# Eight of the hardest CFA questions, and how to answer them The CFA exams are approaching fast. You need to do at least 300 hours of study to pass each one. You can always follow our guide to prepping for the CFA exams. Even so, the questions have a reputation for being extremely difficult.

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.

### Questions for CFA Level I:

1. 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

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. Royal’s valuation is \$5.10 less that Knight’s valuation."

2. A semi-annual pay floating-rate note pays a coupon of Libor + 60 bps, with exactly three years to maturity. If the required margin is 40 bps and Libor is quoted today at 1.20% then the value of the bond is closest to:

A. 99.42

B. 100.58

C. 102.33

Floating rate bonds are pretty difficult to value accurately (in fact we will see this again in Level II Derivatives, as they are an essential component to swaps). However, there is an approximation provided in the CFA curriculum, and a rather neat Quartic short-cut too.

A floating-rate note can be (roughly) valued on a coupon date by discounting current Libor + quoted margin (think of this as the regular coupon) at current Libor + required margin (think of this as the discount rate). In other words, we discount what we get (PMT) at the rate that we need (I/Y).

On the calculator: N = 6, I/Y = (1.2 + 0.4) ÷ 2 = 0.8, PMT = (1.2 + 0.6) ÷ 2 = 0.9, FV = 100 è PV = 100.58.

Quartic shortcut: first note that if a bond is paying exactly what is required (i.e. quoted margin = required margin) then the bond will trade at par on each coupon date. In this question, the bond is paying 20 bps per year more than required. This means that we should pay a 20 bp premium per year. Three year maturity means a 60 bp premium. Hence our quick “guess” is that the bond should trade at 100 plus a 60 bp premium, or 100.60. Answer B is the only possible answer.

3. The following details (all annual equivalent) are collected from Treasury securities:

 Years to maturity            Spot rate 2.0                                 1.0% 4.0                                 1.5% 6.0                                 2.0% 8.0                                 2.5% Which of the following rates is closest to the two-year forward rate six years from now (i.e. the “6y2y” rate)? A 2.0% B 3.0% C 4.0%

Calculating forward rates from spot rates and spots from forwards can be done easily, and quite accurately, with the banana method, described below.

Note that the six-year spot rate (say, z6) is 2% and the eight-year spot rate (z8) is 2.5%. Let’s call the 6y2y rate F, to keep notation easy.

To solve this, draw a horizontal timeline from 0 to 8, marking time 6 on the top. To avoid arbitrage, investing for six years at z6 then two years at F must be the same as investing for eight years at the z8 rate. Mark above your timeline “z6 = 2%” (between T = 0 and T = 6) and “F = ?” (between T = 6 and T = 8), and below the timeline “z8 = 2.5%”.

Algebraically we can say that: (1 + z6)6 x (1 + F)2 = (1 + z8)8.

With a bit of effort, this solves as: F = [(1 + z8)8 ÷ (1 + z6)6]0.5 – 1 = [1.0258 ÷ 1.026]0.5 – 1 = 4.01%.

Quartic banana method: just below the timeline you have drawn, write down how many bananas (or any other inanimate object) you have received if you get 2.5 per year for eight years. Answer: 20. Now write down, above the timeline, how many you get in the first six years, at 2 per year. Answer: 12. Now calculate how many bananas you must have got in the last two years. Answer: 20 – 12 = 8. This is over two years, hence 4 per year, answer C. Banana method gives 4.00%; accurate method gives 4.01%. Close enough!

Questions for CFA Level II:

4. 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

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. Financial information from 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 at 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

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. A share is trading at €35, with a 3% continuous dividend yield and 20% annualized volatility. A one-year call option on this share has strike price €32. The continuous risk-free rate is 2%. Risk factors are: d1 = 0.498, N(d1) = 0.691, d2 = 0.298, N(d2) = 0.617. The value of the call option using the Black-Scholes-Merton model is closest to:

A: €1.51

B: €4.12

C: €4.55

Firstly, the basic calculation from the Black-Scholes-Merton model:

c = SedTN(d1) – Xe–rTN(d2) = 35 x e–0.03 x 0.691 – 32 x e–0.02 x 0.617 = 23.47 – 19.35 = €4.12, answer B.

Now let’s think about this model. BSM gets a bit of a bad press: calculations are relatively new in the curriculum (the learning outcomes used to focus on the assumptions) and the algebra is a little frightening.

However, we need to understand what is required, which is the top-level call calculation, as shown. The risk factors are complex, both to calculate and to understand, but you are almost certainly not going to need these in your exam. Your curriculum provides little explanation and no examples, hence they are safe to put to one side.

You should appreciate how a call is equivalent to “underlying plus financing”, buying part of a share, SedTN(d1), and borrowing money, –Xe–rTN(d2). We can also think of the call as a contingent purchase, contingent of course on the call being in-the-money. The two parts can be explained separately:

• Financing: –Xe–rTN(d2). This is the present value of what we expect to pay for the share. N(d2) is the cumulative normal distribution, the risk-adjusted likelihood that we’ll exercise the option and buy the share. Hence Xe–rTN(d2) is PV(strike) times likelihood of exercise, i.e. our expected cost.
• Underlying: SedTN(d1). This is effectively the expected value of what we buy. S is today’s share price, “discounted” by the continuous dividend yield as we’ll miss out on these dividends between today and the exercise date. N(d1) is a conditional probability, such that SedTN(d1) is the expected value of the stock if and only if it is in-the-money on expiration. If that is a bit much to get your head around then don’t worry as you don’t need to give this explanation – just remember N(d1) is bigger than N(d2) as it has an upward bias. N(d1) is also the hedge ratio or delta of the call.

7. The P&S 400 Index has a current value of 1200. It has a continuous dividend yield of 2% and the risk-free rate is 5% on a continuous basis.

The price of a nine-month forward on the P&S 400 index is closest to:

A: 1173

B: 1227

C: 1237

The basic rule for pricing forward contracts is:

Forward price FP = spot plus cost of carry minus benefit of carry.

The cost of carry includes interest: hence for most contracts the spot is multiplied by (1 + RF)T or eRcT. Other contracts (e.g. commodities) may include storage and insurance. Benefits of carry include dividends (discrete or continuous), coupons, convenience yield (for commodities), or the foreign interest rate (for currency forwards).

In the case of an equity index forward, you may be able to do the entire calculation in your head.

In this question the spot price is 1200. The cost of carry is 5% and the benefit of carry is 2%. Never mind the continuous nature of these rates, for the moment. We can say that the net cost is 3% per year, or 2.25% for nine months. 2.25% of 1200 is 27, hence our estimate of the forward price is 1227, answer B.

If we do this accurately, we get:

FP = S0 x e(Rc - dc)T = 1200 x e(0.05 – 0.02) x 0.75  = 1200 x e0.0225  = 1227.31. Good guess!

Questions for CFA Level III:

7. 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.

“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.)”

8.

Exhibit 2

 Current average bond price Expected average bond price in one year’s time (assuming no change in the yield curve) €97.34 €97.82 Coupon frequency Average bond coupon payment Average bond convexity Average bond modified duration Expected average yield and yield spread change Expected credit losses Expected currency losses (€ depreciating against USD) Annual €2.60 20 4.10 0.35% 0.15% 0.60%

Using the information from Exhibit 2, calculate the total expected return on the bond portfolio assuming no reinvestment income.

The expected return on a bond consists several components.  When answering the question, it is important to incorporate all elements of return in a logical manner. In this example, the elements of return come from:

1. Yield income – this represents the coupon the investor receives as well as any re-investment from the coupon.  Since there is no re-investment income (your questions will usually assume this), the yield income will be equal to the coupon divided by the bond’s current price i.e. (€2.60/€97.34).  This equates to 2.67%.

1. Rolldown return – this is the effect of the bond’s price being ‘pulled to par’ as the bond approaches maturity.  Rolldown return is calculated by taking the difference between the current price of the bond (€97.82) and the expected price in one year assuming no change in yield (€97.34) as a percentage of the current price.

Rolldown return in this example = (€97.82 - €97.34)/ €97.34 = 0.49%

1. Expected change in price based on yield and yield spread change – this reflects investors’ expectations about yield curve changes and is calculated using modified duration and convexity. In this example, modified duration (MD) of 4.1 represents the percentage change in the bond’s price for a 1% change in yields. Recall the inverse relationship between bond prices and yields; if yields are expected to increase, bond prices will fall as a result. Modified duration assumes a linear relationship between bond prices and yields.  Since the relationship is convex in reality, an adjustment for convexity must be made.

The expected change in price based on yield and yield spread change = [- MD × change in yield] + [0.5 × convexity × change in yiled2] = [-4.10 × 0.0035] + [0.5 × 20 × (0.0035)2] × 100 = 1.42%.

Quartic tip: the change in yield is expressed as a decimal when doing the calculation thus 0.35% becomes 0.0035.

1. Expected credit losses – this is the expected loss in principal due to a default.  The number is given here as - 0.15% but may be calculated as probability of default × loss severity.
2. Expected currency loss – only relevant for bonds denominated in a foreign currency.  The amount is given in this example as - 0.60% but can be calculated based on forward rates.

Note that since the question simply asked to calculate, the answer that you need to produce is set out below:

 Yield income (€2.60/€97.34) Rolldown return (€97.82 - €97.34)/ €97.34 Expected change in price based on yield and yield spread change [-4.10 × 0.0035] + [0.5 × 20 × (0.0035)2] Expected credit losses Expected currency losses 2.67% 0.49% - 1.42% - 0.15% - 0.60% Total expected return 0.99%

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