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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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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