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Eight of the hardest CFA questions, and how to answer them

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You need to do at least 300 hours of study to pass each of the three CFA level exams. Or, you can always follow our guide to prepping for the CFA exams. Even so, the chances are that you could be knocked for six by the really tough questions.

We’ve spoken to training companies who coach candidates embarking on the CFA exams. These eight questions – in their opinions – are the toughest questions you are likely to encounter on CFA levels I, II, and III. Helpfully, they have also provided solutions.

1. Level I: Beth Knight, CFA, and David Royal, CFA, are independently analyzing the value of Bishop, Inc. stock. Bishop paid a dividend of $1 last year. Knight expects the dividend to grow by 10% in each of the next three years, after which it will grow at a constant rate of 4% per year. Royal also expects a temporary growth rate of 10% followed by a constant growth rate of 4%, but he expects the supernormal growth to last for only two years. Knight estimates that the required return on Bishop stock is 9%, but Royal believes the required return is 10%. Royal’s valuation of Bishop stock is approximately:

A. $5 less than Knight’s valuation

B. Equal to Knights valuation

C. $5 greater than Knights valuation

Tim Smaby, VP, Advance Designations, Kaplan Professional:

“The correct answer is A.

You can select the correct answer without calculating the share values. Royal is using a shorter period of supernormal growth and a higher required rate of return on the stock. Both of these factors will contribute to a lower value using the multistage DDM.

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Royal’s valuation is $5.10 less that Knight’s valuation.”

2. Level I: John Gray, CFA and Sally Miller are discussing what they think their year-end bonus will be and how they might spend them. Miller is new to working in finance and asks Gray what people usually get and what he has got in the past. Gray explains that the firm prohibits employees from discussing their exact bonus number but also says that 30% of people get ‘good’ bonus’, 50% ‘average’ and 20% ‘low’. Gray says that he really wants a new smart watch recently released by a large tech company and says that he will definitely buy it if he gets a ‘good’ bonus, while there is only a 50% and 10% probability he will get it with an ‘average’ or ‘low’ bonus respectively.

Two weeks later, Miller sees Gray in the office and asks him if he got a good bonus. Gray reminds Miller that the firm’s policy means he cannot say, but Miller notices that he is wearing the new smart watch they were talking about. Miller goes back to her desk and calculates the probability that Gray got a ‘good’ bonus is closest to:

A: 30%
B: 53%
C: 57%

Nicholas Blain, chief executive, Quartic Training:

“Using Bayes’ Formula : P(Event|Information) = P(Event) * P(Information |Event) / P(Information)

In this case, the event is getting a good bonus, and the information is that Gray has bought the new watch.

The probability that he got a good bonus and then bought the watch is given by:
P(Event)*P(Information |Event) = 0.3*1.00 = 0.30

The total probability that he would buy the watch is given by:
P(Information) = 0.3*1.00 + 0.5*0.50 + 0.2*0.10 = 0.57

Therefore, the probability that he got a good bonus is the proportion of probability that he got a good bonus and got the watch, to the total probability he got the watch:
P(Event|Information) = 0.30 / 0.57 = 0.53.”

3. Level I:  For a European Call option on a stock, which of the following changes, (looking at each change individually and keeping all other factors constant) would an analyst be least likely confident about an up or down movement in the price of the option?

A: Share price goes up; dividend goes up
B: The demand for share increases / supply decreases; interest rates fall
C: Share increases in volatility; the firm cancels the next dividend

Nicholas Blain, chief executive, Quartic Training:

“Share price up = Option Price Up
Dividend up = Option Price Down (dividends are benefits of holding the underlying share, when holding the option, you do not receive dividend)
Share in High Demand = Option Price down as this is a benefit in holding the underlying
Interest Rates Fall = Option Price Down as this reduces the cost of carry of holding the underlying
Share increases in volatility = Option Price Up
Cancels next dividend = Option Price Up, as these dividends are not received by the option holder anyway

A is the correct answer; as the increase in the option price due to the share going up could be offset by the decrease in the price due to the dividend going up.

B results in the option price falling for both scenarios and C results in the option price rising in both scenarios.”

4. Level II: Sudbury Industries expects FCFF in the coming year of 400 million Canadian dollars ($), and expects FCFF to grow forever at a rate of 3 percent. The company maintains an all-equity capital structure, and Sudbury’s required rate of return on equity is 8 percent.

Sudbury Industries has 100 million outstanding common shares. Sudbury’s common shares are currently trading in the market for $80 per share.

Using the Constant-Growth FCFF Valuation Model, Sudbury’s stock is:

A. Fairly-valued.

B. Over-valued

C. Under-Valued

Tim Smaby, VP, Advance Designations, Kaplan Professional:

“The correct answer is A.

Based on a free cash flow valuation model, Sudbury Industries shares appear to be fairly valued.

Since Sudbury is an all-equity firm, WACC is the same as the required return on equity of 8%.

The firm value of Sudbury Industries is the present value of FCFF discounted by using WACC. Since FCFF should grow at a constant 3 percent rate, the result is:

Firm value = FCFF1 / WACC−g = 400 million / 0.08−0.03 = 400 million / 0.05 = $8,000 million

Since the firm has no debt, equity value is equal to the value of the firm. Dividing the $8,000 million equity value by the number of outstanding shares gives the estimated value per share:

V0 = $8,000 million / 100 million shares = $80.00 per share

5. Level II: (Excerpt from item set)

Financial information on a company has just been published including the following:

Net income $240 million
Cost of equity 12%
Dividend payout rate (paid at year end) 60%
Common stock shares in issue 20 million


Dividends and free cash flows will increase a growth rate that steadily drops from 14% to 5% over the next four years, then will increase at 5% thereafter.

The intrinsic value per share using dividend-based valuation techniques is closest to:

A. $121

B. $127

C. $145

Nicholas Blain, chief executive, Quartic Training:

“The H-model is frequently required in Level II item sets on dividend or free cash flow valuation.

The model itself can be written as V0 = D0 ÷ (r – gL) x [(1 + gL) + (H x (gS – gL))] where gS and gL are the short-term and long-term growth rates respectively, and H is the “half life” of the drop in growth.

For this question, the calculation is: dividend D0 = $240m x 0.6 ÷ 20m = $7.20 per share.

V0 = $7.20 ÷ (0.12 – 0.05) x [1.05 + 2 x (0.14 – 0.05)] = $126.51, answer B.

However, there is a neat shortcut for remembering the formula. Sketch a graph of the growth rate against time: a line decreasing from short-term gS down to long-term gL over 2H years, then horizontal at level gL. Consider the area under the graph in two parts: the ‘constant growth’ part, and the triangle.

If you look at the formula, the ‘constant growth’ component uses the first part of the square bracket, i.e. D0 ÷ (r – gL) x [(1 + gL) …], which is your familiar D1 ÷ (r – gL). For the triangle, what is its area? Half base x height = 0.5 x 2H x (gS – gL) = H x (gS – gL). This is the second part of the square bracket.

Hence the H-model can be rewritten as V0 = D0 ÷ (r – gL) x [(1 + gL) + triangle].”

6. Level III: A German portfolio manager entered a 3-month forward contract with a U.S. bank to deliver $10,000,000 for euros at a forward rate of €0.8135/$. One month into the contract, the spot rate is €0.8170/$, the euro rate is 3.5%, and the U.S. rate is 4.0%. Determine the value and direction of any credit risk.

Tim Smaby, VP, Advance Designations, Kaplan Professional:

“The German manager (short position) has contracted with a U.S. bank to sell dollars at €0.8135, and the dollar has strengthened to €0.8170. The manager would be better off in the spot market than under the contract, so the bank faces the credit risk (the manager could default). From the perspective of the U.S. bank (the long position), the amount of the credit risk is:

Vbank (long) = €8,170,000 / (1.04)2/12 ˗ €8,135,000 / (1.035)2/12 = €28,278

(The positive sign indicates the bank faces the credit risk that the German manager might default.)

7. Level III: Within the ‘Option Strategies’ section

Option Strike Premium
Call 1 X1 = 20 c1 = 6
Call 2 X2 = 30 c2 = 4
Put 1 X1 = 20 p1 = 0.604
Put 2 X2 = 30 p2 = 8.001


Risk-free rate continuously compounded: 4% annual

Option expiry: 6 months

Using the above data for a box spread, calculate what arbitrage profit can be achieved at the end of 6 months.

Nicholas Blain, chief executive, Quartic Training:

“First, work out the cost of the box spread. Combine the two call options into a bull spread (buy the low strike call and sell the high strike call), and the two put options into a bear spread (buy the high strike put and sell the low strike put). Combining those two gives a box spread. To calculate the initial cost, work out the net premia:

Cost = c1 – c2 + p2 – p1 = 6 – 4 + 8.001 – 0.604 = $9.397

The payoff from the box spread will be the difference between the strike levels, ie 30 – 20 = $10.

If you borrowed $9.397 at the beginning in order to enter the box spread, how much would you have to pay back after 6 months? You need to compound the cost at the risk-free rate:

$9.397 x e0.04 x 0.5 = $9.58683

So the arbitrage profit would be the difference between the payoff and what you have to pay back on the loan, ie $10 – $9.58683 = $0.41317

8. Level III: Assume Felix Burrow is a US investor, holding some euro-denominated assets. Given the information below, calculate the domestic return for Burrow over the year.

Today Expected in 1 year
Euro asset 201.54 203.12
USD/EUR exchange rate 1.1133 1.1424


Nicholas Blain, chief executive, Quartic Training

“The domestic return (return in USD terms) depends on the EUR-return of the asset, as well as on the change in exchange rates:

RDC = (1+RFC) (1+RFX) -1

where RFX is the change in spot rates, using the domestic currency as the price currency (ie we require a USD/EUR quote). In this example, the exchange rate is quoted as such, so we can use the quote provided (otherwise, if the domestic currency was the base currency, we would need to invert the quote first).

RFC = -1 = 0.0078 = 0.78%

RFX = -1 = 0.0261 = 2.61%

RDC = (1 + 0.78%)(1 + 2.61%) – 1 = 3.41%

Burrow’s domestic currency return was higher than the underlying asset return, because he further benefits from the appreciation of the Euro (depreciation of the USD).”

Contact: pclarke@efinancialcareers.com

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