “special
cases”.
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1. So long as various restrictive assumptions apply under these models, we
can be confident that if risk X “dominates” risk Y, then the expected
utility for X is greater than the expected utility for Y ; a result which
obtains for all arbitrarily risk averse decision-makers.
2. Of these two models, the mean-variance model is considerably more
restrictive than the stochastic dominance model. Indeed, the mean-
variance model is not even an appropriate method for risk evaluation
under a variety of circumstances. For example, if one risk has a higher
mean and variance than another risk, then we need further information
about the decision-maker’s utility function in order to determine which
risk is preferred; just knowing the mean and variance is not su�cient in
such a case.
3. Furthermore, the mean-variance model implicitly assumes that risks are
symmetrically distributed and have “thin” tails; examples of such dis-
tributions include the binomial distribution in the discrete setting and
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the normal distribution in the continuous setting. However, if the un-
derlying distribution is 1) positively or negatively skewed, and/or 2)
particularly thin or fat tailed, then it is not appropriate to rank order
risks based upon the mean-variance framework, because variance only
partially captures risk. To illustrate this, we considered a numerical
example (see the “Broader Risk Definitions” class problem, and pp. 5-7
of the Decision Making Under Risk and Uncertainty (Part 4) lecture
note) in which a positively skewed risk with a lower mean and a higher
variance has higher expected utility than a symmetrically distributed
risk with higher mean and lower variance.
4. Stochastic dominance appears to be more “robust” than the mean-
variance model, because (unlike the mean-variance model) the stochastic
dominance model can accommodate broader risk characteristics such as
skewness and kurtosis.
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II. Part 2: February 20 - March 31
A. Insurance economics focuses upon some examples of how risk aversion
influences incentives for risk transfer to a counterparty (in this case, an in-
surer). The three major concepts include the so-called Bernoulli principle,
Mossin’s theorem, and Arrow’s Theorem.
1. Bernoulli principle – if insurance is actuarially fair, risk averters fully
insure.
2. Mossin’s theorem – if insurance is actuarially unfair, risk averters par-
tially insure.
3. Arrow’s theorem – other things equal, the optimal partial insurance
contract is the deductible contract.
B. Asymmetric information, moral hazard, and adverse selec-
tion
1. Asymmetric Information occurs when one party to a transaction has an
informational advantage over the other party.
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theory
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2. The two types of problems that arise when there is asymmetric infor-
mation include (1) moral hazard, which is a problem of hidden action
that occurs after a counter-party relationship has been formed (e.g.,
between a firm and its manager), and (2) adverse selection, which is
a problem of hidden information that occurs prior to the formation
of a counter-party relationship (e.g., between a prospective buyer and
seller).
C. Portfolio Theory
1. Mean-variance e�cient set of portfolios lie along the northwest perimeter
of the feasible set of portfolios, where �p is the X axis variable and E(rp)
is the Y axis variable.
2. Optimal exposure to risk is positively related to the Sharpe Ratio and
risk tolerance; inversely related to market volatility.
D. Capital Market Theory
1. From the portfolio theory, it follows that expected utility maximizing
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investors will (depending on their level of risk tolerance) either lend or
borrow at the riskless rate of interest rf against the (tangent) market
portfolio M.
2. Thus, the mean-variance e�cient set of portfolios in a world with riskless
lending and borrowing is the locus of points that lie along the Capital
Market Line, the equation for which is E(rp) = rf +E(rM)� rf
�M�p.
3. The Security Market Line, also known as the Capital Asset Pricing
Model or CAPM, follows as a logical consequence of the Capital Market
Line. It’s equation is E(ri) = rf +�i[E(rM)� rf ], where �i = �iM/�2M
indicates (on a relative scale) how risky asset i is compared with M.
4. Thus, the expected return on risky asset i is equal to the expected return
on a riskless asset rf , plus a risk premium that is proportional to the
excess expected net return of the market over and above the expected
return on a riskless asset; i.e., E(rm)� rf ; the proportionality factor is
�i. The CAPM implies that only “systematic” (i.e., covariance) risk is
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priced. “Unsystematic” (idiosyncratic, or unique) risk is inconsequential
since investors diversity unsystematic risk away by holding combinations
of the riskless asset and the market portfolio.
5. The notion that only systematic risk matters is belied by the fact that
corporations devote enormous amounts of resources toward managing
idiosyncratic risks; e.g., commercial property-liability insurance premi-
ums paid by U.S. companies in 2017 came to $253.2 billion. Thus, it
appears that management of unique firm-specific risks is important after
all. The question is why, which is a question that we will take up in
Part 3 of Finance 4335.
III. Part 3: April 7 - April 28
A. Risk Management using Financial Derivatives
1. Derivatives such as futures/forwards and options are widely used for the
purpose of managing financial risks.
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2. Limited liability creates option-like payo↵s for corporate stakeholders,
which implies that firm-specific risks a↵ect corporate value.
B. Pricing of Financial Derivatives
1. Pricing of Futures/Forwards - we showed that the “arbitrage-free” price
of a forward contract that expires T periods from today is K = SerT .
If this equality does not hold, e.g., if K > SerT , then one can earn
positive profits with zero net investment and zero risk by selling forward
and buying the replicating portfolio. Similarly, if K < SerT , then one
can earn positive profits with zero net investment and zero risk by buying
forward and selling the replicating portfolio.
2. Similar arguments apply to the pricing of options; specifically, the repli-
cating portfolio for a call option is a margined investment in the un-
derlying, whereas the replicating portfolio for a put option represents a
combination of a short position in the underlying along with the pur-
chase of a bond.
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3. Approaches to pricing options in both the discrete-time and continuous-
time cases include delta hedging, replicating portfolio, and risk neutral
valuation approaches.
4. The arbitrage-free call option price according to the discrete-time “bino-
mial” model (also known as the Cox-Ross-Rubinstein, or CRR model) is
C = SB1�Ke�rTB2, and the arbitrage-free call option price according
to the continuous-time model (also known as the Black-Scholes-Merton,
or BSM model) is C = SN (d1)�Ke�rTN (d2).
5. Prices for otherwise identical (same underlying asset, same exercise
price, same time to expiration) put options in both settings (discrete-
time and continuous-time) are obtained by applying the put-call parity
equation (also shown in the Options Formula Sheet). Consequently,
P = Ke�rT (1 � B2) � S(1 � B1) under the CRR model and P =
Ke�rTN(� d2)� SN(�d1) under the BSM model.
6. As the number of time-steps (n) becomes arbitrarily large and as the
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length of each time-step (�t) becomes arbitrarily for a given time to
expiration (T , where T = n�t), the CRR call and put model prices
converge to the BSM call and put model prices (thanks in no small part
to the Central Limit Theorem).
C. Corporate risk management
1. Tax asymmetries (in the form of progressive marginal tax rates and
incomplete tax-loss o↵sets) motivate firms to employ various corporate
risk management strategies which increase after-tax rates of return to
shareholders by lowering the expected value of taxes.
2. Corporate management adds value by facilitating optimal investment;
e.g., we showed how coordinating corporate financing and risk manage-
ment decisions mitigate the so-called underinvestment problem (where,
in the absence of risk management, it may sometimes be “rational” to
reject a positive net present value project). This agency problem de-
rives from an incentive conflict between corporate owners and creditors
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caused by corporate limited liability.
3. The design of the managerial compensation contract is an important
corporate risk management determinant; e.g., see Moral Hazard Class
Problem and Moral Hazard Class Problem Solutions.
4. Risk management adds value by reducing (the expected value of) finan-
cial distress costs.
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