Mean and variance are going to be two of the most important statistical concepts out there. Now we've looked at different kinds of means and different kinds of variances over previous weeks. If we have the theoretical probability distribution, we could work out the expectation of a random variable x we tended to denote that by μ and this represented a useful measure of central tendency or location i.e. on average. What value do we get for this variable? And we also looked at deriving the population variance or theoretical variance denoted by Sigma squared a measure of the dispersion or spread of the values of said variable. When we introduce some simple descriptive statistics i.e. values calculated based on some observed sample data, the sample equivalence where the sample mean x bar. Add up the observations divide by the total number of observations and also the sample variance which review does the average square deviation about the sample mean. So in the previous section, we introduced a simple decision tree for this ice cream manufacturer problem and we looked at solving this by working out the expected monetary values of the different courses of action. And we simply chose that case to advertise because this led to a greater expected monetary value. But of course that problem neglected to take into account risk. So, there we were simply looking at mean, we didn't really take into account variance. So clearly, if you're an investor one often has to face the risk or return a trade off that in order to achieve a greater return, one typically has to be exposed to a greater degree of risk. So how might we actually model risk? Indeed, what is your personal attitude to risk? Well let's find out. You'll see on the screen, a simple decision tree and let's imagine this is for you to decide what you would do in this situation. So we start off with the square which remember was a decision node and you have a choice between two assets, a safe asset and a risky asset. Now let's assume this safe asset is guaranteed to give you a particular return. An example might be a government bond. Well I say that's an example. In principle, governments shouldn't default and you should be able to trust the government. But as we know some countries have had situations where they decided to default on their debts in the past. Even the US government sometimes debates in Congress about whether the debt ceiling is going to be raised and there's a potential threat of a technical default. So if you can't trust the US government, who can you trust? But regardless, we have a safe asset which will return a particular nominal sum of x here. We also have a risky asset where we either make a £100 or we make £0 and let's assume there is a 50/50 chance of these two possibilities. So it's risky because if you decide to hold it, you don't know yet what return you're going to get. Either you will make nothing or you will make a £100. And let's suppose this is determined by the performance of perhaps the the global economy. So, you can see there's an x in this decision tree. So imagine, you are trying to choose between these two assets. Now the value of x for you is the value which makes you indifferent between the safe asset and the risky asset. Now clearly, different people will have different attitudes to risk and hence your value of x is going to vary across one person to another. So let's think about this in terms of expected values. If you decide to go for the risky asset, what is your expected return? Well, we simply calculate a probability weighted average namely, probabilities times payoffs, summed across all possible options. So in this case we only have two options. Either it's a good outcome and we make a £100 or it's a bad outcome and we make £0. So in expectation, our expected return is not .5 times 100 plus not .5 times zero which will give us £50. So imagine, you were indifferent between this risky asset giving you an expected return of £50 and a safe asset also giving you a guaranteed return of £50. Now, so if your value of x was 50 then you were such that you were indifferent between the safe and risky assets, then you would be a risk neutral individual because you were simply looking at expected returns. I mean we can work out the expected turn of the safe option, it's just here, there's a guaranteed return i.e. with probability 1 that you get x. So if x is 50 then 1 times 50 is 50. So, both the safe and risky assets would lead to an expected return of £50. So if you were indifferent between those two options, then indeed you are risk neutral i.e you are a neutral, you are indifferent about risk because clearly, the safe and risky assets do vary in terms of their risk profile. There is zero risk associated with the safe asset because you have a guaranteed return of £50 where clearly there is risk associated with the risky asset, hence the name because maybe you end up with nothing but you could end up with as much as a £100. So a risk neutral investor will pay no attention to risk and simply compare different possible investment opportunities based on expected returns only. That's really the approach we adopted in the ice cream manufacturer problem in the previous section. We did not take into account the risks of getting particularly good or particularly bad outcomes. We simply based our to advertise or not to advertise decision based on expected monetary values. Now let's suppose, your value of x in this safe risky asset problem is somewhat below 50, then this would indicate that you are a risk averse individual because let's say your value of x was £40 such that then you are indifferent between these two courses of action. So that means, you would be indifferent between receiving £40 with certainty versus getting £50 in expectation by undertaking a risky activity. So, if let's say your value of x is £40, you are somewhat at risk averse. If your value of x is above £50 then you would be exhibiting some risk-loving behavior because you get excited by undertaking risk. So clearly, there are different degrees of risk aversion and risk loving behavior. And that would affect or be determined by your value of x and to what extent it deviates from the expected return of 50 for that risky asset. So if we think back to our ice cream manufacturer problem, we kept our lives very simplistic by effectively assuming risk neutrality and paying no attention to how likely we are to get the bad outcome of a low level of sales. Of course in practice in the real world, people do tend to care about risk and different people have different attitudes to risk. Some people will be very much risk loving and undertake risky activities. Others prefer to play it safe, they dislike risk. They're scared of the bad outcome and hence, they will tend to choose more risk averse situations. Recall that diagram of two hypothesized stock returns of the black and the red chairs and we saw how they have the same expected return, they have that same measure of central tendency the same mean but they did differ in terms of the dispersion of the returns that could be obtained. I may mentioned at that time depending really on your attitude to risk, you would either go for the riskiest stock giving the possibility of a very high return, but being exposed to the risk of a very negative return or if you're a more risk adverse in your investments, you would opt for the one with the smaller variants. And hence, you less likely to lose lots of money, but of course you're also less likely to make lots of money. So our risk, we are perhaps familiar with it from finance but we can easily relate it to the statistical concept of variance because variance is a very simple statistical measure but it's arguably one of the most widely used in finance to act as a quantification of risk.
