L2. Risk

TOC

1. Ranking Prospects 1.1 State-by-state dominance 1.2 First Order Stochastic Dominance 1.3 Second Order Stochastic Dominance 1.4 Stochastic Dominance Summary 2. Utility Function 2.1 Axioms of Choice 2.2 Utility Function 3. Risk 3.1 Risk Measures 3.2 Risk Premium 4. Asset Allocation 4.1 Participate or not 4.2 Allocation and risk-aversion level 4.3 Allocation and wealth level Asset Allocation: Summary

1. Ranking Prospects

1.1 State-by-state dominance

Example

The comparison can be more convenient represented if we describe investments in term of their returns.

State-by-state dominance: asset pays as much in all states of nature and strictly more in at least one state. For the above example, Asset 3 > Asset 1, 2. But for Asset 1 and Asset 2, there is no ranking possible on the basis of the dominance criterions. Hidden requirement here is that investors are all non-satiation.

1.2 First Order Stochastic Dominance

Intuition for A FSD B: We are likely to have more wealth from investment A as comparing to investment B.

Decision rule: the cumulative distribution curve of B is above or equal A

Consider two assets A and B, security A FSD security B iff

Which means: in every ordered state [a, b] of investment A and B, we receive higher wealth in investment A.

An example is shown as below

Therefore, investment A FSD investment B, which is equivalent to “someone’s choice of A over B”.

1.3 Second Order Stochastic Dominance

Intuition for C SSD D: C and D have the same return, but D has extra risk over C.

Decision rule: The area below the cumulative distribution curve of C is smaller than D

Consider 2 assets C and D, security C SSD security D iff

All non-satiation and risk-averse agents will prefer the 2nd stochastic dominant asset.

An example is shown as below,

All non-satiation and risk-averse agents will prefer Invest. C in this example.

1.4 Stochastic Dominance Summary

When an investor is non-satiation, he will choose invesment according to state-by-state dominance and first order stochastic dominance. When an investor is non-satiation and risk-aversion, he will choose investment according to second order stochastic dominance.

Note that the above three criteria are only asset-level and do not include investors’ preferences.

Assume :

FSD: For any s.t. , A FSD B means

Proof

Assume there exist , where when , . We only need to prove that s.t. . Construct a s.t.

Thus

which is contradict with .

SSD: For any s.t. , , A SSD B means

2. Utility Function

How people make choices?

Economic theory: individual behavior is the results of a process of optimization under constraints.

The objective is determined by individual preference

The constraints is a function of the person’s income or wealth level and of market prices

2.1 Axioms of Choice

The notion of economic rationality can be summarized by the following assumptions:

Complete: individuals can specify their preferences for each ones

or or

Transitivity: if an individual prefers A to B and B to C, it has to prefer A to C

if and , then

Continuous

2.2 Utility Function

Existence of Utility Function

Theorem: There exists a continuous, time-invariant, real-valued utility function, such that for any two objects of choice a and b:

Utility Function and Preference

Preference is non-satiationutility function is non-decreasing

Preference is risk aversionutility function is concave

VNM Utility Function

Theorem: under some basic axioms, there exists a utility function defined on the uncertain space, which is called von Neumann-Morgensten (VNM) utility function

where is a utility function over monetary payments.

Note that VNM is preserved under affine transformation (linear transformation)

Utility function of monetary payments uniquely characterizes an investor. Investors are concerned only with final payoffs and the cumulative probabilities to achieving them, but not the process. Individuals choose the action to maximize their expected utilities.

Example

A lottery ticket costs $20. The probability of winning the $10,000 prize is 1%. A person has 100 cash with a monetary utility function

If he does nothing:

If he buys the lottery:

Therefore, he will not buy the lottery.

3. Risk

3.1 Risk Measures

Risk Aversion

Risk aversion refers to the desire of investors to smooth their consumptions.

For example, for an investor with wealth , there is a financial contract with following payoff

Assume and , the utility without the contract is

The utility with this contract

, the difference is due to risk aversion.

Utility function of risk aversion is concave as shown above. Concave means the marginal utility decreases and the twice differentiable utility function . Besides, the indifference curve is convex.

The degree of risk-aversion is related to the curvature of the utility function. Second derivative is not a sufficient measure of risk-aversion because it’s not invariant to an affine transformation.

The Arrow-Pratt measures of absolute and relative risk-aversion is defined as below.

Absolute Risk Aversion (ARA)

Definition:

Define as the probability at which the agent is indifferent between accepting or rejecting an investment.

By Taylor’s theorem:

thus

Rearrange it and we can get

is linked with the minimum probability of success above 1/2 necessary for investor to take on such a prospect.

Investor will demand more favorable odds if the investor has higher ARA (more risk averse) and more absolute wealth at risk.

Example: Constant ARA

Thus

Therefore, the odds requested are independent of the level of initial wealth .

Relative Risk Aversion (RRA)

Definition:

is defined as the probability at which the agent is indifferent between accepting or rejecting an investment.

Investor demand more favorable odds:

If investor has higher RRA (more risk averse)

More relative wealth at risk

Example: Constant RRA (CRRA)

Thus

Therefore, the odds requested are independent of the level of initial wealth, but depend on the , the fraction of wealth that is at risk.

Risk Neutral Investors

They are indifferent to risk and are concerned only with an asset’s expected payoff. Their utility if of a linear form:

ARA=RRA=0 for them. They do not demand better than even odds when considering risky investments.

3.2 Risk Premium

Jensen’s Inequality: for a concave function

If an uncertain payoff is available for sale, a risk-averse agent will only be willing to buy it at a price less than its expected payoff, that is

Certainty Equivalence (CE): the maximal certain sum of money a person is willing to pay to acquire an uncertain opportunity

The difference between the CE and the expected value of the prospect is a measure of the uncertain payoff’s Risk Premium

Assume an investment with , find its risk premium :

where

thus

Ignoring the terms of higher order

Investors will demand more risk premium if investors have higher ARA (more risk averse) or the investment has higher volatility (more risk).

Example

An investment with original price and payoff , find its certainty equivalence:

Assume the investor’s utility function is

Then the return for this investment is

The certainty equivalence can be expressed as

Plug in all these numbers

Therefore,

Assume , then

The “risk-free” rate is

The risk premium (demanded by this investor when considering this investment) is

4. Asset Allocation

Asset allocation is an investment strategy that aims to balance risk and return by apportioning a portfolio’s assets. When constructing asset portfolios, individuals’ goals, wealth level and risk tolerance are considered. The main asset classes include: Cash and equivalents, Fixed-income and Equities.

Consider an investor with wealth level , who is deciding what amount, , to invest in a risky portfolio with uncertain rate of return , to maximize his expected utility.

End of period wealth is

4.1 Participate or not

Under the risk aversion , F.O.C. w.r.t

Theorem: Assume and and let denote the solution, then

Proof

Define

F.O.C.

By risk aversion , then

therefore, is everywhere decreasing.

As is decreasing, if , will have to be increased from 0 to achieve

Since , thus

Similarly, we can derivate that

Example CRRA

Assume return with . Then

Since

thus

finally, we have

Assume utility function is log utility (), then

Consider three cases:

, let’s say , then . Thus

It means that, the investor will allocate 25% of its wealth in the risky asset.

, the numerator is then . Thus no wealth will be invested in the risky asset.

, let’s say , then . Thus

It means that, the investor will short risky asset and purchase risk-free asset.

4.2 Allocation and risk-aversion level

In above example, if we change to , then .

For investor , denote

: the measure of absolute or relative risk aversion

: the investor optimal allocation on risky asset

Theorem (Arrow, 1971)

For all wealth levels , if the coefficient of investor’s risk averse , then optimal risky allocation .

4.3 Allocation and wealth level

ARA and Risky Amount

Theorem (Arrow)

Let be the optimal choice:

DARA:

Investors are willing to risk larger amounts of wealth as they get wealthier

CARA:

The amount invested in risky asset is unaffected by the agent’s wealth

IARA :

Example: CARA

Utility function is , .

The choice problem is , the first order condition is

It means that the amount invested in risky assest is unaffected by the agent’s wealth.

Example: IARA

Utility function is , and . Calculating the first order condition, we can obtain that . It means that agents become less willing to accept greater bets as they become wealthier.

RRA and Elasticity

Let be the optimal choice, the elasticity is the measure of how the fraction investred in the risky asset changes as wealth changes

Theorem:

DRRA:

CRRA:

IRRA :

DRRA investors are willing to risk larger proportion of wealth, as they get wealthier.

Example: Risk-Neutral Agent

Utility function:

The solution is: