 Let's look at question 2 where it says the chief executive officer of a company benefits from an executive reward plan that includes company shares currently worth 100,000 euros. The shares currently trade at 1 euro each. The CEO wishes to retire in 4 years time and hopes that the share fund value at that time will be at least a target value of 150,000 euros. The share price ST at time T measured in years follows the stochastic differential equation where WT is a standard Brownian motion and the surplus amount is defined as the difference between the shares fund values in 4 years time and the CEO's target value. What we need to do is calculate the standard deviation of the surplus amount. And what we need to maybe do is just take a step back and remember that if a random variable let's call it X follows a log normal distribution then we know from the actuarial formula book that the variance is given by the following formula. Now we also know from the theory that the share price is distributed log normally. And we have this from our notes and we can therefore see that this equates to what we've been given in the question. And I've kind of taken the green chunk to link it up to that 0.6875 and the purple you see goes to 0.25 that's the sigma and then sometimes we'll see Brownian motion it's sometimes to know by ZT WT BT there's quite a lot of different notations that can be used. But the important thing here is that we're able to identify what is the sigma and it's going to be 0.25. We don't need to figure out mu on its own because whenever we're going to be calculating and needing mu it's going to be in that full expression which is including the minus half sigma squared which we know from the question is 0.06875. I've just shown the colors to show how this links up and of course S0 is falling away because we're told that the current share price is 1 euro. So because we know this and we know the formula above we know that the variance of the stock price at time t is going to be equal to the expected stock price at time t squared times the exponential of 0.25 squared t minus 1. Which means we first need to look at the expected value of the share price at time t which is given by the following formula and we know that t is going to be 4 because that's what the question is selling us the CEO is retiring in 4 years time and we get 1.49182 and you should be thinking to yourself hold on the CEO wanted to get 150,000 euros which means they needed the share price to go up to 1.5 or more. This actually looks like it's quite possible the fact that it's going at 1.49. So you want to be just thinking about the numbers while you're doing these calculations and just making sure that they make sense. Now that we have the expected value we can plug that in and we can get our variance for the stock price in 4 years time. Now what we need to do is we can look at it relative to the target. So we have the variance of 100,000 times the share price in 4 years time minus 150 that's the goal amount and because it's the variance and we're dealing with the constant if it's constant on its own it falls away if it's constant attached to a random variable it's going to get squared. So we basically go through these two steps and we've got that 100,000 squared times the variance of S4. We know what the variance of S4 is because we've just calculated it which means if we want to figure out the standard deviation we're going to take this expression take the square root of it and we see that we get our final answer of 79,505 euros. Now that is the first part of the question. The second part says the CEO is considering buying put options on the shares to protect against the risk that the share price is lower than required at retirement. We now need to suggest four issues that would need to be considered before deciding whether to proceed with this hedge. And this is a classic actuarial exam question where the first chunk is very, very mathematical but the second part requires that actuarial judgment and discernment and this is something that we all need to do and you don't want to leave out these last two marks because this could be the difference between passing and failing. So you definitely want to also answer the judgment questions. So yeah what are some issues that would need to be considered before deciding whether to proceed with this hedge. And the first thing is it's all great going out and buying a put option but what is the price of the premium? Maybe it's too expensive you know is this really worth it? And every time you take out an option or a derivative that is hedging you also lose a bit of the upside and some of that upside is going to be lost to cover the price of the premium and that is one thing that you definitely want to consider. Now you also want to consider whether these put options on this type of share even exist on an exchange. And they're saying that a lot of students take for granted they just assume that every single possible derivative that could exist does exist on listed exchanges and that's not always the case. If this is a very, very small company this isn't going to be necessarily the case. Which means you might have to look at doing it over the counter but whenever you're trading derivatives over the counter then there's the counterparty risk that also needs to be considered. Now we also want to look at who's buying these derivatives. We're dealing with a CEO. Now first off does the CEO know how to trade derivatives especially if this isn't a financial company and it might be something else you know it's important to appreciate that derivatives are sophisticated instruments and they do require a little bit of expertise and experience before you just jump in and start playing around with them. But then there's a more serious consideration. Is the CEO allowed to trade derivatives in his own company? Especially a put option because the put option gains if the company share price tanks which means the shareholders are going to be really, really concerned if the CEO is going to make a lot of money if the company fails and the CEO is in a position that can influence that possibility. So that's a very big consideration that you may want to bring up. Look this question is only for two marks you only need it to bring up four and I mean there are all the classic other ones that you can talk about you know the fees and the taxes and other regulations and all those other things but I think this was the critical one was to realize whether a CEO should be able to trade a derivative especially if it gains if the company suffers and that concludes question two.