## 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: $ E[X] = \sum_{all~x} P[X=x]\times x $

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: $ Var[X] = \sum_{all~x} P[X=x]\times (x-E[X])^2 $

For example,

$ \begin{array} {|c|c|c|c|} Outcome & P(Win) & Opt~A & Opt~B\\ \hline Good & \frac{1}{10} & 50,000 & 26,000\\ Middle & \frac{22}{25} & 12,500 & 15,000\\ Bad & \frac{1}{50} & 0 & 10,000\\ E[X] & & 16,000 & 16,000\\ Var[X] & & 131,500,000 & 11,600,000\\ \end{array} $

NOTE: $ E[A]=\frac{1}{10}50,000+\frac{22}{25}12,500+\frac{1}{50}0=16,000 $

And: $ Var[A]=\frac{1}{10}(50,000-16,000)^2+\frac{22}{25}(12,500-16,000)^2 $ $ +\frac{1}{50}(0-16,000)^2=131,500,000 $

Now, since the $ Var[B] $ is less than that of $ Var[A] $, 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: $ U(\bullet) $ which is the Utility Function.

Note, the following conditions hold:

- Each individual has OWN $ U(\bullet) $

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

- ASSUME companies have NO $ U(\bullet) $
- If wealth increases, $ U(\bullet) $ increases

- Individual prefers X more than Y:

$ X \succ Y $

$ U(X) > U(Y) $

## Expected Utility Edit

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

Then: we use Expected Utility $ U(W) = E[v(W)] $

$ U(W) = \sum_{i=1}^n P(W=w_i)\times v(w_i) $

- $ n $ is n states of the world
- $ w_i $ is wealth in state i
- $ v(\bullet) $ is:
- The expected $ U(\bullet) $
- Continuous non-decreasing

## Expected Utility Axioms Edit

All preferences are ASSUMED to be:

**Complete**

All risks can be compared, ranked

**Reflexive**

$ X \succsim X $ meaning risk is at least as good as itself

**Transitive**

If $ X \succsim Y $ and $ Y \succsim Z $ then $ X \succ Y \succ Z $

**Independent**

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

## Risk Aversion Edit

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

$ E[W] \succ W $ or $ v(E[W])>U(W) $ where $ U(W)=E[v(W)] $ so $ v(E[W])>E[v(W)] $

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** - $ {\partial\over\partial W}v(W)>0 $
- $ {\partial^2\over\partial W^2}v(W)<0 $

Side Note: Prove algebraically (not graphically) that $ v(E[W])>E[v(W)] $ This is called JENSEN'S INEQUALITY

We know that $ v(x) \approx v(y) + (x-y)v'(y) + \frac{1}{2} (x-y)^2 v''(y) $

Let $ y = E[W] $ and also $ x=W $

So, $ v(W) \approx v(E[X]) + (W-E[W])v'(E[W]) + \frac{1}{2} (W-E[W])^2 v''(E[W]) $

The condition is that $ \partial ^2 \over \partial W^2 U(W) < 0 $ so that $ v''(W) <0 $.

This entails that $ \frac{1}{2} (W-E[W])^2 v''(E[W]) < 0 $

Thus, $ v(W) \approx v(E[X]) + (W-E[W])v'(E[W]) + \frac{1}{2} (W-E[W])^2 v''(E[W]) $ $ <= v(W) \approx v(E[X]) + (W-E[W])v'(E[W]) $

## Risk Loving Edit

This is when the risk is preferred to the expected risk: eg. Lottery $ E[W] \prec W $ or $ v(E[W])<U(W) $ where $ U(W)=E[v(W)] $ so $ v(E[W])<E[v(W)] $

## Risk NeutralityEdit

This is when the risk is indifferently preferred to the expected risk. $ E[W] = W $ or $ v(E[W])=U(W) $ where $ U(W)=E[v(W)] $ so $ v(E[W])=E[v(W)] $

This means that $ {\partial^2\over\partial W^2}v(W)=0 $.

Thus, $ v(W) = aW + b $

## Insurance Applications Edit

Some notation:

- Loss is $ I $
- Sum insured is $ I $
- Insurance premium is $ \pi $
- Insurance premium paid is $ \pi I $
- Individual has wealth $ W $
- $ p $ is loss probability

$ \begin{array} {|c|c|c|c|c|c|} Wealth & Outcome & P & If~Lose & Wealth(No Ins) & Wealth(Yes Ins)\\ \hline W & Loss & p & -I & W-I & W-\pi I \\ W & No~Loss & 1-p & 0 & W & W-\pi I \\ \end{array} $

So, with FULL Insurance, Loss: $ W = W-I+I-\pi I~at~p $

No Loss: $ W = W-\pi I~at~1-p $

Thus, Expected Utility is just addition: $ v(W- \pi I) = p \times v(W- \pi I) + (1-p) \times v(W- \pi I) $

So, with NO Insurance, Loss: $ W = W-I ~at~p $

No Loss: $ W = W~at~1-p $

Thus, Expected Utility is just addition: $ p \times v(W-I)+(1-p)\times v(W) $

Thus, someone would prefer Insurance if:

$ v(W- \pi I) > p \times v(W-I)+(1-p)\times v(W) $

Now, what is the optimum amount of insurance premiums at $ \pi \times X $? Where $ X $ is the amount of insurance.

Loss: $ W = W-I+X - \pi X ~at~p $

No Loss: $ W = W - \pi X~at~1-p $

Thus, Expected Utility is just addition: $ p \times v( W-I+X - \pi X) + (1-p) \times v(W - \pi X) $

To get the optimum amount, find the turning point: So find $ {\partial v\over\partial X}=v'(\bullet) $

So $ p(1-\pi) \times v'( W-I+X - \pi X) - \pi^2(1-p) \times v'(W - \pi X)=0 $

Thus, $ \frac{p \times v'( W-I+X - \pi X)}{(1-p) \times v'(W- \pi X)} = \frac{\pi}{(1-\pi)} $

- 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 $ \pi = p $, then Risk Averse individual will choose $ X=I $

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

We need to show that $ \partial v \over \partial X < 0 $

So, $ p(1-\pi)^2 \times v''(W-I+X- \pi X) + \pi^2 (1-p) \times v''(W- \pi X) < 0 $

## Common Utility Functions Edit

Quadratic Utility F(x)

- $ v(w) = w-\tau w^2 $
- $ v'(w) = 1- 2\tau w $
- $ v''(w) = -2\tau $

Exponential Utility F(x)

- $ v(w) = -e^{\alpha w} $
- $ v(w) = \alpha e^{-\alpha w} > 0 $
- $ v(w) = - \alpha^2e^{-\alpha w} < 0 $

## Time Preference Edit

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

- $ C_0 $ is consumption
- This means $ W-C_0 $ is used for investment
- Now invest: $ C_1=[W-C_0](1+r) $
- Since $ W = C_0 + (W-C_0) $

Then, $ W=C_0 + \frac{C_1}{(1+r)} $

Thus, $ W=U(C_0)+U(C_1) $

To find maximum utility, find $ {\partial U(C_n)\over\partial C_0} $

So, $ \frac{U'(C_1)}{U'(C_0)}=\frac{1}{(1+r)} $

Eg: $ U(w)=-e^{-0.005w} $ If Individual has $ W=$100,000 $ and $ r=8% $, find optimal consumption.

$ \frac{1}{1.08}=\frac{U'(C_1)}{U'(C_0)} $ $ U'(w)=0.005e^{-0.005w} $

Knowing $ C_1=[W-C_0](1+r) $ Then, $ \frac{1}{1.08}=e^{-540+0.0104C_0} $

Thus, $ C_0(max)=51,915.68 $

So how are the $ r $ 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.