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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])<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 $## Insurance Applications Edit Some notation: 1. Loss is$ 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 $## Common Utility Functions Edit Quadratic Utility F(x) 1.$ 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.