Finance: State Prices and Risk Neutral Probabilities

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Questions and Answers

What does the term $β_{i,t}$ represent in the expected return equation?

The variance of the market return

The expected return of the risk-free asset

The proportionality constant in the regression

The covariance between asset return and market return (correct)

In the context of the content, what relationship does the expected return have with the risk-free rate?

The expected return is equal to the risk-free rate.

The expected return can be less than or equal to the risk-free rate. (correct)

The expected return is always greater than the risk-free rate.

The expected return is independent of the risk-free rate.

What does the term $λ_{t}$ represent according to the derived relationships?

The adjustment factor for the risk-free rate

The expected value of the market return

The expected value of asset returns

The measure of risk premium per unit of risk (correct)

What does dividing both sides by $E_t[M_{t+1}]$ accomplish in the given equations?

It normalizes the expected returns for the market factor. Signup and view all the answers

What is implied about the expected return $E_t[R_{i,t+1}]$ when there is a high beta $β_{i,t}$?

It suggests a direct relationship to a higher expected return. Signup and view all the answers

What does the equation p = B π̃s ys represent?

The expected value of cash flow under risk neutral probabilities, discounted by the risk-free rate. Signup and view all the answers

What does the stochastic discount factor relate to according to the provided content?

State prices divided by physical probabilities. Signup and view all the answers

In the context of pricing risky assets, what are Arrow-Debreu (AD) securities used for?

To bundle payoffs or cash flows state-by-state. Signup and view all the answers

How are risk neutral probabilities calculated?

By dividing state prices by the price of the risk-free asset. Signup and view all the answers

Which of the following statements about option prices and state prices is correct?

Option prices with different strikes can be used to infer state prices. Signup and view all the answers

What is the main takeaway from the valuation under physical and risk-neutral measures?

They can result in different pricing strategies. Signup and view all the answers

The expression Ẽ (ys) represents what in the context of cash flow valuation?

The expected value of cash flow under risk neutral probabilities. Signup and view all the answers

What is the significance of the risk-free rate (rf) in the pricing model discussed?

It serves as a benchmark to evaluate the cost of risk. Signup and view all the answers

What does a high coefficient of relative risk aversion imply about time preference?

It corresponds to a low or negative rate of time preference. Signup and view all the answers

How does RRA(2) calculate the implied risk aversion?

By assuming a correlation equal to 1. Signup and view all the answers

What is the implication of the equity premium puzzle according to the content?

It is linked to high coefficients of relative risk aversion. Signup and view all the answers

What does the Hansen-Jagannathan Bound help to establish?

A lower bound on expected excess returns based on risk characteristics. Signup and view all the answers

What is the formula for calculating the expected excess return on a risky asset?

Et[ri,t+1] = -σi,m + rf,t+1 + σi² Signup and view all the answers

What is the role of the correlation coefficient ρ(ere, ∆c) in the covariance equation?

It indicates the strength of the relationship between stock returns and consumption. Signup and view all the answers

In the context of risk aversion, what does the term 'equity premium puzzle' refer to?

The inability to explain high stock returns despite risk aversion. Signup and view all the answers

What does it mean if the correlation ρi,m is less than 0?

It shows a weak or negative relationship. Signup and view all the answers

What does λt represent in the context of asset pricing?

The price of risk common to all assets Signup and view all the answers

What does the equation 0 = Et ri,t +1 + Et mt +1 + (σi2 + σm 2 + 2σi,m) imply?

It shows the relationship between expected returns and the stochastic discount factor. Signup and view all the answers

Which property is associated with a random variable X that is conditionally lognormally distributed?

The logarithm of expected value of X is the average of log X plus half the variance of log X. Signup and view all the answers

What does equation (12) suggest about the relationship between the risk premium and covariance?

Negative covariance with the stochastic discount factor results in higher expected returns. Signup and view all the answers

In the equation for the log return on the risk-free asset, what does the term σm 2 represent?

The unconditional variance of the log stochastic discount factor Signup and view all the answers

What does the last statement regarding assets with negative covariance highlight?

They must yield higher expected returns. Signup and view all the answers

What does the term βi,t signify in the asset pricing context?

The quantity of risk specific to an asset Signup and view all the answers

What effect does increasing log-normality have on joint distributions of asset returns?

It simplifies the handling of variance and expectations. Signup and view all the answers

What does the symbol Mt+1 represent in the context of asset pricing?

Intertemporal marginal rate of substitution Signup and view all the answers

What does a high value of Mt+1 indicate about consumption levels?

Low level of current consumption Signup and view all the answers

According to the fundamental asset pricing equation, how does the expected discounted return behave?

Always equals 1 Signup and view all the answers

What is required for the existence of a positive stochastic discount factor in markets?

Absence of arbitrage opportunities Signup and view all the answers

What can the equation 1 = Et [Mt+1 (1 + Ri,t+1)] be manipulated to show?

The expected price of an asset equals its future payoffs Signup and view all the answers

What is indicated by low expected returns across time and assets?

High marginal utility Signup and view all the answers

What does the fundamental asset pricing equation assume about investor behavior?

Investors maximize utility through time-separable functions Signup and view all the answers

Which of the following best describes the fundamental asset pricing equation?

It provides a method for asset valuation based on expected future payoffs Signup and view all the answers

Study Notes

State Prices and Risk Neutral Probabilities

The price of a random cash flow Y is calculated as the expected value of cash flow under the risk neutral probabilities, discounted by the risk-free rate
The value of an asset is the expected value of discounted future payoffs by a stochastic discount factor (SDF).

Summary

There is a set of elementary contingent claims (AD securities) each paying one dollar in one specific state of nature and nothing in any other states
Valuation can be done under both physical and risk neutral probability measure.
Stochastic discount factor equals state prices divided by physical probabilities
Risk neutral probabilities are state prices (the prices of AD securities) divided by the price of the risk-free asset.

Inferring State Prices From Option Prices

State prices can be inferred from prices of options with different strikes
The state of nature is summarized by the value of the market portfolio which has a discrete probability distribution with possible values.

The Risk-Free Rate Puzzle

One response to the equity premium puzzle is to consider larger values for the coefficient of relative risk aversion.
High values of the coefficient of relative risk aversion would imply high values of the risk-free rate.

Hansen-Jagannathan Bound

The expected excess return on any risky asset is given by the negative of the covariance of the asset with the stochastic discount factor.
An asset with a negative covariance with the stochastic discount factor must have higher expected returns.

Imposing Log-normality

With joint conditional lognormality and homoskedasticity of asset returns and consumption, a convenient property emerges for a random variable X
The log return on the risk-free asset is equal to the negative of the expected value of the log stochastic discount factor minus one-half of the variance of the log stochastic discount factor.

The Fundamental Asset Pricing Equation

The fundamental equation in asset pricing states that the expected value of the product of the stochastic discount factor and the gross return on any asset is equal to one.
The Stochastic discount factor is a positive random variable that is high when consumption is low.
The value of an asset is equal to the expected value of discounted future payoffs (by Mt +1 ).

Expected Return - Beta Representation

The expected return should be proportional to beta in a regression of that returns on the SDF M.

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Description

This quiz explores the concepts of state prices, risk neutral probabilities, and their role in asset valuation. It covers the calculation of expected cash flows using stochastic discount factors and the relationship between state prices and option prices. Test your understanding of these fundamental financial principles.