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Ito’s lemma is like the first few terms of a Taylor series. It can be informally derived by taking the Taylor expansion of the function up to its...

Why do you think a risk discount rate would always be higher than the risk-free rate? (It need not be, which I believe solves your confusion.)

No, it is not. Bear in mind what you’re reading is on the internet.

Yep.

It would actually be pretty unintuitive if positive homogeneity did not hold - that's why it's a good property for a risk measure, and why it's a...

You can be 100% confident that you won’t get a loss bigger than 20,000 because the probabilities there add up to 1.

Assets are things we own or things we are owed. If we use £10 of cash to buy £10 of some other asset, we haven’t made a profit or loss. We’ve just...

There is a good discussion here about it: https://quant.stackexchange.com/questions/16693/why-is-brownian-motion-merely-almost-surely-continuous...

In my example I tried to keep things simple by saying the risk-free rate would not change, so there will be absolutely zero market risk for the...

What you described is a decent rule of thumb for most 'nice' probability distributions, in particular continuous ones, but I wouldn't be confident...

Each cash-flow needs to be discounted using an appropriate risky rate, otherwise (if you discount everything at risk-free) you need to make...

Some of the what you wrote sounds sort of right but I think it's a bit muddled. I'll try to help, but this is partly for my own benefit as I...

Intuitively: Asset A is less likely to experience big negative returns than asset B, since it is positively skewed, even though the mean and...

In the event of a very large loss (e.g. an SCR-sized one), you might be able to put a deferred tax asset (or reduce the size of a deferred tax...

Question 1 Recall that P{A|B} = P{A, B} / P{B} = 1 / P{B} * P{A,B}. Here, B is X < -VaR_a, therefore 1 / P{B} = 1 / a. This is where the factor of...

The key phrase is "They aim to link default events explicitly to the fortunes of the issuing corporate entity". In other words, they are explicit...

You have used different interest rates for different payments. Is that correct? Your overall method sounds fine to me.

The position is, in a nutshell: most people who work outside insurance / pensions aren’t very aware of actuaries. People who are qualified or...

I’m afraid I don’t have any tips to share. It’s a bit unfortunate, but these are the standard parameters in the literature about this stuff....

The MGF of a random variable X is E[exp(tX)]. X is normal if and only if exp(tX) is lognormal. Hence the link between the mean of a lognormal and...

My guess is the word "actual" is to differentiate the ex-post returns which are actually realised from the ex-ante expected returns. Cash and...

You will make a lower margin from those customers, but not necessarily a loss. (Also even if you make an underwriting loss you can make some...

There isn’t a correct answer here, it’s not black and white. It’s both an advantage of insurance that some people subsidise others and also a...

The first equation holds (perhaps have a go at proving it). The second doesn’t, e.g. say we’re rolling a dice, and A is ”lands on 1” and B is...

You can create a probability transition matrix P from a transition rate matrix Q, but not vice versa. Think about it like this: if you know the...

You’re confusing different notation here. The mu and sigma in the notes’ formula for E[St] are the drift and volatility parameters for a Brownian...

It means you need to choose how much of B to hold and how much of C to hold such that the resulting portfolio is immunised. This will involve...

Response variable distribution choice There are 'standard' choices for choice of distribution based on the properties you expect your response...

Perhaps notation is getting you bogged down a bit here. When you write E[E[Y|X]], the two expectations are being taken over different variables....

All of which is to say yes in practice swap trading is now very low on credit risk (there is probably close to zero), but that’s because the swaps...

I’ve read this before, but not sure it’s correct. If you enter into a swap agreement then the curve changes massively in your favour, it’s not...

This piece of core reading smells like it was taken from regulation: https://www.eiopa.europa.eu/rulebook/solvency-ii/article-2457_en So the...

Also, a tutor (the man, the myth, the @Ian Senator) can probably chip in with more authority here, but I don't think this is a great topic to...

Yes you're correct, neither standard deviation nor variance satisfy translation invariance or monotonicity. If X~N{2,100} then the standard...

I don't know what the core reading says, but monotonicity does not say that if one portfolio starts more valuable than another then the risk for...

These conditions are all about modelling the future returns of the portfolio as random variables. Your mistake is only thinking about the initial...

I have no idea what you’re referring to as I don’t have any of the notes, but it sounds like perhaps x in the course notes is referring to the sum...

This is actually a very interesting point. Maximising the variance here is exactly the same thing as minimising the projection error, and it boils...

Yeah I see your point but I think you’re missing the case where the rate changes over time. If you fully specify a formula for the force of...

When I said “We can ignore the k because it's just a constant multiplier, and focus on when exp((t-2)x) will converge.” above I meant the integral...

We say that an MGF for X exists if E[exp(tX)] converges for t in some neighbourhood of zero. i.e. there exists some (potentially very small)...

I'm afraid not, as I'm not sure which fraction you're talking about or the context or the question. Happy to help if you type out more details.

I don't have the course notes, but there is no requirement that MGFs converge. There isn't even a requirement for an MGF to exist :)

Yes, under CAPM all investors have the same market expectations hence hold the same market portfolio, so it must be market cap weighted. I don’t...

I think you’re confusing terms here. You’ve said that X and Y are the “risks” of two portfolios, i.e. the results of calculating a coherent risk...

Just had a look at the examiner's report (I assume that's what you mean by the solution?) That's not what it says. It says that Cov(Y_t,...

Your interpretation is basically right. In practice if a book of business gets significantly bigger you would probably change the distribution...

Time value can be negative for puts but not calls because your upside is capped on a put but not on a call. Your payoff on a put can be at most...

What I said here was poorly typed and confusing. I should have said: So, in conclusion, when the CMP said futures have no time value and hence...

If a market is not weak form efficient, that means you can use information from past prices to make money trading the markets. That would...

Your maths is right, but the way you're using the terminology doesn't line up with the way people speak about this stuff in practice. Theta is...

Market risk quant

Your formulas for normal and Poisson X+Y hold if X and Y are independent, but not in general. That’s the source of your confusion. Read this...

There are definitely lists in the core reading where there are essentially duplicate points, and points which are sub-points of each other. I...

Carry back: Company makes £100 last year, so owes £20 tax (assuming 20% rate). If it loses £10 the next year it might carry back the loss,...

> Why would the PV of fixed rate payments = PV of floating rate payments? Assuming this is talking about interest rate swaps: The fixed rate is...

It could be paid out as a dividend to shareholders, or the company could retain these earnings on the balance sheet. I.e. they’ve made a profit,...

Some more (perhaps contrived but hopefully illustrative) examples of when you might change reserves and/or prices: If regulation is announced...

If you’re trying to price something using a costs plus approach, this is quite similar to some parts of reserving. In both you are trying to...

Capital can be used to mean either of those things. It can have a pretty broad array of meanings, some of which are themselves very broad. In my...

Reserves are a liability item on the balance sheet. If you increase the value of the reserves on the end of period balance sheet, it will decrease...

Yes. Some roles in London won’t pay that much but it’s certainly possible to get newly qualified life roles in London paying £70k+.

It varies based on location (in London or not). You'd want somewhere between £60k and £75k though.

Unless the acted notes have changed a lot since I took the exams, I seem to recall there being quite a good explanation of filtrations in there. A...

The covariance of independent random variables is 0. The independent increments property of Brownian motion hence tell us the covariance of dW_t...

The way the word 'stationary' is used usually means 'strictly stationary' in my experience. Do bear in mind stationary is not the same thing as...

The integral is shown to be equal to the sum of 2 independent random variables with distributions N(0, 0.5) and N(0, 2). You know they are...

A stationary process is one where the joint distribution doesn't change through time. An i.i.d. stochastic process has the same joint distribution...

For upper/lower tail dependence measures and 'coefficients', see sections 11.6 and 11.8 here:...

I should again re-iterate: I have no idea whether you would get any marks in CS2 for any of these sorts of points (I almost suspect not!). Just...

Statistical fitting A copula can be fit to data just like any other CDF of a random variable. E.g. given a vector of observations x_1, x_2, ...,...

This is perhaps outside the scope of your exam, but I thought it might be useful add a few factors which inform how people generally pick a copula...

Consider the lifetime of a claim on a policy: 1) the policy is underwriting 2) an accident happens 3) a claim is reported to the insurer in...

yep. SP6 is what you’re after if you’re looking for the most quantitative SP. The rest have quantitative elements but are mainly qualitative.

No problem. Why is it not helpful? Do you not understand why the binomial theorem is true? I assumed you'd be ok with that, but you just hadn't...

See here: https://en.wikipedia.org/wiki/Binomial_theorem#Statement It's just a standard piece of maths you'd be expected to spot, much like...

Under Solvency II you'd profit $5m straight away. You'd record $10m of premiums as either cash if received up front, or as cash inflows as part of...

I don't have the reading I'm afraid, thought I'd try and point you in the right direction anyway. XY will be the product of two random variables...

You aren't given information about the specifics of utility for the consumer so you should just sketch an indifference curve, no need to do any...

It's probably already clear to you from Steve's explanation above, but for clarity's sake: the hedging and replicating approaches to valuation of...

Have you had a go at this yourself yet? If so, where did you get stuck? Starting with (i) for example - do you know what a budget line is? (The...

UPR is the unearned portion of your premium. Under GAAP accounting you hold this on your balance sheet as a liability, because even though you've...

It doesn’t really make sense for the capital market line to pass above the efficient frontier. It gives the risk-return profile of the risk free...

You assume all the investors are rational and risk averse, so within CAPM they’re all going to come to the same conclusions about what to hold....

So, you’re saying that if we don’t write any new business *and* we don’t revalue our reserves at all, then the new reserve is just the part of the...

That's actually not the case. For example, if when I set the reserves at 1/1/20 I thought there was a 25% chance of a £100 claim on 1/1/2030, I...

You just explained yourself why the company doesn’t make a profit or loss immediately on purchasing a building with credit. There is both an...

P&L just breaks down the different between one balance sheet and the next. An increase (decrease) in assets is a profit (loss) and a decrease...

I'll paste an answer I've made before about assets, liabilities and capital. This might help clear some things up for you. It's a question that...

Let's say I raise £10m from investors, who have all bought shares, to start an insurance company. Now I have a balance sheet with £10m of cash on...

I think there is plenty of time to learn R sufficiently well between now and September in order to pass the exam, starting from scratch.

It sounds to me like we're not given enough information to get to a definitive answer here, so there are a few options open to us. But we can make...

Not a brilliantly worded line. Let's break it down: (1) "a low value of rho" (2) "reduces risk" (3) "relative to" (4) "uncertainty in the...

Any of that's fine, doesn't make much difference to most actuarial employers. Also they don't really care about exemptions, but if you'd like...

I don't have the course notes, but thought I'd try to help anyway. The variance V of a distribution is an integral (or sum, same thing) ranging...

Which are you more interested in: investment or health? Choose the one you're most interested in learning about. If you're not interested in...

Your steps look right. The point of hedging is to reduce risk, and standard deviation of the strategy is one way to measure the risk involved. So...

I agree with you that some of those points aren't addressing the question precisely. I would argue stress and scenario testing quite directly...

This doesn't directly address the question, but I saw a good post on LinkedIn recently, which included a visualisation of insurance accounting...

I've known people who mark the papers and discussed things like this with them. This shouldn't cause you any issues, don't worry about it.