To accompany **The Actuary’s Free Study Guide for Exam 3F / Exam MFE**, I offer a list of general studying methods, techniques, and insights that have guided my own preparation for the financial economics half of the third actuarial exam. While my methods may not be suited to every type of actuarial student – and it is ultimately your decision to embrace them or to reject them, based on your estimation of your abilities and ways in which you learn most efficaciously – I have found them tremendously helpful in making sense out of an immense exam syllabus. I will first discuss general study approaches and then address ideas to keep in mind for this exam in particular.

**General Studying Approaches**

**1. Begin studying early and study regularly.** The actuarial exams, as you are likely well aware, are not comparable to final exams in college, to which you might allot a few hours of study and get an A as a result. These exams require *months* of preparation in order to adequately learn and apply the material. I recommend starting at least 2.5 months ahead of the exam date and reading, solving practice problems, and even writing practice problems yourself every day.

**2.****Set daily goals and develop a system to quantify your studying.** To make sure that you are putting in the necessary effort every day, it is not enough to have a subjective feeling that you have worked sufficiently. The exam material is quite difficult and, in my personal experience, after doing *any* work, one feels like one has done plenty. It is much wiser to set an objective goal in advance for each day and attempt to meet it. Of course, goals need not be rigid and can respond to any unforeseen challenges posed by the course material. Setting up a point system that needs to be met every day rather than insisting on highly specific and unalterable objectives.

Here, I will outline the point system I have used to prepare for Exam 3F/MFE as well as for exams 1/P and 2/FM. I require myself to accumulate at least 100 points per day. Here is how the points may be accumulated.

I receive 5 points for every page of exam-relevant text that I read.

I receive 10 points for every exam-relevant problem that I solve.

I receive 20 points for every exam-relevant problem I formally write myself and then solve.

I am, of course, allowed and encouraged to go over my 100-point target. I keep a running total of all the points I have accumulated during the course of studying as well as a current arithmetic average of points for all days thus far. My average acts like a grade in a course, and so I have an incentive to strive to keep my grade in Actuarial Studying high – above 100, if at all possible. I also sometimes amuse myself by trying to deliberately raise my overall “course” average on a particular day by doing extra exam preparation.

A point system helps one establish a common denominator by which to measure one’s efforts at tasks that can often vary considerably. Just like money provides a convenient way to measure economic value while dealing with millions of goods, so does a point system allow one to compare disparate studying approaches and have some rough measure of accomplishment on a particular day. Like money, any point system is also highly imperfect at measuring the desired objectives – effort and learning. Develop a point system that you consider to be the most accurate and the most relevant to the approaches that work best for you.

The incentives provided by my point system have shown to be effective in my own experience. In preparing for Exam 1/P, I maintained an average of 114.95 points per day over 98 days. In preparing for Exam 2/FM, I maintained an average of 128.24 points per day over 105 days. I am doing even better thus far in preparing for Exam 3F/MFE; I have a current average of 143.35 over 79 days. My studying for the prior two exams seems to have been sufficient; I passed both on the first try, with scores of 10/10 and 9/10, respectively. And I still have detailed records of the studying I did on every day that I prepared for each exam.

**3. Do as many practice problems as possible. This includes writing your own!** The way to truly learn each concept relevant for the exam is to immerse oneself in it and repeatedly expose oneself to ways in which it might be applied. For me personally, a mathematical formula or idea does not stay in my mind if I just read about it. If I write down the relevant formulas, my memory functions somewhat better, but it is in the *use* of the formulas that I truly memorize them; after I have used a formula five or six times, I no longer have any difficulty recalling it.

I am a highly frugal individual and pride myself on only spending money when it is absolutely necessary for the improvement of my well-being. When I began to study for this exam, I noticed that there was a dearth of freely available practice problems for it – problems that would help students to make sense of the theoretical concepts described in McDonald’s *Derivatives Markets* and to see how those concepts apply in situations relevant to them – i.e., in exam situations. Granted, some admirable efforts have been made by parties like Dr. Ewa Kubicka, Dr. Sam Broverman, Dr. Bill Cross, Dr. Abraham Weishaus, and Actuarial Brew to offer free practice problems – but even those combined were not enough to enable students to learn the entire syllabus from them alone. This is why I wrote **The Actuary’s Free Study Guide for Exam 3F / Exam MFE**, which now accounts for 418 of the 625 free Exam 3F/MFE practice problems with free available solutions of which I am aware.

I encourage you to write any problems you think might be helpful in teaching actuarial students important exam-related concepts or skills. It is even possible to earn some small passive income from these problems if you render them available online on sites like Associated Content, which will pay you $1.50 per 1000 page views to your articles. While this is not a living, it might earn you more than the interest in your bank account and may even compensate for a significant fraction of your exam fees. If you choose to publish your practice problems via Associated Content, submit them under the “Display-Only” terms, which will enable you to edit your material, albeit with substantial time delays of about a week for the edits to get implemented. As is virtually inevitable with problems of this level of difficulty, any author will sometimes make substantial errors or oversights – and indeed, virtually all textbooks and practice manuals for this exam are accompanied by substantial sections of errata. The advantage of publishing problems online is that user feedback regarding any errors is extremely fast and contributes to an ultimately better product. Freely available online practice problems and solutions can be assured to have high quality in a manner similar to open-source software, which also gets continually improved on the basis of user feedback.

**Exam-Specific Approaches**

**4. Assure yourself some points by mastering the easier material.** Exam 3F/MFE is comprised of material of highly varying difficulty and open-endedness. Fortunately, several major topics on the syllabus – mostly represented in the earlier chapters of McDonald’s *Derivatives Markets* – are quite systematic and straightforward in their application. Answering questions regarding them properly only requires you to thoroughly memorize a few formulas and problem-solving techniques. The easier topics on this exam include **put-call parity, binomial option pricing, and the Black-Scholes formula.** Questions related to these three topics combined have in the past accounted for over 50% of both the Casualty Actuarial Society’s and the Society of Actuaries’ exams – and there is no reason why you should get any of these questions wrong.

**5. Learn as much as possible about the intermediate-level material.** Approximately another 25% of the exam should be composed of material pertaining to **delta-hedging** and **exotic options.** All of this material is also manageable, although the variety of problems that could be asked is somewhat greater.

Much of your success at hedging questions will be assured if you memorize the formula for the **delta-gamma-theta approximation**: C_{new} = C_{old} + єΔ + (1/2)є^{2}Γ + hθ. The delta-gamma approximation is just the delta-gamma-theta approximation without the hθ term. You should also be able to answer questions regarding what it takes to **delta- and gamma-neutralize a portfolio**. In those questions, you will be given delta and gamma values for different kinds of options. Always work with gamma-neutralization first, because once you have rendered a portfolio’s gamma zero, you can always compensate for the leftover delta by adding or subtracting shares of stock (as each stock has a delta of 1). The mathematics behind these problems is simple arithmetic, and again, there is no reason to get any of them wrong. Sometimes, you might be given a delta in disguise; remember that Δ = e^{-δT}N(d_{1}). If you are given δ, T, and N(d_{1}), you can find Δ using this formula.

Also remember the formulas for option elasticity (Ω = SΔ/C) and option volatility (σ_{option} = σ_{stock}*│Ω│). These are easy to memorize and apply.

Less likely but quite possible are questions regarding the return and variance of the return to a delta-hedged market-maker. To be able to answer those questions, memorize the formulas

R_{h,i} = (1/2)S^{2}σ^{2}Γ(x_{i}^{2}-1)h for the return and Var(R_{h,i}) = (1/2)(S^{2}σ^{2}Γh)^{2} for the variance of the return. Also remember that you can model the stock price movement in any time period h as σS_{t}√(h). You might be asked to determine said stock price movement, given h, S_{t}, and some function of σ – probably N(d_{1}) – from which you will be able to figure out σ.

It also would not hurt to memorize the Black-Scholes partial differential equation: rC(S_{t}) = (1/2)σ^{2}S_{t}^{2}Γ_{t} + rS_{t}∆_{t} + θ.

For exotic options, each type of option is not difficult to understand conceptually, but there are a lot of types to keep in mind. For **Asian options**, make sure you remember that they are path-dependent and that you know how to take a geometric average. Pay attention to the difference between geometric average *price* options and geometric average *strike* options. For **barrier options**, remember the parity relationship Knock-in option + Knock-out option = Ordinary option. Know about rebate options as well; the concept is not difficult. For **compound options**, remember the parity relationship CallOnCall – PutOnCall + xe^{-rt_1} = BSCall. **Gap options** are priced just like ordinary options via the Black-Scholes formula, with the exception that the *trigger price* rather than the strike price is used in the formula for d_{1}, while the strike price is still used in the formula for the overall option price. For pricing **exchange options**, the Black-Scholes formula holds as well, with two significant differences. Sigma in the formula for exchange options is σ = √[σ_{S}^{2} + σ_{K}^{2} – 2ρσ_{S}σ_{K}]; you will be given the individual volatilities of the underlying asset and the strike asset, as well as their correlation ρ. Also, S is your underlying asset price, K is your strike asset price, and you will be given the dividend yields δ_{S} and δ_{K} pertaining to these assets. Then your Black-Scholes exchange call price will be

C= Se^{-(δ_S)T}N(d_{1}) – Ke^{-(δ_K)T}N(d_{2}).

**6. Memorize shortcut approaches to Brownian motion and interest rate models.** The most difficult – and most open-ended – topics on the exam will be based on Chapters 20 and 24 of McDonald’s *Derivative Markets –* dealing respectively with Brownian motion and interest rate models. The best way to approach these topics is *not* from the vantage point of theory or derivation from first principles. Rather, you will save yourself a lot of time and stress by memorizing the general form of results obtained by doing specific kinds of problems.

You *do* need to know what arithmetic, geometric, and mean-reversion (Ornstein-Uhlenbeck) Brownian processes look like. You should also be familiar with terminology pertaining to them, including *drift, volatility,* and *martingale* (a martingale is a process for which E[Z(t+s)│Z(t)] = Z(t)).

**Ito’s Lemma** can be easily memorized in the form

dC(S, t) = C_{S}dS + (1/2)C_{SS}(dS)^{2} + C_{t}dt, which is applicable to geometric Brownian motion. It is doubtful that you will be asked to apply Ito’s Lemma to non-geometric kinds of Brownian motion. When you apply Ito’s Lemma, remember the multiplication rules, which state that (dZ)^{2} = dt, and any other product of multiple dZ and dt is 0. If you have two correlated Brownian motions Z and Z’, then dZ * dZ’ = ρ.

If you memorize some results derived using Ito’s Lemma, then you will not have to go through the derivation on the test. For example, if dX(t)/X(t) = αdt + σdZ(t) (i.e., X follows a geometric Brownian motion, then d[ln(X)] = (α – 0.5σ^{2})dt + σdZ(t) (i.e., ln(X) follows an arithmetic Brownian motion *with the exact same volatility.*) Just memorizing the latter formula can make several currently known prior exam questions extremely easy to solve.

Do not forget about the formula for the Sharpe ratio (φ = (α – r)/σ) and the fact that the Sharpe ratios of two perfectly correlated Brownian motions are equal.

To figure out *whether a particular Brownian motion has zero drift* (or what kind of drift factor it has), you will need to apply Ito’s Lemma to the given Brownian motion and see what the resultant dt term is.

Also recall that the **quadratic variation** of a Brownian process is expressible as_{
n→∞}lim_{i=1}^{n}Σ(Z[ih] – Z[(i -1)h])^{2} = T from time 0 to time T. Here, Z is the Standard Brownian Motion. In X is some other Brownian motion with volatility factor σ, then

_{n→∞}lim_{i=1}^{n}Σ(Y[ih] – Y[(i -1)h])^{2} = σ^{2}T.

Other useful facts to know are as follows.

Var[σdZ(t)│Z(t)] = σ^{2}Var[dZ(t)│Z(t)] = σ^{2}dt.

For any arithmetic Brownian motion X(t), the random variable [X(t + h) – X(t)] is normally distributed for all t ≥ 0, h > 0, and has a mean of X(t) + αh and a variance of σ^{2}h.

The Black-Scholes option pricing framework is based on the assumption that the underlying asset follows a geometric Brownian motion:

dS(t)/S(t) = αdt + σdZ(t).

When X(t) follows an arithmetic Brownian motion, the following equation holds:

X(t) = X(a) + α(t – a) + σ√(t-a)ξ.

When X(t) follows a geometric Brownian motion, the following equation holds:

X(t) = X(a)exp[(α – 0.5σ^{2})(t – a) + σ√(t-a)ξ].

If given either of those two equations, you should be able to recognize arithmetic and geometric Brownian motion.

Be sure to know how to value claims on *power derivatives* (of the formS(T)^{a}). This is a newly added topic and is thus likely to be tested on at least one question. Memorize the following formulas.

d(S^{a})/S^{a} = (a(α – δ) + 0.5a(a-1)σ^{2})dt + aσdZ(t)

γ = a(α – r) + r, where r is the annual continuously compounded risk-free interest rate.

δ* = r – a(r – δ) – 0.5a(a-1)σ^{2}

F_{0,T}[S(T)^{a}] = S(0)^{a}exp((a(r – δ) + 0.5a(a-1)σ^{2})T)

F^{P}_{0,T}[S(T)^{a}] = e^{-rT}S(0)^{a}exp((a(r – δ) + 0.5a(a-1)σ^{2})T)

While many actuarial students might think that Brownian motion is the most difficult topic on the exam, I believe that the interest rate models covered in Chapter 24 are in fact harder, because a virtually endless variety of practically impossible questions can be asked regarding them. I hope that the SOA and CAS will be reasonable in what they choose to test. For instance, asking students to use the explicit bond-price formulas for the Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models would be uncalled for, as these formulas take tremendous effort to memorize. The past exam questions I have seen have not asked for these formulas. Instead, they tended to involve various “auxiliary” formulas for these models.

Of course, it is essential to know the Brownian motions associated with the Vasicek and CIR models. For the Vasicek model, dr = a(b – r)dt + σdZ. For the CIR model, dr = a(b – r)dt + σ√(r)dZ. Note that the only difference is a √(r) factor in the dZ term. This has important implications, however, as discussed in Section 72 of my study guide. Make sure you read this section to find out about the essential similarities and differences between these models as well as why the time-zero yield curve for either of these models cannot be exogenously prescribed (a learning objective on the exam syllabus).

For both the Vasicek model, the following “auxiliary” equations are important to know, as questions involving them have appeared on prior exams.

P[t, T, r(t)] = exp(-[α(T – t) + β(T – t)r]), where α(T – t) and β(T – t) are constants that stay the same whenever the difference between T and t is the same, even if the values of T and t are different.

Now say you have a Vasicek model where

dP[r(t), t, T]/P[r(t), t, T] = α[r(t), t, T]dt + q[r(t), t, T]dZ(t). You are given some α(r_{1}, t_{1}, T_{1}) and are asked to find α(r_{2}, t_{2}, T_{2}). All you need to do is to solve the following equation:

**[α(r _{1}, t_{1}, T_{1}) – r_{1}]/(1 – exp[-a(T_{1} – t_{1})]) = [α(r_{2}, t_{2}, T_{2}) – r_{2}]/(1 – exp[-a(T_{2} – t_{2})])**

The derivation of this formula is quite involved and is provided in Section 71. You would be well advised to simply memorize the end result (and read the derivation, if you are interested in the reasoning, but do not repeat the derivation each time unless you enjoy pain and suffering).

(It would not hurt to also memorize that q[r(t), t, T] = -σB(t, T) in the case above. A question using this formula is not likely to be asked, and if it is, I hope that σ and B(t, T) will be given.)

The Black-Derman-Toy (BDT) interest rate model has easy and difficult applications. The easy applications involve pricing caplets and interest rate caps, provided that you have had practice with the BDT discounting procedure. Remember that the risk-neutral probability for the BDT model is always 0.5 and the interest rates at the nodes in any given time period differ by a constant multiple of exp[2σ_{i}√(h)]. To discount the expected value of a payment at any node, divide it by 1 plus the interest rate at that node. Remember that, for interest rate caps, you will need to account for payments made *each* time period when the interest rate exceeds the cap strike rate.

The difficult parts of using the BDT model involve finding yield volatilities and actually constructing BDT binomial trees. I am aware of a scant few exam-style questions on the former and of none on the latter – which I hope will be indicative of the composition of future MFE exams. There is no way around memorizing the formula for yield volatility in the BDT model:

Yield volatility = 0.5ln(y[h, T, r_{u}]/y[h, T, r_{d}]) = 0.5ln([P[h, T, r_{u}]^{-1/(T-h)} – 1]/[P[h, T, r_{d}]^{-1/(T-h)} – 1])

You will likely be given a multi-period BDT tree and asked to calculate the volatility in time period 1 of the bond used in constructing the tree. I am hard-pressed to see how the formula above might be used in finding the volatilities for time periods greater than 1, since the formula relies on the presence of two and only two nodes in the binomial tree.

For constructing binomial trees, I hope that, if a question is asked, it will involve constructing a tree for a one-period model. To do this, learn the following formulas:

If P_{h} is the price of an h-year bond, where h is one time period in the BDT model, then

P_{h} = 1/(1 + r_{0}) and thus r_{0} = 1/P_{h} – 1. If P_{2h} is the price of an 2h-year bond, then R_{h} = r_{d} and σ_{h} meet the following conditions: r_{0} = σ_{h}and P_{2h} = 0.5P_{h}(1/(1 + R_{h}exp[2r_{0}]) + 1/(1 + R_{h})). Anything beyond this is intense algebraic busy work, and I hope that the SOA and CAS are kind enough not to embroil students in it.

Also remember to review the Black formula for pricing options on futures contracts (Section 37) and caplets (Section 75).

**7. Do not forget the “miscellaneous” topics!** I can think of two “miscellaneous” topics that appear seldom or not at all in McDonald’s text but are highly likely to be tested. One of these is computing **historical volatility.** Fortunately, this is a straightforward and systematic procedure, and you should have no difficulty with it after working through Section 81. The other topic, **equity-linked insurance contracts**, can involve applications of virtually anything else on the syllabus. Nonetheless, you are likely to be asked to use one or both of the following formulas to discover option payoffs in disguise:

**Formula 80.1:** max(AB, C) = B*max(A, C/B)

**Formula 80.2:** max(A, B) = A + max(0, B – A)

Section 80 gives some equity-linked insurance problems that you might find beneficial.

**8. Take practice tests under exam conditions several days before the actual exam.** Taking practice exams will give you an idea of your current knowledge level as well as areas to which you might need to devote additional attention. If you have studied well and you do well on the practice exams, this will give you a lot of confidence coming into the actual test. I expect the passing threshold for future sessions of Exam 3F/MFE to be around 13 or 14 out of 20, since the Fall 2007 exam – a particularly difficult test – had a passing threshold of 12. To conveniently find free practice exams online, see **A Comprehensive List of Free Study Materials for Exam 3F / Exam MFE** and **A Comprehensive List of Free Study Materials for Exam 3F / Exam MFE: Part 2**. You can also compile many practice exams from the exam-style questions offered in various sections of my study guide.

This is as much essential studying advice as immediately comes to mind. Remember, however, that the more you know, the better your chances are on this exam, and that the information I have provided here does not substitute for comprehensive months-long preparation. Use all the resources at your disposal and learn everything you can. I hope that the free resources provided in this study guide will aid you in passing the exam and achieving a high mark.

**See other sections of** **The Actuary’s Free Study Guide for Exam 3F / Exam MFE****.**