Tag Archives: Sum of IID

Approximating Aggregate Losses

An aggregate loss S is the sum of all losses in a certain period of time.  There are an unknown number N of losses that may occur and each loss is an unknown amount X.  N is called the frequency random variable and X is called the severity.  This situation can be modeled using a compound distribution of N and X.  The model is specified by:

\displaystyle S = \sum_{n=1}^N X_n

where N is the random variable for frequency and the X_n‘s are IID random variables for severity.  This type of structure is called a collective risk model.

An alternative way to model aggregate loss is to model each risk using a different distribution appropriate to that risk.  For example, in a portfolio of risks, one may be modeled using a pareto distribution and another may be modeled with an exponential distribution.  The expected aggregate loss would be the sum of the individual expected losses.  This is called an individual risk model and is given by: 

\displaystyle S = \sum_{i=1}^n X_i

where n is the number of individual risks in the portfolio and the X_i‘s are random variables for the individual losses.  The X_i‘s are NOT IID, and n is known.

Both of these models are tested in the exam; however, the individual risk model is usually tested in combination with the collective risk model.  An example of a problem structure that combines the two is given below.

Example 1: Your company sells car insurance policies.  The in-force policies are categorized into high-risk and low-risk groups.  In the high-risk group, the number of claims in a year is poisson with a mean of 30.  The number of claims for the low-risk group is poisson with a mean of 10.  The amount of each claim is pareto distributed with \theta = 200 and \alpha = 2.
Analysis: Being able to see the structure of the problem is a very important first step in being able to solve it.  In this situation, you would model the aggregate loss as an individual risk model.  There are 2 individual risks– high and low risk.  For each group, you would model the aggregate loss using a collective risk model.  For the high-risk, the frequency is poisson with mean 30 and the severity is pareto with \theta = 200 and \alpha = 2.  For the low-risk group, the frequency is poisson with mean 10 and the severity is pareto with the same parameters.

For these problems, you will need to know how to:

  1. Find the expected aggregate loss.
  2. Find the variance of aggregate loss.
  3. Approximate the probability that the aggregate loss will be above or below a certain amount using a normal distribution.  
    Example: what is the probability that aggregate losses are below $5,000?
  4. Determine how many risks would need to be in a portfolio for the probability of aggregate loss to reach a given level of certainty for a given amount.
    Example: how many policies should you underwrite so that the aggregate loss is less than the expected aggregate loss with a 95% degree of certainty? 
  5. Determine how long your risk exposure should be for the probability of aggregate loss to reach a given level of certainty for a given amount.

Problems that require you to determine probabilities for the aggregate loss will usually state that you should use a normal approximation.  This will require the calculation of the expected aggregate loss and the variance of the aggregate loss.

MEMORIZE
Expected aggregate loss for a collective risk model is given by:

E[S] = E[N]E[X]

For the individual risk model, it is

\displaystyle E[S] = \sum_{i=1}^n E[X_i]

Variances under the collective risk model are conditional variances.

Var(S) = E[Var(X|I)] + Var(E[X|I])

When frequency and severity are independent, the following shortcut is valid and is called a compound variance:

Var(S) = E[N]Var(X) + Var(N)E[X]^2

Variance under the individual risk model is additive:

\displaystyle Var(S) = \sum_{i=1}^n Var(X)

Example 2: Continuing from Example 1, calculate the mean and variance of the aggregate loss.  Assume frequency and severity are independent.
Answer: This is done by

  1. Calculating the expected aggregate loss and variance in the high-risk group.
  2. Calculating the expected aggregate loss and variance in the low-risk group.
  3. Adding the expected values from both groups to get the total expected aggregate loss.
  4. Adding the variances from both groups to get the total variance.

I will use subscript H and L to denote high and low risk groups respectively.

E[S_H] = E[N_H]E[X_H] = 30\times 200 = 6,000

\begin{array}{rll} Var(S_H) &=& E[N_H]Var(X_H) + Var(N_H)E[X_H]^2 \\ &=& 30 \times 40,000 + 30 \times 200^2 \\ &=& 2,400,000 \end{array}

E[S_L] = E[N_L]E[X_L] = 10 \times 200 = 2,000

\begin{array}{rll} Var(S_H) &=& 10 \times 40,000 + 10 \times 200^2 \\ &=& 800,000 \end{array}

Add expected values to get

E[S] = 6,000 + 2,000 = 8,000

Add variances to get

Var(S) = 2,400,000 + 800,000 = 3,200,000

Once the mean and variance of the aggregate loss has been calculated, you can use them to approximate probabilities for aggregate losses using a normal distribution.

Example 3: Continuing from Example 2, use a normal approximation for aggregate loss to calculate the probability that losses exceed $12,000.
Answer:  To solve this, you will need to calculate a z value for the normal distribution using the expected value and variance found in Example 2.

\begin{array}{rll} \Pr(S > 12,000) &=& 1- \Pr(S< 12,000) \\ \\ &=& \displaystyle 1-\Phi\left(\frac{12,000 - 8,000}{\sqrt{3,200,000}}\right) \\ \\ &=& 1 - \Phi(2.24) \\ \\ &=& 0.0125 \end{array}

CONTINUITY CORRECTION
Suppose in the above examples the severity X is discrete.  For example, X is poisson.  Under this specification, we need to add 0.5 to 12,000 in the calculation for \Pr(S > 12,000).  So we would instead calculate \Pr(S > 12,000.5)  This is called a continuity correction and occurs when we have a discrete severity random variable.  If we were interested in \Pr(S<12,000), we would subtract 0.5 instead.  This has a greater effect when the domain of possible values is smaller.

Another type of problem I’ve encountered in the samples is constructed as follows:

Example 4: You drive a 1992 Honda Prelude Si piece-of-crap-mobile (no, that’s my old car and you are driving it because I sold it to you to buy my Mercedes).  The failure rate per year is poisson with mean 2.  The average cost of repair for each instance of breakdown is $500 with a standard deviation of $1000.  How many years do you have to continue driving the car so that the probability of the total maintenance cost exceeding 120% of the expected total maintenance cost is less than 10%?  (Assume the car is so crappy that it cannot deteriorate any further so the failure rates and average repair costs remain constant every year.)
Answer:  For one year,

E[S_1] = 1,000

\begin{array}{rll} Var(S_1) &=& 2 \times 1,000^2 + 2 \times 500^2 \\ &=& 2,500,000 \end{array}

For n years, we have

E[S] = 1,000n

Var(S) = 2,500,000n

According to the problem, we are interested in S such that \Pr(S > 1,200n) = 0.1.  Under normal approximation, this implies

\begin{array}{rll} \Pr(S>1,200n) &=& 1-\Pr(S<1,200n) \\ \\ &=& \displaystyle 1- \Phi\left(\frac{1,200n - 1,000n}{\sqrt{2,500,000n}}\right) \end{array}

Which implies

\displaystyle \Phi\left(\frac{200n}{\sqrt{2,500,000n}}\right) = 0.9

The probability 0.9 corresponds to a z value of 1.28.  This implies

\displaystyle \frac{200n}{\sqrt{2,500,000n}} = 1.28

Solving for n we have n = 1024 years.  LOL!

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The Lognormal Distribution

Review: If X is normal with mean \mu and standard deviation \sigma, then

Z = \displaystyle \frac{X-\mu}{\sigma}

is the Standard Normal Distribution with mean 0 and standard deviation 1.  To find the probability Pr(X \le x), you would convert X to the standard normal distribution and look up the values in the standard normal table.

\begin{array}{rll} Pr(X \le x) &=& Pr\left(\displaystyle \frac{X-\mu}{\sigma} \le \frac{x-\mu}{\sigma}\right) \\ \\ &=& \displaystyle Pr\left(Z \le \frac{x-\mu}{\sigma}\right) \\ \\ &=& \displaystyle \mathcal{N}\left(\frac{x-\mu}{\sigma}\right) \end{array}

If V is a weighted sum of n normal random variables X_i, i = 1, ..., n, with means \mu_i, variance \sigma^2_i, and weights w_i, then

\displaystyle E\left[\sum_{i=1}^n w_iX_i\right] = \sum_{i=1}^n w_i\mu_i

and variance

\displaystyle Var\left(\sum_{i=1}^n w_iX_i\right) = \sum_{i=1}^n \sum_{j=1}^n w_iw_j\sigma_{ij}

where \sigma_{ij} is the covariance between X_i and X_j.  Note when i=j, \sigma_{ij} = \sigma_i^2 = \sigma_j^2.

Remember: A sum of random variables is not the same as a mixture distribution!  The expected value is the same, but the variance is not.  A sum of normal random variables is also normal.  So V is normal with the above mean and variance.

Actuary Speak: This is called a stable distribution.  The sum of random variables from the same distribution family produces a random variable that is also from the same distribution family.

The fun stuff:
If X is normal, then Y = e^X is lognormal.  If X has mean \mu and standard deviation \sigma, then

\begin{array}{rll} \displaystyle E\left[Y\right] &=& E\left[e^X\right] \\ \\ \displaystyle &=& e^{\mu + \frac{1}{2}\sigma^2} \\ \\ Var\left(e^X\right) &=& e^{2\mu + \sigma^2}\left(e^{\sigma^2} - 1\right)\end{array}

Recall FV = e^\delta where FV is the future value of an investment growing at a continuously compounded rate of \delta for one period.  If the rate of growth is a normal distributed random variable, then the future value is lognormal.  The Black-Scholes model for option prices assumes stocks appreciate at a continuously compounded rate that is normally distributed.

S_t = S_0e^{R(0,t)}

where S_t is the stock price at time t, S_0 is the current price, and R(0,t) is the random variable for the rate of return from time 0 to t.  Now consider the situation where R(0,t) is the sum of iid normal random variables R(0,h) + R(h,2h) + ... + R((n-1)h,t) each having mean \mu_h and variance \sigma_h^2.  Then

\begin{array}{rll} E\left[R(0,t)\right] &=& n\mu_h \\ Var\left(R(0,t)\right) &=& n\sigma_h^2 \end{array}

If h represents 1 year, this says that the expected return in 10 years is 10 times the one year return and the standard deviation is \sqrt{10} times the annual standard deviation.  This allows us to formulate a function for the mean and standard deviation with respect to time.  Suppose we write

\begin{array}{rll} \displaystyle \mu(t) &=& \left(\alpha - \delta -\frac{1}{2}\sigma^2\right)t \\ \sigma(t) &=& \sigma \sqrt{t} \end{array}

where \alpha is the growth factor and \delta is the continuous rate of dividend payout.  Since all normal random variables are transformations of the standard normal, we can write R(0,t) =\mu(t)+Z\sigma(t) . The model for the stock price becomes

\displaystyle S_t = S_0e^{\left(\alpha - \delta - \frac{1}{2}\sigma^2\right)t + Z\sigma\sqrt{t}}

In this model, the expected value of the stock price at time t is

E\left[S_t\right] = S_0e^{(\alpha - \delta)t}

Actuary Speak: The standard deviation \sigma of the return rate is called the volatility of the stock.  This term comes from expressing the rate of return as an Ito process. \mu(t) is called the drift term and \sigma(t) is called the volatility term.

Confidence intervals: To find the range of stock prices that corresponds to a particular confidence interval, we need only look at the confidence interval on the standard normal distribution then translate that interval into stock prices using the equation for S_t.

Example: For example z=[-1.96, 1.96] represents the 95% confidence interval in the standard normal \mathcal{N}(z).  Suppose t = \frac{1}{3}, \alpha = 0.15, \delta = 0.01, \sigma = 0.3, and S_0 = 40.  Then the 95% confidence interval for S_t is

\left[40e^{(0.15-0.01-\frac{1}{2}0.3^2)\frac{1}{3} + (-1.96)0.3\sqrt{\frac{1}{3}}},40e^{(0.15-0.01-\frac{1}{2}0.3^2)\frac{1}{3} + (1.96)0.3\sqrt{\frac{1}{3}}}\right]

Which corresponds to the price interval of

\left[29.40,57.98\right]

Probabilities: Probability calculations on stock prices require a bit more mental gymnastics.

\begin{array}{rll} \displaystyle Pr\left(S_t<K\right) &=& Pr\left(\frac{S_t}{S_0} < \frac{K}{S_0}\right) \\ \\ \displaystyle &=& Pr\left(\ln{\frac{S_t}{S_0}} < \ln{\frac{K}{S_0}}\right) \\ \\ \displaystyle &=& Pr\left(Z< \frac{\ln{\frac{K}{S_0}} - \mu(t)}{\sigma(t)}\right) \\ \\ \displaystyle &=& Pr\left(Z<\frac{\ln{\frac{K}{S_0}} - \left(\alpha - \delta - \frac{1}{2}\sigma^2\right)t}{\sigma\sqrt{t}}\right) \end{array}

Conditional Expected Value: Define

\begin{array}{rll} \displaystyle d_1 &=& -\frac{\ln{\frac{K}{S_0}} - \left(\alpha - \delta + \frac{1}{2}\sigma^2\right)t}{\sigma\sqrt{t}} \\ \\ \displaystyle d_2 &=& -\frac{\ln{\frac{K}{S_0}}- \left(\alpha - \delta - \frac{1}{2}\sigma^2\right)t}{\sigma\sqrt{t}} \end{array}

Then

\begin{array}{rll} \displaystyle E\left[S_t|S_t<K\right] &=& S_0e^{(\alpha - \delta)t}\frac{\mathcal{N}(-d_1)}{\mathcal{N}(-d_2)} \\ \\ \displaystyle E\left[S_t|S_t>K\right] &=& S_0e^{(\alpha - \delta)t}\frac{\mathcal{N}(d_1)}{\mathcal{N}(d_2)} \end{array}

This gives the expected stock price at time t given that it is less than K or greater than K respectively.

Black-Scholes formula: A call option C_t on stock S_t has value \max\left(0,S_t - K\right) at time t.  The option pays out if S_t > K.  So the value of this option at time 0 is the probability that it pays out at time t, discounted by the risk free interest rate r, and multiplied by the expected value of S_t - K given that S_t > K.  In other words,

\begin{array}{rll} \displaystyle C_0 &=& e^{-rt}Pr\left(S_t>K\right)E\left[S_t-K|S_t>K\right] \\ \\ &=& e^{-rt}\mathcal{N}(d_2)\left(E\left[S_t|S_t>K\right] - E\left[K|S_t>K\right]\right) \\ \\ &=& e^{-rt}\mathcal{N}(d_2)\left(S_0e^{(\alpha - \delta)t}\frac{\mathcal{N}(d_1)}{\mathcal{N}(d_2)} - K\right) \end{array}

Black-Scholes makes the additional assumption that all investors are risk neutral.  This means assets do not pay a risk premium for being more risky.  Long story short, \alpha - r = 0 so \alpha = r.  So in the Black-Scholes formula:

\begin{array}{rll} \displaystyle d_1 &=& -\frac{\ln{\frac{K}{S_0}} - \left(r - \delta + \frac{1}{2}\sigma^2\right)t}{\sigma\sqrt{t}} \\ \\ \displaystyle d_2 &=& -\frac{\ln{\frac{K}{S_0}}- \left(r- \delta - \frac{1}{2}\sigma^2\right)t}{\sigma\sqrt{t}} \end{array}

Continuing our derivation of C_0 but replacing \alpha with r,

\begin{array}{rll} \displaystyle C_0 &=& e^{-rt}\mathcal{N}(d_2)\left(S_0e^{(r - \delta)t}\frac{\mathcal{N}(d_1)}{\mathcal{N}(d_2)} - K\right) \\ \\ &=& S_0e^{-\delta t}\mathcal{N}(d_1) - Ke^{-rt}\mathcal{N}(d_2)\end{array}

For a put option P_0 with payout K-S_t for K>S_t and 0 otherwise,

P_0 = Ke^{-rt}\mathcal{N}(-d_2) - S_0e^{-\delta t}\mathcal{N}(-d_1)

These are the famous Black-Scholes formulas for option pricing.  When derived on the back of a cocktail napkin, they are indispensable for impressing the ladies at your local bar.  :p

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Normal Approximation

If a random variable Y is normal, you can map it to a standard normal distribution X (useful for finding probabilities in the standard normal table) by the following relationship:

Y = \mu_y + \sigma_yX

Example 1:  Y is normal.  E[Y] = 100 and Var(Y) = 49  Then

\begin{array}{rl} P(Y \leq 111.515) &= P(X \leq \frac{111.515 - 100}{\sqrt{49}}) \\ &= P(X \leq 1.645) \\ &= 0.95 \end{array}

Example 2:  Y has the same distribution as example 1.  Then P(Y \leq y) = 0.9 implies 

P(X \leq \frac{y - 100}{\sqrt{49}}) = 0.9

Which implies:

\frac{y - 100}{\sqrt{49}} = 0.8159

Hence y = 105.7113.

With regard to Central Limit Theorem:

By the Central Limit Theorem, the distribution of a sum of iid random variables converges to a normal distribution as the number of iid random variables increases.  This means that if the number of iid random variables is sufficiently large, we can get approximate probabilities by using a normal distribution approximation.

 

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Variance and Expected Value Algebra

Linearity of Expected Value: Suppose X and Y are random variables and a and b are scalars.  The following relationships hold:

E[aX+b] = aE[X]+b

E[aX+bY] = aE[X] +bE[Y]

Variance:

Var(aX+bY) = a^2Var(X)+2abCov(X,Y)+b^2Var(Y)

Suppose X_i for i=\left\{1\ldots n\right\} are n independent identically distributed (iid) random variables.  Then Cov(X_i,X_j) = 0 for i\ne j and

\displaystyle Var\left({\sum_{i=1}^n X_i}\right) = \sum_{i=1}^n Var(X_i)

Example:

X is the stock price of AAPL at market close.  Y is the sum of closing AAPL stock prices for 5 days.  Then

\begin{array}{rl} Var(Y) &= \displaystyle \sum_{i=1}^5 Var(X_i) \\ &= 5Var(X) \end{array}.  

Contrast this with the variance of Z = 5X.  In other words, Z is a random variable that takes a value of 5 times the price of AAPL at the close of any given day.  Then

\begin{array}{rl} Var(Z) &= Var(5X) \\ &=5^2Var(x) \end{array} 

The distinction between Y and Z is subtle but very important.

Variance of a Sample Mean:

In situations where the sample mean \bar{X} is a random variable over n iid observations (i.e. the average price of AAPL over 5 days), the following formula applies:

\begin{array}{rl} Var(\bar{X}) &= \displaystyle Var\left(\frac{1}{n} \displaystyle \sum_{i=1}^n X_i\right) \\ &= \displaystyle \frac{nVar(X)}{n^2} \\ &= \displaystyle \frac{Var(X)}{n} \end{array} 

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