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Let Jon be an EUM with the utility function $u(x) = \exp( - \beta x)$. He wants to compare stock prices of two mutual funds. The today price for each share is $a$ dollars. The first share returns
The second share returns $Y = a + U$, where $U \sim \Uniform[-1,1]$.
Compute $\E[u(X)]$ and $\E[u(Y)]$. Which investment should Jon prefer if $\beta = 1, a = 100$.
Without computing the expected utitlies, can he decide whether or not he should invest in $X$ at all? Hint Remember condition-Z.
Compute the certainity equivalent of $X$ when $\beta = 1, a = 100$.
Suppose a customer has two models for the age of a computer component, $X \sim \Pareto(\alpha)$ and $Y \sim \Exponential(\lambda)$. Use the following tail function for the Pareto random variable:
Compute the moment generating functions of $X$ and $Y$. What does this tell you about the tails of $X$ and $Y$?
Compute the survival functions and hazard rates (or force of mortality) of $X$ and $Y$.
For both $X$ and $Y$, find the conditional distributions (cdfs of the age to death) given the component has already survived for $8$ years.
A customer of a company suffers a car accident with probability $0.1$ every year. The damage to the car is $\xi \sim \Exponential(1/500)$ distributed. The payment function is one with a standard deductible,
What is the average payment of the company as a function of $d$?
If the company wants to pay $$50$ per client per accident on average, what should $d$ be?
Given that an accident has happened, what is the average percentage of the total loss to the client that the company pays? Simply write down an integral and don’t evaluate it.
An insurance portfolio consists of two homogeneous groups of clients $N_i = 1,2$. Assume $N_i \sim \Poisson(10i^2)$ and are independent.
If $N$ is the total number of all claims in the portfolio, what distribution does $N$ have? Compute $\E[N]$, and $\Var(N)$. Write the expression for $\Prob(N \leq 50)$.
| Estimate $\Prob(N_1 < 11 | N = 50)$. |
Let the amount of individual claim in the first group be $$100$ (nonrandom) and suppose the second group is $$300$. Find $\E[S]$, $\Var(S)$ and $M_S(z)$.
On average, what proportion of the total claim comes from the first group?
Suppose an insurance portfolio has one homogeneous group of clients, and receives $N \sim \Poisson(1000)$ claims every year. Suppose the individual claim distribution has mean $\mu = \E[X_i] = 1$ and standard deviation $\sigma = \sqrt{\Var(X_i)} = 0.5$. Let $c$ be the surplus money that the company has in store to pay off its clients.
Assume that the claims process $S_t$ is represented by a compound Poisson process. The mean time between adjacent claims is 0.5 hours and the random value $X$ of a particular claim is $\Exponential(0.1)$. The insurance company sets premiums according to $c_t = 1.2 \E[S_t]$.
A life insurance company offers a 20 year term life insurance to all 5 year old children (payable by the state of utah). It is estimated that 40% of the newborns in Utah have a constant hazard rate $\mu_1(x) = 0.1$ and the rest have $\mu_2(x) = 0.2$.
What is the probability that a newborn with hazard rate $0.1$ survives to 5 years. What is the probability that a newborn with hazard rate $0.2$ survives to 5 years?
What is the probability that a randomly selected $5$ year old has hazard rate $0.1$?
Use the answer from the last part to write down $\Prob(T(5) > t)$.
Compute the actuarial present value for this type of insurance, when the payoff is $$1$ payable at the moment of death and discount rate (force of interest) $\delta$. (It is enough to just write down an integral for the formula). Recall that for term life insurance, the APV is given by
You need not simplify this integral.
General problem about insurance and stuff
Deductible
EUM customer, random loss. Condition Z
Term life insurance to all 10 year old children.
$n$ year certain and life annuity is structured in the following way. Actuarial present value.
Non homogeneous Poisson, interarrival time.
Survival function.
Surplus process, adjustment coefficient. Lundberg’s inequality.
Solutions and rubric may be found here
(15 pts) Let $X$ be a Pareto random variable. Its tail function is given by for $\alpha > 1$,
(15 pts) Suppose a customer has wealth $$1$ (he’s a student at Utah), and faces a random loss $\xi$. The loss happens with probability $q$ and if it happens, it’s gamma distributed $\GammaDist(3)$. In other words, where $X \sim \GammaDist(3)$.
Recall that $\GammaDist(\alpha)$ has the following density function.
(10 pts) An insurance company is trying to decide on the best kind of payout/payment function $r(X)$, where $X$ is a random loss that a customer might experience. To maximize their profits, the company requires that on average, they pay $$500$ on each claim: $\E[r(X)] = 500$.
(15 pts) Let $N$ be Poisson with parameter $\lambda$, and $X_1, X_2, X_3, \ldots$ be iid Normal($0,1$) random variables. Assume that the $N$ and the $X_i$ are all independent. Then let Compute $\E[S_N], \operatorname{Var}(S_N), M_{S_N}(t)$. A useful formula for the variance is where $\Var(S_N|N) = \E[S_N^2 |N] - \E[ S_N | N]^2$.
Topics to be covered:
(25 points total) (EUM)
Suppose that a customer has wealth $1000, faces a random loss $\xi$, and uses utility function $u_1(x) = -e^{-x/50}$ to make investment decisions. At the same time, the insurance company has wealth $100,000, and uses utility function $u_2(x) = -e^{-x/500}$ to make investment decisions. If $\xi$ is exponential with mean $250:
(a) What is the maximum amount that the customer will pay for the insurance?
(b) What is the minimum amount that the insurance company will accept to insure the loss $\xi$?
Solution
Let $g_{\max}$ be the maximum premium that the customer is willing to pay. Then Let’s compute the expected utility in case the person is uninsured. This means that if the person his uninsured, his expected utility is always $-\infty$. But whatever $g_{\max}$ is his expected utility is a finite number if he’s insured. This would always be bigger than $-\infty$. So this means that he’d be willing to pay whatever premium the insurance company might charge. This is unrealistic, but it’s what the EUM criterion tells us nevertheless.
For part $(b)$ we do the same thing, but with Here $h_{\min}$ is the smallest premium that the insurance company would accept.
(25 points total) (Propotional insurance)
If $Z$ is a Normal random variable with mean $\mu$ and variance $\sigma^2$ then $e^{Z}$ is called a log-normal random variable with parameters $\mu$ and $\sigma^2$. Suppose $e^Z$ is a random loss, and that a customer is covered by a proportional insurance policy that pays $k e^{Z}$ where $k < 1$. Show that, given that the loss occurred, the actual payment is also log-normal. Calculate the parameters of the payment. Are they smaller or larger than the parameters for the loss $e^Z$?
Solution We may write the payout as
But $Z + \log k \sim N(\mu + \log k, \sigma^2)$ which means that $Y$ is log normally distributed.
(25 points total) (EUM criteria)
Jim makes his investment decisions via expected utility maximization and uses utility function $u(x) = \sqrt{x}$. Right now he is trying to choose between stock A and stock B. Both are currently valued at $100. Stock A has a 50% chance of going up by 10% and a 50% chance of going down by 10%. Similarly, stock B has a 50% chance of going up by 12% and a 50% chance of going down by 12%.
Using his EUM criteria, which stock should Jim prefer?
What is the certainty equivalent for stock $A$ and what is the one for stock $B$? Does this also tell you which stock he should prefer?
If stock B was currently sitting at $101 instead then which one would he prefer?
(25 points total) (Basic conditional probability)
A company insures the cost of injuries for a group of customers. The probability that a particular customer will be injured is $0.05$. The cost incurred by a customer is random but always has a loglogistic distribution. This is a random variable $\xi$ whose cdf is given by
The values $\theta > 0$ and $\gamma > 0$ are parameters. Now for 40% of the injuries their financial cost follows a loglogistic distribution with $\theta = 5$ and $\gamma = 3$. For the remaining 60% of the injuries the financial cost follows a loglogistic distribution with $\theta = 3$ and $\gamma = 2$. The deductible on each policy is 6.
(a) Given that an injury occurs, what is the probability that it will result in a claim?
(b) What is the probability that a particular customer will make a claim?
Solution
Let $L$ be the event that loss occurs. Let $\xi$ be the loss of a customer. Let $X_1$ be loglogistic with parameters $\theta=5,~\gamma=3$, and let $X_2$ be loglogistic with $\theta=3,~\gamma=2$. They represent the losses due to injury type 1 and injury type 2. Let $T_1$ and $T_2$ be the events the injuries of type 1 and type 2 have occured respectively.
The customer does not make claim if $\xi \leq 6$, since $6$ is the deductible.
In the last step, you will have to substitute for $\Prob(X_i \geq 6) = 1 - \Prob(X_i \leq 6)$ for $i=1,2$ using what you’ve been given about the loglogistic function.
| Finally, we need to find $\Prob(\xi \geq 6) = \Prob(\xi \geq 6 | L)\Prob(L)$ since this represents the probability that the customer makes a claim. We know $\Prob(\xi \geq 6 | L)$ from the previous. $\Prob(L) = 0.05$ is given in the problem. |
(25 points total) (MGFs, convolution, independence)
(a) Find the distribution of the sums of the following independent random variables:
**Solution** We did something very similar on the HW, so it's left to you.
(b) Let $N$ be Poisson with parameter $\lambda$, and $X_1, X_2, X_3, \ldots$ be iid Normal($0,1$) random variables. Assume that the $N$ and the $X_i$ are all independent. Then let Compute $\E[S_N], \operatorname{Var}(S_N), M_{S_N}(t)$.
**Solution** Again we did all three problems in class by conditioning on $N$.
Topics covered: Probability review and basic conditional probability. Mean-variance criteria, value-at-risk, monotonicty/strict-monotonicity of criteria. Convex and concave functions, utility functions.
Consider two random variables $X$ and $Y$. $X$ is discrete and takes values in ${1,\ldots,20}$ with uniform probability. We have twenty random variables $Y_1,\ldots,Y_{20}$, independent of $X$. Each $Y_i$ is normally distributed with mean $i$ and variance $1$. We know that $Y = Y_i$ if $X = i$.
Compute the conditional distribution (cdf) of $Y$ given $X$. That is, for each $i=1,\ldots,20$, compute
Solution:
Compute the expectation of $X$ and $Y$.
Solution: Clearly
By conditioning, we clearly have The tower property gives
Compute the covariance between $X$ and $Y$. That is, compute
Hint: It might be easy to condition on $X$ to compute this. You do not need to simplify expressions, as long as you have the correct idea. You may write all answers in terms of the standard normal distribution’s cdf $\Phi(x)$ or its pdf $f(x) = (2\pi)^{-1/2} \exp(-x^2/2)$.
Solution: Again using the tower property
Suppose you have 10 independent assets whose returns $X_1,\ldots, X_{10}$ that are approximately normally distributed with mean $10$ and variance $1$. In other words, if you invest $10$ dollars in investment $i$, you will get a return of $10 X_i$ dollars. Suppose you have $10$ billion dollars to begin with. Compare the following two strategies using the value-at-risk criterion (Use $q_{0.05}$):
Recall that
Proceed by answering the following questions
Solution:
The question is a little ambiguous. So I’ll accept both of the following answers. In units of billions
Answer 1
Answer 2
Let me do the first case. The second case is similar. Notice that $X_1 - 1$ is $N(9,1)$ distributed. Therefore $S_1 \sim N(90,100)$ since
Similarly
Since the normal distribution is continuous, we have $q_{0.05}(S_1) = s^*$ where
There is a number $t^$ such that $\Prob(N \leq t^) = 0.05$ where $N$ is a standard normal distribution. We know from the tables or otherwise $t^* = -1.64$ (you need not remember this number $t^*$). So if we standardize $S_1$, we get
This gives us
Repeating the computation for $S_2$ we get
Suppose your happiness is proportional to the amount of money you earn at the end of the year, but only until $200,000. If you earn any more than $200,000, your happiness neither increases nor decreases. Let $u(x)$ be your “happiness” or utility function, where $x$ is the amount of money you earn.
Solution Draw a graph that looks like this
<img src=http://www.math.utah.edu/~arjunkc/wordpress/wp-content/uploads/plot_for_midterm1.png style=”width:500px”>