## RisksEdit

Risk is future uncertainty, split into two classifications:

**Pure risk**

Exposure to *adverse* outcomes where insurance should have been normally used.

**Speculative risk**

Exposure to BOTH *adverse* AND *favourable* outcomes.

In life, both are seen, and is managed by:
**Holistic Risk Management**

**Enterprise Risk Management** is used by banks, insurers, companies to manage risks rigorously.

## Expected ValuesEdit

The expected value of a risk is called the *Actuarial Value of Risk* or the *Pure Premium*

For DISCRETE Random Variables:

But, Expected Value just returns the average of return. It doesn't show any variation in results. So, Variability is used instead.

## Variance Edit

For DISCRETE Random Variables:

For example,

NOTE:

And:

Now, since the is less than that of , even though both expected values are = to 16,000, the lower the variability, the better the choice.

So B is better. [Lower variability around expected value]

## Utility Edit

In Economics, to represent preferences over alternatives, we use: which is the Utility Function.

Note, the following conditions hold:

- Each individual has OWN

Eg: Very rich and very poor. Very poor has higher to get $1 when compared to rich person (lower )

- ASSUME companies have NO
- If wealth increases, increases

- Individual prefers X more than Y:

## Expected Utility Edit

But before, we dealt with known utilities. How about if they are UNKNOWN?

Then: we use Expected Utility

- is n states of the world
- is wealth in state i
- is:
- The expected
- Continuous non-decreasing

## Expected Utility Axioms Edit

All preferences are ASSUMED to be:

**Complete**

All risks can be compared, ranked

**Reflexive**

meaning risk is at least as good as itself

**Transitive**

If and then

**Independent**

Considering other risks to be equal, then if , then any other risks attached to either option (but same), will still result to

## Risk Aversion Edit

This is when the expected risk is preferred to the real risk. Remember is Wealth

or where so

So, the graph is steeper at the beginning, then becomes smoother at higher wealth values

- So higher utility for $1 then goes down at $1,000.

Risk aversion shows that:

- Utility function is
**CONCAVE**

Side Note: Prove algebraically (not graphically) that This is called JENSEN'S INEQUALITY

We know that

Let and also

So,

The condition is that so that .

This entails that

Thus,

## Risk Loving Edit

This is when the risk is preferred to the expected risk: eg. Lottery or where so

## Risk NeutralityEdit

This is when the risk is indifferently preferred to the expected risk. or where so

This means that .

Thus,

## Insurance Applications Edit

Some notation:

- Loss is
- Sum insured is
- Insurance premium is
- Insurance premium paid is
- Individual has wealth
- is loss probability

So, with FULL Insurance, Loss:

No Loss:

Thus, Expected Utility is just addition:

So, with NO Insurance, Loss:

No Loss:

Thus, Expected Utility is just addition:

Thus, someone would prefer Insurance if:

Now, what is the optimum amount of insurance premiums at ? Where is the amount of insurance.

Loss:

No Loss:

Thus, Expected Utility is just addition:

To get the optimum amount, find the turning point: So find

So

Thus,

- LHS is individual's trade-off between Marginal Utility if Loss occurs and Marginal utility if Loss DOESNT Occurs.
- RHS is insurer's trade-off between loss and no loss.
- If , then Risk Averse individual will choose

Side Note: How do we know whether the turning point is in fact a Maximum Turning Point?

We need to show that

So,

## Common Utility Functions Edit

Quadratic Utility F(x)

Exponential Utility F(x)

## Time Preference Edit

Now, how do we consider the time value of money?

- is consumption
- This means is used for investment
- Now invest:
- Since

Then,

Thus,

To find maximum utility, find

So,

Eg: If Individual has and , find optimal consumption.

Knowing Then,

Thus,

So how are the or interested determined?

- Determined by marginal Utility of additional consumption in future
- Market IR is det by D for Investment. Reflects preferences of individuals for future vs current consumption.