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## 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 $U(W) = E[v(W)]$ $U(W) = \sum_{i=1}^n P(W=w_i)\times v(w_i)$ 1. $n$ is n states of the world 2. $w_i$ is wealth in state i 3. $v(\bullet)$ is: 1. The expected $U(\bullet)$ 2. 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:

1. Utility function is CONCAVE
2. ${\partial\over\partial W}v(W)>0$
3. ${\partial^2\over\partial W^2}v(W)<0$

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]) where $U(W)=E[v(W)]$ so $v(E[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:

1. Loss is $I$
2. Sum insured is $I$
3. Insurance premium is $\pi$
4. Insurance premium paid is $\pi I$
5. Individual has wealth $W$
6. $p$ 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, $\frac{p \times v'( W-I+X - \pi X)}{(1-p) \times v'(W- \pi X)} = \frac{\pi}{(1-\pi)}$

1. LHS is individual's trade-off between Marginal Utility if Loss occurs and Marginal utility if Loss DOESNT Occurs.
2. RHS is insurer's trade-off between loss and no loss.
3. 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

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

Exponential Utility F(x)

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

## Time Preference Edit

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

1. $C_0$ is consumption
2. This means $W-C_0$ is used for investment
3. Now invest: $C_1=[W-C_0](1+r)$
4. 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?

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