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What is asset pricing theory about ?

Asset is anything that promises you a stream of future cash flows. Think it as a stock, a bond, a piece of real estate etc.

TIME and RISK is all about pricing theory. How much it ends up and when ?

Art of the risk management is guessing how things are going to change if the environment is changing.

- Investment vs equilibrium
- what dose the market look like after everybody invested ?

## Basic Asset Pricing Formula

p = E(mx)

where x is the payoff.

For example if we buy a stock at time t for P_t, then it’s payoff at time t+1 will be X_{t+1} = P_{t+1} + d_{t+1} , where d is the dividend.

The payoff is a random variable. It has multiple status of nature with each of this has associated probabilities.

Question: what’s a worth ?

Once we understand the possibilities of the payoff, what is the investor willing to pay to get that random payoff.

Asset pricing is modeling human behavior, we need to write down a simplified model about how people feel about stuff there is no way avoiding psychology and that’s point of Utility function.

Utility functions is a way of in a very simple consistent way capturing the evident fact that people prefer money now and people prefer money that isn’t so risky, and so what we need is a mathematical structure to help us capture those features of human psychology and that’s what the utility function dose.

So in simplicity, price is equal to Expected discounted payoff.

So we can then do the optimization problem of, starting at your initial consumption (c+) pay some today lose the price today of the security ( p+) but in exchange , you get a little bit more of the payoff tomorrow

So today’s price is expected discounted by marginal utility payoff tomorrow.

This characterizing how the consumer prices and feels about the value.

Take the random payoff, multiply with discount factor, take expectation and that is the price.

## Classic Issue

Before we do the classic ideas of the theories of finance, you got to meet the players. we got to think about returns, present values, the other ways of stating the basic representations.

Alternative representations of our **price equals expected discounted payoff** ideas. Returns, present values and how are those stuff look in continuous time.

So let’s start with rate of return, return is just a price one security. You put in one dollar today and you get back what you get in the future , threfore price equals expected discounted payoff just reads one equals expected discounted return.

Low price today is the same thing as a high return from today till tomorrow. It kind expressing things in terms of returns sort of hides that but that’s what’s going on. If you have a security whose price is driven down today, that’s the way that it gets a high expected return from today until tomorrow.

In the return concept that we are using here, the Capital R is the gross return, price plus dividend over initial price, it’s a number like 1.1 not a number like 0.1 or 10% or annualized. People uses a lot of different units, when you deal with data make sure that you are using the right set of units.

### Risk free rate

Most common kind of return is the risk free rate, it is the benchmark for many things.

Another way of looking at risk free rate is consider it as a bond. On bond pricing a zero coupon bond pays $1 for sure tomorrow, so it’s payoff is 1, and its price is E(mx1).

### Excess return

Excess return is the R minus Risk free rate or the differences of any two returns.

Excess return has the price of zero. Means you have no money today, you borrow a dollar and invest and return that one dollar tomorrow and you still make excess money left.

Exess return focuses on the risk element.

### Present values

### Classic Issue in Finance

note: transformation is done by Ito’s lemma.

Where dose interest rates come from ?

**Interest rates**: are about what dose it take to get people to save some of their income not consume it all today and consume some tomorrow. Well if people are more impatient you have to pay them a higher interest rates to consume tomorrow rather than today.

**Intertemporal substitution**: this says that interest rates are higher when consumption growth is higher. gamma controls how much do I have to bribe you with a higher interest rate t get you to consume less today and have a big consumption growth from today till tomorrow.

**precautionary savings**: if you live in an economy with a lot of volatility ( a lot of uncertainty ) people are nervous about the future and they try to save a lot today. If people are trying to save a lot today that’s going to drive down today’s interest rate.

### Risk Premiums

Covariance drive the risk premium.

### Consumption + Risk Premium

### Risk Premiums & Betas

The beta the coveriance that makes the asset returns different.

lambda is the market price of return.

lambda is the slope coefficient. alpha is the deviation.

We trying to explain Expected return with beta. ( 2)

The market price of the risk should be higher if you live in a risky economy or if the risk aversion is higher naturally.

Only systematic risk is priced.

### Mean Variance Frontier and Roll Theorem

- all securities lies inside the green zone