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The guarantee ensures that the customer never loses the monetary value of their investment - inflation however could be an issue after 10 years!...

There's no recognition of diversification benefits. So the capital amounts required to cover the risks from each business unit are summed together...

Looks good to me.

Yes, they're essentially the same thing. Contributions are made to fund the scheme. The scheme is funded by contributions (and investment return)....

There isn't a unique pair of portfolios that can be used, but here's how I would tackle the problem: Portfolio 1 should contain the derivative in...

The current (September 2023) IFoA Examinations Handbook says that "You cannot use functions only available in versions of Microsoft Office...

There is no requirement to do this in the current (September 2023) IFoA Examinations Handbook.

We can - that's how I would solve it.

1) The single factor models are driven by a single standard Brownian motion engine, and so there just isn't enough flexibility to generate yield...

Hi 1) The grossing-up factor gives the proportion of the ultimate claim amount that has been paid so far. 2) The grossing-up factor from Accident...

Yes, you're correct, they would make a loss on the sale of the call option if the share price is above 1.3S_t. However, the society will also have...

=NORM.S.DIST(x,TRUE)

Hi. The reason for selling the call is because of the phrase "up to a maximum level of 30% growth over the year". If the share price exceeds...

From your first expression, collect all of the Ep terms: (Ep-Eb)/(Ea-Eb) * sigmaA + (Ea-Ep)/(Ea-Eb)* sigmaB. This will give you the first half of...

Hi Your understanding is correct; P(B_{t}<0)=1/2. But if you keep zooming in and magnifying the Brownian motion around the origin, we can also say...

Hi Ep[AT*VT] gives the value at time 0 of a derivative with a payout at time T, so V0=Ep[AT*VT]. Ep[At*Vt] gives the value at time 0 of a...

Hi. You need the apply the chain rule to a(T-t), but it looks as though you've gone for the product rule instead. The question says that a is a...

Check out the question on page 5 of the Asset pricing models chapter - that covers it entirely. Let me know if you have problems with it.

The IFoA Examinations Handbook says that you MUST include ALL working as part of your answer (their caps, their bold). Using examples is great,...

Hi. The Core Reading says that "the method is not uniquely defined as using a particular form of grossing-up factors or development factors....

Here's the chain rule (function-of-a-function rule): if f(t) = g(h(t)), then df/dt = dg/dh * dh/dt. In the current question we have h(t) = a(T-t)...

The function f is given as: exp(a(T-t)-x). The partial derivative with respect to t requires the chain rule because we're taking the exponential...

Ah, I see what you mean. Given that the dividends are certain, the usual approach is to deduct their present value from the starting share price....

Consider an up-step in the binomial tree. The share price moves from S(t) to S(t)*u, but then the dividend is paid and the share price needs...

In the exam question the X represents a win, whereas in the factsheet the X represents a loss.

Check out the comment in cell H11: "The SQRT(2) term is included as the distribution of W(2) - W(0) is N(0,2), ie SQRT(2) times N(0,1)." Hope that...

Hi The return on a portfolio with two risky assets is given by: RP = xA*RA + (1-xA)*RB If the returns RA and RB are independent then the variance...

Hi You're correct to start with d < e^r < u, but this is only true when the interest rate r is continuously compounded. In this question the...

Hi The "annuity rate" is usually the percentage rate at which a pension fund can be converted into an annuity. So an 8% annuity rate means that a...

Hi I don't think that's how it works. In Accident Year X Development Year 0, the figure in the run-off triangle shows the claims paid in calendar...

That's a neat way of handling the integral! The limits do indeed remain the same. It might be clearer to say p=int(x=0;x=inf): PHI(x) dPHI(x)....

Because the return rates are list in ascending order, and we're looking for the probability of the return attaining or exceeding that level. If...

The issue of having numerous risk managers sits with the company rather than the IRC provider. IRCs are bespoke arrangements, so the company can...

Hi The required interval here is the range of share prices that could occur with 99% confidence - this excludes the 1% of extreme outcomes. But...

Hi 1. With integrated risk covers you get exactly what you want because they're bespoke products. This avoids the possibility of overlaps. 2....

Hi Eitan I understand your frustration. Unfortunately, it's not possible for us to comment on the marking of specific exam scripts. Sorry

Hi Here are couple of earlier forum posts about structural risk. Let me know if these don't clear the issue for you....

Try this. Let's say you want to buy a share (on the presumption that you've got the cash to do so), but you can choose to either purchase it now...

Hi Holding a call option means that the investor has effectively delayed the (potential) purchase of the share until the expiry date. The cash...

Another way of expressing this point would be to say that a good model can be run in a number of different modes. Maybe there's a "debug" mode...

OK, I see what you mean. Start with the definition for \(q\), and replace \(s_{up}\) and \(s_{down}\) with their respective values found on page...

The risk-neutral up-step probability \(q\) is chosen so that the share price is expected to grow that the risk-free rate over each time-step....

Hi I think I see what you mean. Let me know if I've not understood you correctly. The first term in the definition of \(P(t,T)\) includes...

Check out page 68 of Baxter and Rennie - the full explanation is given there. Briefly, \(M_t\) represents the change in the measure up to time...

Hi Thanks for pointing this out. Leptokurtic definitely refers to having higher kurtosis. I'll make sure this is corrected in the next edition of...

The market portfolio is defined as only containing risky assets.

When there's a probability of success, p, and an experiment is run n times, the number of successes follows a binomial distribution binom(n,p) by...

You’re absolutely right; a long position in the underlying share is required to get that total delta equal to zero. Unfortunately the Examiner’s...

In the IFoA solution spreadsheet, run this Goal Seek and see if you get 2,296 put options. [ATTACH]

The "ud=1" assumption only applies when there are zero dividends. When implementing dividends in the binomial model, the share price needs to be...

Rather than rolling over the whole position one go, the stated assumption in this question is to roll over 20% of the contracts each day over a...

Candidates should answer exam questions as if they were written on the day in which the exam is sat. There is no requirement to second-guess when...

The second interpretation of ruin at time t=2 is correct; it's conditional on ruin not having already occurred at time t=1.

Unfortunately these country reports are very expensive and so the links that the IFoA website provides are the only feasible ways of accessing...

"Relevant" is dependent on the scenario. In the Core Reading the word is in quotes marks to indicate that whilst the investor/consumer thinks that...

I think the examiners were just being helpful in giving the hint! The difference in approach is between whether Ito's Lemma or Taylor's formula is...

Hi Yes, the expectation of a function of a random variable follows this pattern. Very commonly seen in Chapter 2, where the expected utility is...

The issue here is the non-standard way in which the VaR has been calculated in parts (i) and (iii). For example, it's not clear why the "1-" is...

Hi When presented with a number of choices we usually conform to answering the stated question rather than challenge the options given. For...

Hi The April 2019 question asks for the VaR that is "relative to the expected portfolio value", which is why the £544,740 is deducted from the...

Correct answers expressed in different units are still correct. For example, 1 = 100% = 10,000%%, although some notation might feel more natural...

Notice that it's the same standard Brownian motion, Wt, that's powering St and At. This is what it means for the processes to have the same risk...

To check for second-order stochastic domination you need to calculate the integrals for the cumulative probability functions, each of which is a...

That's a really neat way of approaching the question - nice one. Here are your calculations: del,B(t)/del,sigma = - F(t)*N'(d1)*(T-t)^0.5...

As you've switched to using tau, your Taylor's expansion will need to have a term that looks like @f/@tau*d(tau). Notice that d(tau) = - dt, which...

Hi Laura You're absolutely right. There's been a long history of ambiguity in Value at Risk exam questions around whether "p" or "1-p" should be...

1st Solution: The conditions are that X>0 and that Y>-X. This will ensure that B1>0 and B2>0. 2nd Solution: You've got the right reason. The same...

Thanks @CapitalActuary for the great post about Brownian motion. Question 1: Ft will contain the entire history of the process up to and including...

Hi. Actually, it's the probability density function which is being multiplied by the width of the increment - this produces the probability mass....

Yep, you've got it.

Every time you find a del,f/del,sigma in your equation you should replace it with "V" for vega, this will give you: del,V/del,t + rSt*del,V/del,s...

1) Yes, imagine it to be money in a savings account that offers an interest rate of r(t) over the small interval of time between t and t + dt. A...

Great, I'm glad it worked out. Taylor's formula is used to derive Ito's Lemma - so you can use either method to arrive at the same result.

Yes, I would expect it. Maybe you could email the IFoA to promote them to update it?

Here it is.

The question does tie itself in knots by not specifying the size of ud relative to 1. I don't really want to bounce you to another form post...

Hi This question is definitely challenging, and there are already a number of forum posts which I think cover your concern. Check these out:...

Hi I can't see the statement in Chapter 7 you're referring to, but in general, typing maths is probably quicker than using the equation editor....

Hi Q1: d1 and d2 don't have any financial significance on their own. They're just intermediate calculations without any interpretation. Q2: Think...

Hi It comes from the "dX_s" and "dsdX_s" terms from the previous line. Note that both of these have a dX_s in them. Let me know if that doesn't...

Hi The Core Reading is essentially saying that given the correct final answer, we can demonstrate that it's the right answer by working backwards...

Remember that Excel/R is just a calculator. You don't have to explain in Paper A how the maths was computed (ie which functions you used to get...

Hi For part (iii), the integrand is of the form f'(x)f(x), therefore the integral is 0.5*f(x)^2. The reverse operation is easier to see:...

Hi. The calculation of the mean accumulation (ie the average fund value at the end of the projection) uses the mean rate of return (ie the average...

E_p is the expected return of any portfolio made from combinations of Asset 1 and Asset 2. So one legitimate portfolio is when only Asset 1 is...

Think of "shifting the probability" as sliding the distribution function to the right. So whilst the outcomes are still split 50/50 in terms of...

The "1/15" is the pdf of the uniform distribution U(0,15) This comes from the standard result that the variance of the sum of two random...

By definition. The market portfolio is made up of only the risky assets held in the same proportions as their market capitalisations. Since the...

If there was cash in the portfolio to start with I can't help but think that the investor should use this to repay part of the loan :)

This is a rebalancing exercise. The investor starts with a portfolio worth $10,076 and then looks to rebalance the option holding, with a final...

Sorry - I've revised my earlier post. It meant to reference the market portfolio rather than the y-axis. The investor has the freedom to choose...

In the two summations, i and j take the same values, ie 1,2,3,...,N. If you write all of the terms out it might be clearer: \[\beta_P =...

Hi i and j are just the indices used to iterate over all of the securities, so we have: \[ \beta_P=\sum_{i=1}^{N}x_i\beta_i =...

Along the efficient frontier there are lots of efficient portfolios - and one of those efficient portfolios will hold zero in the risk-free asset....

Only the IFoA can say what is and what isn't acceptable - everyone else can provide only interpretation and speculation. Copying values from Excel...

Hi. Thanks for your question. We don't get very many etymological queries on this forum :) Geometric Brownian motion is called such because from...

Yes, that's right. The components at and bt remain constant over the interval between t and t+dt and yet the value of the portfolio changes to...

Hi Sorry, I missed your latest post. I've revised my earlier response to try and clarify. I didn't make a clear distinction between when we need...

Yes. It's the share price that's being modified not the option. It's more complicated when the dividend isn't paid at expiry! Yes, the share...

No reason. They evaluate to same thing :)