Financial Economics

Index

• Introduction
  • The Fama-Shiller controversy
  • Introduction to finance and investment planning
• Options and financial strategies
• Option valuation
• Pricing by arbitrage
• An introduction to the economic analysis of asset markets
• Choice under uncertainty
• Demand for risk
• Risk sharing and insurance
• Mean variance analysis
• CAPM
• Risk sharing and asset prices in a market equilibrium (CCAPM)

Suggested readings:

• Investments, Z. Bodie, A. Kane and A. Marcus, Fifth Edition, Parts IV and VI, for lectures 1 and 2 (later editions are available).
• Finance and the Economics of Uncertainty by G. Demange and G. Laroque, for the rest of the course.

0.1 Introduction - The Fama-Shiller controversy

In 2013, Eugene Fama, Bob Shiller and Lars Hansen jointly received the Nobel Prize in economics for their economic analysis of asset prices but, ironically, Fama and Shiller represent an interesting and controversial case for the subject.

Eugene Fama is most often considered as the father of the Efficient Market Hypothesis (EMH) which supports the Capital Asset Pricing Model (CAPM), which states that a stock's returns over time should be commensurate with its riskiness in relation to the overall market.

In formulas:

Where $R_f$ is the risk-free rate, ${\beta }_i$ (Beta) the sensitivity of the asset to the market, and $E(R_m)-R_f$ the risk-premium (the extra return investors demand for picking risky assets over the risk-free).

Moreover, Fama argued that any test of EMH is actually testing two things at once:

• That the market is efficient.
• That the model is a correct way to measure risk.

Bob Shiller, on the other hand, published in 1981 the paper Do stock prices move too much to be justified by subsequent changes in dividends? that, answering Yes! to the question, claims that the EMH was one of the remarkable errors in the history of economics.

Regarding Fama's theories, Shiller's opinion was something like...

• I think that maybe he has a cognitive dissonance
• It's like being a Catholic priest and then discovering that God doesn't exist [...], you can't deal with that, you've got to somehow rationalise it.

In particular, Shiller’s empirical result is that stock prices move way more than the dividends.
If a stock price is truly the "discounted value of all future dividends", it should be stable because dividends don't change that much, but Shiller showed the opposite using the Cyclically Adjusted Price-to-Earnings (CAPE) ratio.

In the following years, empirical data gave credit to Shiller's critic.

But what does the CAPM actually say?

• how the market should price financial assets in function of the risk
• it shows that a complete market (perfect information and full pricing) will lead to a market equilibrium with optimal risk allocation

However, the CAPM relies on very heavy and simplifying assumptions, some are technical and could be relaxed, others are foundamental and not often realistic:

• the complete market assumption:
  • it would require infinitely many assets
  • it would require complex products (derivatives)
  • complexity implies expertise and potential moral hazard
• other assumptions are unrealistic:
  • investors can lend and borrow unlimited money under risk free rates
  • all assets are divisible and liquid
  • all agents have identical beliefs

So, was Fama simply wrong?

• Not really, he didn't give up and he refined the model, abandoning CAPM and replacing it with a multi-factor risk model
• Even Shiller still endorses a loose version of it.
• For Shiller, asset prices can overshoot in the short term (due to emotion and irrationality), but they show reversion to the mean in the long period.
• Additionally, according to Shiller markets are still the best tools we have for aggregating information.

To sum up, the Fama-Shiller case is typical from a social science and economics: two theories can contradict each other and still retain intellectual value.
"All Models are wrong, but some are useful" applies perfectly here: Fama gave an excellent structure to think the financial markets and, even if his work is imperfect, it's yet a solid starting point.

0.2 Introduction to finance and investment planning

A financial system is a set of institutions and markets that have the primary purpose of allowing the desynchronization of income and consumption.

A financial system allows to match different financial needs of different agents in two dimensions:

• Time (borrow and save):
  • wish of continuous consumption vs discrete income stream
  • smooth consumption of time/the life-cycle
• Risk (diversify, insure, hedge):
  • smooth consumption across state of nature and possible market scenarios.

Asset: something valuable with well-defined property rights (a contract, a good).

Real Asset: something that has an intrinsic value due to its substance and properties.

Financial Asset: something that has NO intrinsic material value but which can be traded.

(Note that the distinction between financial and non-financial asset is not binary and well defined: for example, "is gold a financial or physical/real asset?")

Financial assets can be categorized as follows:

• Riskless assets: assets whose future values can be known with certainty (a government bond that yields 100 CHF in 5 years, assuming no default)
• Risky assets: assets whose future value may depend on some events (insurance, stocks, options)
• Derivatives are risky assets whose future values depend on the price of other assets.

A bond is often considered a riskless asset (depending on the issuer).

A zero-coupon (ZC) bond with maturity $t$ yields no payment before period $t$ and pays $1$ in period $t$.
They can vary only in terms of maturity.

Denote by $p_{0,t}$ the price of a ZC bond with maturity $t$ at time $0$ (usually the first subscript is current time, the second the maturity). In absence of arbitrage, we have:

where $r_t$ is the discrete per period forward interest rate between $t-1$ and $t$. By iteration we have:

In relation to this, we can distinguish:

• Discounting: the process of determining the prevent value of a future payment $x_t$ at time $t>0$:

• Compounding: the process of determining the current value at time $t$ (in the future) of a monetary amount $x_0$ invested at $t=0$

The discrete spot interest rate ${\bar{r}}_t$, for a zero coupon bond with maturity $t$ such that:

Or, in other words, the spot rate is the geometric average of the per period forward rate of interest:

We now consider a bond with price $p_0$ in $t=0$ that yields a series of positive coupon payments $a_t$ until the maturity date, so for $t=1,\dot{T}$, plus the final payment of par value in $t=T$.

Its yield to maturity(YTM) is the unique rate $y$ for which the present value of these payments is equal to $p_0$:

The same analysis can be formulated in continuous time, in this case the instantaneous forward interest rate is:

or, equivalently:

By integration, we get:

This interest rate $r(\tau )$ is called "instantaneous" since it represents the evolution of the bond's price process for an infinitesimally small period of time $dt$.

Moreover, in continuous time we can define:

The continuously compounded forward interest rate between $t_1,t_2$, denoted by ${\bar{r}}_{t_1}(t_2)$ is defined as:

The continuously compounded spot interest rate for a bond with maturity $T$ is:

which is the arithmetic average of the instantaneous forward interest rates between $0,t$.

The function $t\to {\bar{r}}_0(t)$ provides the yield curve.

We now consider risky assets. To evalute an uncertain stream of future payments, we still used an additive process:

where the second formulation is needed if we want risk to be taken into account, and we write $r_i=r_t^r+{\pi }_t$, where $r_t^f$ is the risk-free rate and ${\pi }_t$ the risk premium.

1. Options and financial strategies

Derivatives are assets whose values mechanically depend on the values of other financial assets (the underlying).
For example:

• Forward contracts: OBLIGATION to purchase (long position) or sell (short position) the underlying at a specified future price at a specified delivery date.
• Options contracts: RIGHT to purchase or sell a specified amount of the underlying at a specified exercise price at or before a specified expiration date.

Options offer an advantage: the transaction does not have to occur if it is not profitable for the owner of the option. This advantage comes at a price:

• Forwards are entered at NO cost
• Options are purchased or sold at positive price that represents the cost of the right to buy/sell.
• Selling (or writing) an option implies an obligation (if the option is exercised, the seller must sell or buy the stock and incurs a loss) $\to$ the seller receives a compensation.

Derivatives can help shaping the risk exposure:

• Hedging means insuring against market price volatility
• Speculation means expoiting market price volatility

• $X$ agreed delivery price
• $S_t$ spot market price of the underlying at maturity $T$

Profits at maturity $T$ are:

• Payoff long position $=S_t-X$
• Payoff short position $=X-S_t$

For this reason, forwards are great for speculation. At a given time $t$, an underlying has a price $S_t$, with no clear evolution for the future. If a speculator expects:

• $S_T>S_t⟹$ long forward with $X<S_T$
• $S_T<S_t⟹$ short forward with $X>S_T$

On the other hand, forwards are used in hedging to avoid risks.
Suppose, for example, that Giacomo's GmbH needs to sell an activity (example: some running shoes) at maturity $T$, with value $S_T$ at given time $T$.
Giacomo faces then a risk management problem: how can he insure against price fluctuations of the running shoes?

Solution: use the forward contract to construct a hedged portfolio that fixes the profits from the future sale at a desired level ${\pi }^{\ast }$:

To fix the profits at ${\pi }^{\ast }=X$, Giacomo can open a short position on the forward contract with delivery price $X$.

(The opposite - long forward contract - applies if you need to buy and you want to avoid risk).

An option is a right to purchase/sell a certain amount of the underlying at/before future expiration $T$ at exercise price (strike) $X$.
Options differ by:

• Position
  • Long: option holder (right to exercise)
  • Short: option writer (obligation to facilitate the option's exercise)
• Type of right:
  • Put: long position has the right to sell an asset for $X$
  • Call: long position has the right to purchase an asset for $X$
• Possibility of early exercise:
  • American option: exercise at or before expiration
  • European option: exercise only at expiration.

Consider now:

• $X$ exercise price of an option
• $S_T$ market price of the underlying at expiration $T$
• Premium: purchase price of an option (market value):
  • $C_0$: premium for a call at $t=0$
  • $P_0$: premium for a put at $t=0$.

Payoff/value at expiration:

Profits are then given by:

• Long position: $\text{payoff}-C{V}_T(premium)$
• Short position:$\text{payoff}+C{V}_T(premium)$

If the agent has an expectation about the price development of the underlying:

• Bullish strategies: generate a profit when the underlying's price increases
• Bearish strategies: generate a profit when the underlying's price decreases
• Non-directional strategies: generate a profit depending on the underlying's actual volatility.

But... what are the reasons for trading in options rather than trading in the underlying directly?

• Leverage effect: option values respond more than proportionately to changes in the underlying's value.
• Tailoring risk exposure of investment: options can be less risky due to limitation of the downside risk.
• Portfolio insurance: it can be insured by long on a put for that stock.

Consider an agent that:

• invest in a stock with price $S_0$
• wants to insure against potential declines in the stock price

$⟹$long put on the stock (exercise price $X$ in $T$ with premium $P_0$)

Consider an agent that wants to take advantage of higher future price of the underlying.
He could buy the stock, buy a call or short a put, or as an alternative, construct a portfolio that allows to reduce the risk at the cost of _giving up part of the profits:

This is called Bull spread strategy:

• 1 long call with strike $X_1$ and premium $C_{0,1}$
• 1 short call with strike $X_2>X_1$ and premium $C_{0,2}$

The opposite also applies, i.e. consider an agent that wants to take advantage of lower future price of the underlying.

• 1 short call with strike $X_1$ and premium $C_{0,1}$
• 1 long call with strike $X_2>X_1$ and premium $C_{0,2}$

Consider an agent that expect large move in the stock's price but he's uncertain about the direction.

Straddle Strategy:

• 1 long call with strike $X$ and premium $C_0$
• 1 long put with strike $X$ and premium $C_0$

Moreover, if he thinks:

• a stronger bullish move is expected: $⟹$ buy 2 calls and 1 put (long strap)
• a stronger bearish move is expected: $⟹$ buy 2 puts and 1 call (long strip)

2. Option valuation

In this chapter we focus on European options, please refer to the above sections for formal definition.

We also use the following notation:

• $S_0$: price of the underlying asset at toime $t=0$
• $C_0,P_0$: prices (premia) of the call and the put option at time $t=0$.
• $r_f$: rate of the risk-free asset.

An Arbitrage Opportunity is defined as a financial strategy that yields a (sure) cash flow $A_0\ge 0$ at time $0$ and a cash-flow $A_T\ge 0$ at time $T$ and there is at least one stat of the word where one inequality is strict.

• $A_0>0,A_T\ge 0$ almost surely is an arbitrage
• $A_0\ge 0,A_T\ge 0$ almost surely with $\text{Prob}[A_T>0]>0$ is an arbitrage

In this chapter, we assume that the market is arbitrage-free (the market clears out arbitrage opportunities).

• Upper Bound: $C_0\le S_0$

Proof:
Consider two portfolios:

• Portfolio $A$: $1$ call + cash $PV(X)

  • Price in $t=0$: $ [C_0 + PV(X)]
  • Value in $T$: $X + \max [S_t. X, 0] = \max [S_t, X]

• Portfolio $B$: $1$ stock $S_0$

  • Price in $t=0:S_0$
  • Value in $T$: $S_T$

Portfolio $A$ is never less valuable than portfolio in $T$, so $A$ cannot be less expensive in $t=0$, otherwise we have an arbitrage opportunity, therefore we conclude:

and $C_0\ge 0$ since the call is an option.

• Lower Bound: $C_0\ge \max [S_0-PV(X),0]$ where $PV$ is the present value operator.

• Upper bound: $P_0\le PV(X)$
• Lower Bound: $P_0\ge \max [PV(X)-S_0,0]$

Proof:
Consider two portfolios:

• Portfoio $C$: buy $1$ put + $1$ stock
  • Price in $t=0$: $C_0+S_0$
  • Value in $T$: $\max (X-S_T,0)+S_T=\max [S_t,X$]
• Portfolio $D$: cash $PV(X)$
  • Price in $t=0$: $PV(X)$
  • Value in $T$: $X$

$C$ is never less valuable than $D$, so $C$ cannot be less expensive in $t=0$, otherwise we have an arbitrage opportunity, therefore:

and we add $P_0\ge 0$ since the put is an option.

To understand this, consider the following:

• Portfolio $A$: $1$ stock and $1$ put
• Portfolio $B$: $\frac{X}{(1+r_f{)}^T}$ bonds and $1$ call

the values of the two portfolio are equal in $t=T$, therefore they have to be equal also in $t=0$, leading to the Put-Call parity formula.

Determinants of the value (price/premium) of a call option $C_0$ (European, no dividend):

• stock price $S_0$ (+), exercise price $K$ (-)
• volatility of underlying stock (+) (intuition: volatility increases the probability that the stock value at expiration is greater than $K$, increasing potential gains).
• time to expiration $T-t$ (+) (intuition: the more time to expiration, the higher the range od stock price increases, similar to volatility), also, the present value of exercise price falls)
• interest rate (+) (intuition: present value of exercise price falls).

Consider a market with:

• a stock that currently trades at $S_0$, and which value in $t=1$ can be either $S_1^u,S_1^d$, with $S_1^u>S_1^d$
• call option on the stock with exercise price $X\in (S_1^d,S_1^u)$ and expiration date $T=1$
• a risk free asset with yearly interest rate $r_f$

Then it is possible to replicate the payoffs of the call using a combination of the risk-free asset and the stock. The replicating strategy can be found as:

which gives:

In absence of arbitrage, since the price of the replicating strategy and the option must be the same $\forall t$, we can conclude that:

Intuition: in case of arbitrage, given the put-call parity formula $\text{C}+PV(K)=\text{P}+\text{S}$ we can construct an arbitrage opportunity by shorting the "cheapest" side and going long on the "expensive" one.

Note that the same reasoning (with different derivations) can be applied to any payoff.

We can now apply the same reasoning in continuous-time.
Considering a given time range $T$, we split the time range in (infinitely many) periods $dt$ and we assume that in any period the stock price is multiplied by either $(1+\alpha dt),(1+\beta dt)$.
We assume the distribution of the stock price at time $T$ is log-normal.
Applying (infinitely many times) the two state valuation model makes it possible to compute the value of the call.

Formally, the value of the stock at time $t$ denoted $S(t)$ is assume to vary as follows:

where $B(t)$ is a standard Brownian Motion, so for any $y$ the distribution of $S(t)$ is log-normal.

We then assume:

• constant risk-free interest rate
• no arbitrage

and we get the famous Black-Scholes formula:

• $d_1=\frac{\ln (S_0/X)+(r+{\sigma }^2/2)T}{\sigma \sqrt{(}T)}$

• $d_2=d_1-\sigma \sqrt{(}T)$

• $N(d)$ probability that a random draw from a standard normal distribution will be less than $d$

• $r$: risk free interest rate, (annualized continuously compounded rate with same maturity as the expiration of the option, note that is different from $r_f$),

• $\sigma$: standard deviation of the annualized continuously compounded rate of return of the stock.

Intuition: $N(d)$ can be interpreted as the risk-adjusted probabilities that the call option will expire in-the-money ($X<S_T$)

3. Pricing by arbitrage

We consider two periods: ex-ante, ex-post, denoted by $t=0,1$ and $\Omega$ contingent states at time $t=1:\omega \in \{1,\dots ,\Omega \}$.
There are $K$ assets available at time $t=0:k\in 1,...K$.

• $a_k^{\omega }$ is the value of asset $k$ in state $\omega$ at time $t=1$
• we denote $p_k$ the price of asset $k$ at time $t=0$
• a market is the data: $(a_k^{\omega },p_k)$

A portfolio $(z_k)$ is a vector of asset quantities: $z_k$ is the quantit of asset $k$ in the portfolio.

• the price of portolio is: ${\sum }_k{p}_k{z}_k$
• the value of the portfolio in state $\omega$ is: ${\sum }_k{a}_k^{\omega }{z}_k$

Given now a market with:

a portfolio with asset quantities $(z_1,...z_k)$ has price:

and values

We then introduce the followig definitions:

• an asset is risk-free if $a_k^{\omega }$ is independent of $\omega$
• an asset $k$ can be replicated by a subset of assets, indexed $i\in S$, if there exists a constant $(z_i)$ such that:

(in such cas the asset $k$ is called redudant).

A market $(a_k^{\omega },p_k)$ is complete if for any asset $b^{\omega }$ there exists a constant $(z_k)$ such that:

In other words, the market is complete if the payoff matri has max rank:

where $n$ is the number of states.

On the other hand, markets are complete when all revenue configurations are replicable through some portfolio.

Consequences of complete markets:

• there are at least as many assets at the states: $|\Omega |\le K$
• a market is complete if and only if the payoff matrix $a$ ith dimension $(K\times \Omega )$ has rank $\Omega$.
• if a market is complete with $K>\Omega$, there are $\Omega$ independent assets, then we can eliminate $K-\Omega$ assets which are linear combinations of the independent ones.

An arbitrage portfolio is a portfolio $(z_k)$ such that:

at least one of these $\Omega +1$ inequalities being strict.

A market is arbitrage free if there is no arbitrage portfolio.

A market $(a_k^w,p_k)$ without arbitrage opportunity satisfies the Law of One Price:
If two assets $i,j$ satisfy $a_i^{\omega }=a_j^{\omega }$ for every state $\omega$, then the two assets have the same price $p_i=p_j$.

More on the law of one price [Investopedia].

Assume that $p_i>p_j$, then the portfolio such that $z_i=-1,z_j=1,z_k=0$ for $k\ne i,j$ satisfies the following:

that is an arbitrage portfolio.

A market $(a_k^w,p_k)$ is arbitrage-free is and only if there exists a vector $q=(q_1...q_{\Omega })\in {\mathbb{R}}^{\Omega }$ such that:

• $q^{\omega }>0$ for every state
• $p_k={\sum }_{\omega }{a}_k^{\omega }{q}_{\omega }$ for every assets

$q$ is then called state-price vector.

In matrix form, the state-price vector with positive components solves:

There may exists several state price vectors in a market.

Intuition:

• the state price ($q_{\omega }$) is simply the price you must pay today to receive that $1$ in state $\omega$ tomorrow (regardless of the asset).
• the state-price vector is just the collection of these prices for every possible future state.
• $q$ doesnt tell you which assets to buy to receive $1$
• once you have the state-price vector $q$, you can price any asset $k$ using the sum above (the price of an asset $k$ equals the sum of its payoff in each state weighted by the state prices)

Assume a state price vector $(q_{\omega })$.

Denote ${\pi }_{\omega }^{\ast }=\frac{q_w}{{\sum }_{\omega }{q}_{\omega }}>0$, with ${\sum }_{\omega }{\pi }_{\omega }^{\ast }=1$.
Therefore ${\pi }_{\omega }^{\ast }$ can be interpreted as a probability. In particular, $({\pi }_{\omega }^{\ast })$ is called the risk neutral probability.

• if a market admits several state price vectors, it admits also as many risk neutral probabilities.

If we define: $r_k^{\omega }=\frac{a_k^{\omega }}{p_k}$ the return of asset $k$ in state $\omega$, using $p_k={\sum }_{\omega }{a}_k^{\omega }{q}_{\omega }$:

If we substitute the definition of ${\pi }_{\omega }^{\ast }=\frac{q_{\omega }}{{\sum }_j{q}_j}$:

If we extract the constants ($p_k$ and the total sum of state prices) out of the summation:

Since ${\sum }_{\omega }{q}_{\omega }{a}_{k\omega }$ is exactly the definition of $p_k$, the terms cancel out:

All asset returns have the same-risk neutral expectation. (This does not imply that all assets have the same returns expectations according to physical probability).

For any state ${\omega }_0$, the corresponding Arrow-Debreu security, denoted by $a_{{\omega }_0}^{\omega }$, is the asset with values given, for all $\omega$, by:

• If the market is arbitrage free and $q$ is a state-price vector, then the price of $a_{{\omega }_0}$ is $q_{{\omega }_0}$.
• A market that contains all Arrow-Debreu securities is complete.

Intuition: a contract that pays exactly $1$ if a specific state of the world and $0$ in every other possible scenario.

Assume now that a market includes a risk-free asset that pays $1$ in all state of the world and denote by $p_0$ its price ($p_0=\sum q_{\omega }$) and $r_f=1/p_0$, then:

• for all assets $k$: ${\mathbb{E}}_{{\pi }^{\ast }}[r_k^{\omega }]=r_f$
• if the market contains the Arrow-Debreu security $a_{{\omega }_0}$, its price is $p_0{\pi }_{{\omega }_0}^{\ast }$

Given an arbitrage-free market $M$, define $Q_M$ the set of state price vectors for this market:

• $Q_M$ is a convex set
• $Q_M$ empty $⟺M$ offers arbitrage opportunities
• $Q_M$ is reduced to a point $⟺$ M is complete and arbitrage-free
• if $q,q^{\prime }\in Q_M$, then $q-q^{\prime }\perp M$, that means:

For amy claim $c=(c_1,...c_{\Omega })$, not necessarily replicable in $M$, the set of prices $p_c$ for such a claim to not generage arbitrage opportunities is:

• this set is singleton $⟺$ $c$ is replicable in $M$
• if $c$ is not replicable, the adding it to the market with a price within arbitrage bounds will reduce $Q_M$ (dimension decreases by one)

4. An introduction to the economic analysis of asset markets

Some introductionary definitions for the following chapters:

We begin this chapter with a simple example about market equilibrium.

Consider two agents $A,B$ and two-states of the world (ex-post) $w=1,2$:

• in state $w=1$, agents get income $w_A^1,w_B^1$
• in state $w=2$, agents get income $w_A^2,w_B^2$ Denote then by $c_A^1,c_A^2,c_B^1,c_B^2$ their consumption.
  The feasible allocations must satisfy:

The no-trade situation corresponds to $c_A^1=w_A^1,c_B^1=w_B^1$ and $c_A^2=w_A^2,c_B^2=w_B^2$.

In this chapter, we rely on the standard specification for the Expected Utility Theory:

where ${\pi }_1,{\pi }_2$ are the probability for scenario $w=1,2$ and $u_i:{\mathbb{R}}_{+}\to \mathbb{R}$ is the utility index function.

An allocation is Pareto-optimal $⟺$ there is no alternative feasible allocation at which every individual in the economy is at least as well off and some individuals is strictly better off.

Graphically, Pareto-optimal allocations are the ones where the indifference curves are tangent.

The contract curve (or core) is the part of the Pareto set for which both agents do at least as well as their initial endownets. On the core, both agents gain from trade and the outcome is Pareto-optimal.

Agents interact by trading assets on a market:

• assets are purchased/sold ex-ange (before uncertainty realizes)
• payoffs are given ex-post
• market equilibrium occurs when asset prices are such that individual strategies are globally compatible (demand = supply).

We consider two assets (Arrow-Debreu Securities) in a two-states world:

• A risky asset $1$:
  • it can be purchased/sold ex-ante at $p_1$
  • ex-post it pays $1$ for $w=1$ and zero otherwise
• A risky asset $2$:
  • it can purchased/sold ex-ante at $p_2$
  • ex-post it pays $2$ for $w=2$ and zero otherwise.

$z_A^1,z_B^1,z_A^2,z_B^2$denote the quantities of assets for agents $A,B$ (they can be positive or negative depending on purchase/sell).

Ex-ante, each agent has zero initial wealth and a budget constraint:

Ex-post, assets' pay-offs and income and realize and agent $i$ consumes $w_i^i+z_i^i$.

Therefore, agent $i$ solves the following optimization problem:

This problem generates the demand functions:

A market is balanced (cleared) when demand equals supply for all assets (Market Clearing Conditions):

At equilibrium, we then have 2 equations in 2 unkwnowns:

Considering also the budget constraints, the previous two equations are then equal, so we have $1$ independent equation and $2$ unknowns.
$⟹$ Equilibrium prices are determined up to a multiplicative scalar.

The following plot shows risk transfer: A has NO income risk, but rather takes some risk from B.

The allocaitons $(c_A^1,c_A^2),(c_B^1,c_B^2)$ are comonotonic if and only if:

All market equilibrium allocations are comonotone, this is also the case for Pareto-optimal allocations.

Consider now the case where $B$'s endowment in state $2$ decreases, so $p_1/p_2$ is smaller.

In this case, an increase in the risk of $B$ leads $A$ to take more risk.

In general, low risk averse agents take the most of the risk.

An interactive simulation of risk preferences can be found here.

5. Choice under uncertainty

Consider $Z$ a set of possible consequences. Let $\Omega =\{1,2,|\Omega |\}$ the set of possible states of the world and $L(\Omega )$ the set of lotteries with consequences in $Z$.

A lottery is a list of pairs $(x^{\omega },{\pi }_{\omega })$ where ${\pi }_{\omega }$ gives the probability that $\omega$ will occur, and $x^{\omega }$ is the outcome.

Example: $((0,0.5);(1,0.5))$ is the lottery that pays $0,1$ with equal probabilities.

A compound lottery is a lottery whose prizes are themselves lotteries.

Consider two lotteries $L,L^{\prime }$.
The compound lottery with $\lambda \in [0,1]$ corresponds to the lottery which consists in playing the first lottery with probability $\lambda$ and the second with probability $(1-\lambda )$.

The resulting lottery is denoted by $(\lambda L+(1-\lambda )L^{\prime })$. The operation + is called a mixture operation.

Example:

We want to define a theory of preference defined over $L(Z)$.

A relation of preferences $\ge$ on $L(Z)$ is a binary relation which is:

1.Complete

• for any $L,L^{\prime }\in L(Z)$, we have $L\ge L^{\prime }$ or $L^{\prime }\ge L$

2. Transitive
   • $L_1\ge L_2$ and $L_2\ge L_3⟹L_1\ge L_3$

Von Neumann and Morgenstern suggested to assume the following about the relation of preference:

• Independence: for all $L,L^{\prime },L"$ and $\lambda \in (0,1)$:

• Continuity: if $L\ge L^{\prime }\ge L"$, then there exists $\lambda \in [0,1]$ such that $L^{\prime }\sim \lambda L+(1-\lambda )L"$.

Graphically, the independence axiom:

A relation of preferences is represented by a utility function $V:L(Z)\to \mathbb{R}$ if and only if: $L\ge L^{\prime }⟺V(L)\ge V(L^{\prime })$.

If $\ge$ is a preference relation on $L(Z)$ that satisfies the previous axioms, then there exists $u:Z\to \mathbb{R}$ such that:

that can also be rewritten as:

In this case, we say that the agent is said to have "VNM preferences".

A little note on terminology:

• $V:L(Z)\to \mathbb{R}$ is the utility function (expected utility function).
• $u:Z\to \mathbb{R}$ is just a function that appears when looking at the structure of $V$ (utility index). Furthermore, for the rest of the chapters, we assume that$u$ is strictly increasing and twice continuously differentiable.

We denote the average of lottery $L$ by ${\mathbb{E}}_{[}x]={\sum }_{\omega }{\pi }_{\omega }{x}^{\omega }$.
We represent with ${\delta }_{{\mathbb{E}}_{[}x]}$ the degenerate lottery that gives the expected payoff of $L$ with probability one.

An agent is said to be risk averse if and only if for all $L\in L(Z)$:

An agent with VNM preferences is risk averse if and only if her utility index is concave: $u^{\prime }\ge 0,u^{\prime \prime }\le 0$.

To prove it, we consider the Jensen Inequality: if a function $u$ is concave, then we have:

Recall: for any lottery $L$ we can define the cumulative distribution function $F_L:\mathbb{R}\to [0,1]$ as:

A lottery $L$ is said to second-order stochastically dominates $L^{\prime }(\text{SOSD})$ if and only if: ${\mathbb{E}}_L[x]\ge {\mathbb{E}}_L^{\prime }[x]$ and:

(Note that the first order stochastic dominance implies the second one)

Clearest Explanation of Second-Order Stochastic Dominance! [VIDEO]
Understanding First-Order Stochastic Dominance[VIDEO].

Consider $L_1,L_2$ with the same mean and with density functions $f_1(x),f_2(x)$. $L_2$ is said to be a mean preserving spread of $L_1$ if there exists an interval $I$ such tat:

There are different ways to define an increase in risk:

• $L$ is riskier than $L^{\prime }$ if $L^{\prime }\text{ SOSD }L$ ($L^{\prime }$ second order stochastically dominates $L$).
• $L$ is riskier than $L^{\prime }$ if $L$ is a mean preserving spread of $L^{\prime }$.
• Adding a white noise: $L$ is riskier than $L^{\prime }$ if $L=L^{\prime }+ϵ$, where $\mathbb{E}[ϵ]=0$. (Note that $L,ϵ$ do not need to be independently distributed).

An agent with VNM preferences dislikes increases in risk if and only if her utility index $u$ is concave.

For a lottery $L$, its certainty equivalent is amount $e_L\in \mathbb{R}$ such that: ${\delta }_{e_L}\sim L$ or, in other terms:

Intuition: $e_L$ is the amount of money for which the individual is indifferent between playing $L$ and a fixed amount of money.

• For a concave utility (risk averse agent): ${\mathbb{E}}_L[u(x)]-e_L>0$
• For a linear utility (risk neutral agent): ${\mathbb{E}}_L[u(x)]=e_L$

For any lottery $L$ with payoff $X$, define the certainty equivalent $e_L$ as:

$u(e_L)={\mathbb{E}}_L[u(X)]$

The risk premium is then:

${\pi }_L={\mathbb{E}}_L[X]-e_L$

For a VNM agent, this is equivalent to:

$u({\mathbb{E}}_L[X]-{\pi }_L)={\mathbb{E}}_L[u(X)]$

• A risk neutral individual has zero risk premium
• A (strictly) risk-averse agent associates a positive risk premium to any non-degenerate lottery.

Intuition (preference rule): a VNM agent (risk-averse) always prefers the lottery with the higher certainty equivalent. If the lotteries have the same mean $\mathbb{E}[X]$, the agent prefers the one with the smaller risk premium.

Consider $x\in \mathbb{R}$, $ϵ$ small and the following lottery:

Using Taylor expansions, we can show that the (absolute) risk premium ${\pi }_a$ when $ϵ$ is small (Arrow-Pratt approximation) is such:

The absolute risk aversion coefficient $R_a(x)$ for wealth level $x$ is defined as:

For small risks of the form $x+\widetilde{ϵ}$:

Note that $\mathbb{E}[{\widetilde{ϵ}}^2]$ is the variance of the risk.

Consider $x\in \mathbb{R}$ and the same lottery as before, $L+\widetilde{ϵ}$.
We can define the relative risk premium ${\pi }_r$ by:

and, using the Taylor expansion when $\widetilde{ϵ}$ is small:

The relative risk aversion coefficient $R_r(x)$ for wealth level $x$ is then defined as:

For a small risk of the form $(1+\widetilde{ϵ})x$:

For two agents $A,B$ all the following sentences are equivalent:

• $\forall L$ the certainty equivalent of $A$ is smaller or equal than the one of $B$.
• $\forall L$ the risk premium for $B$ is smaller or equal than the one for $A$
• $u_A$ is more concave than $u_B$: there exists and increasing concave functio $\varphi$ such that: $u_A=\varphi (u_B)$.
• $\forall x$ we have:

-$A$ is at least as risk averse as $B$

• Costant Absolute Return Aversion (CARA)

• Constant Relative Risk Aversion (CRRA)

• Hyperbolic Absolute Risk Aversion (HARA)

• Quadratic Utility Index

Note that for a quadratic utility index:

then the utility of a lottery only depends on its mean and variance.

6. Demand for risk

Consider an agent with wealth $W$ who can invest at time zero $t_0$ in a risk-free asset with return $r_0$ and in a risky asset with random return $\tilde{r}$.

If the agent invests $\alpha$ in the risky assets, he gets in period $1$:

with $\omega =W{r}_0$ and $\tilde{R}=\tilde{r}-r_0$, the agent's problem rewrites as:

Let's start deriving some properties:

Then if the agent is risk-averse, $V$ is concave and therefore $V^{\prime }$ is non-increasing, so we can conclude that:

• if $\mathbb{E}[\tilde{r}]>r_f⟹\mathbb{E}[\tilde{R}]>0$: the agent should purchase some risky asset (regardless of how much risk averse he migh be).
• if $\mathbb{E}[\tilde{r}]<r_f$: it is never optimal to purchase the risky asset.

We derive the first order conditions to quantify how much the agent should invest.

• If $\tilde{R}$ is always of the same sign $⟹$ there is no interior solution.
• Even if the agent is risk-averse and $\tilde{R}$ changes sign, there may not be an interior solution.

In the following, we assume that there exists an interior solution and that $\mathbb{E}[\tilde{R}]>0$, so that the solution is positive (${\alpha }^{\ast }>0$).

We assume $\tilde{R}=ϵ+{\tilde{R}}^{\prime }$, with $\mathbb{E}[{\tilde{R}}^{\prime }]=0$ and $ϵ>0$.
We denote by $\alpha (ϵ)$ the solution to:

from which we can derive the first-order condition:

We know that $\alpha (0)=0$, so using a Taylor Approximation, we expand $\alpha (ϵ)$ around $ϵ=0$.

Since $\alpha (0)=0$ (at zero mean excess return, the agent doesn't buy the risky asset), the first-order Taylor expansion is: $\alpha (ϵ)\approx \alpha (0)+ϵ{\alpha }^{\prime }(0)=ϵ{\alpha }^{\prime }(0)$.