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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)]

Expected Utility 1
Expected Utility 2
Expected Utility 3
Expected Utility 4













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.

Marginal Utility and Pricing Edit

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