As you move into the final stages of your revision for the CFA exam, Nicholas Blain, CFA, and Dianne Ramdeen, CFA, explain some useful and intuitive shortcuts on some challenging exam-style questions at Levels I, II and III of the CFA Program.

1. Level I Fixed Income: valuing floating rate bonds

Question	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

 

Worked solution

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 → 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.

2. Level I Fixed Income: calculating forward rates from spot rates

Question	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%

 

Worked solution

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!

3. Level II Equity: using the H-model

Question		(Excerpt from item set) 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

 

Worked solution

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 – g_L) x [(1 + g_L) + (H x (g_S – g_L))] where g_S and g_L 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 g_S down to long-term g_L over 2H years, then horizontal at level g_L. 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 – g_L) x [(1 + g_L) …], which is your familiar D₁ ÷ (r – g_L). For the triangle, what is its area? Half base x height = 0.5 x 2H x (g_S – g_L) = H x (g_S – g_L). This is the second part of the square bracket.

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

4. Level II Derivatives: Black-Scholes-Merton

Question		(Excerpt from item set) 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

Worked solution

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

c = Se^–dTN(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, Se^–dTN(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: Se^–dTN(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 Se^–dTN(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.

5. Level II Derivatives: pricing forward contracts

Question		(Excerpt from item set) 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

 

Worked solution

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 + R_F)^T or e^RcT. 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 e^0.0225  = 1227.31. Good guess!

6. Level III Equity: fund performance

Question		(Excerpt from item set) Eliza’s current equity allocation has been invested in a large cap equity fund offered by a high-profile asset management firm.  Eliza wishes to understand why the returns of the fund she invested in were not the same as the benchmark against which its performance was measured.  To assist with her understanding, Maya, her assistant, provides the following information in Exhibit 1 for the four largest sectors in which the fund invested. Exhibit 1 Sector Sector Returns Fund Sector Weightings Benchmark Sector Weightings Manufacturing 15.20% 22% 25% Pharmaceuticals 10.15% 12% 13% Information Technology 3.10% 10% 15% Financials – 4.29% 20% 15%   Using the information in Exhibit 1, which sector has contributed the most to the fund’s underperformance relative to the benchmark?
	A	Manufacturing
	B	Pharmaceuticals
	C	Financials

 

Worked solution

In terms of calculating each sector’s performance and determining their contribution to the fund’s performance, there is a long way and then there is the Quartic way. First the long way.

The table below calculates the return on the fund and the return on the benchmark using the sector weights and the sector returns.  This is, in effect, a weighted average calculated by multiplying each sector weighting by the respective return and summing the results.

	Return using fund weightings (A)		Return using benchmark weightings (B)		Difference (A) – (B)
Manufacturing	(15.20% × 0.22) =	3.344%	(15.20% × 0.25) =	3.800%	– 0.456%
Pharmaceuticals	(10.15% × 0.12) =	1.218%	(10.15% × 0.13) =	1.320%	– 0.102%
Information Technology	(3.10% × 0.10) =	0.310%	(3.10% × 0.15) =	0.465%	– 0.155%
Financials	(- 4.29% × 0.20) =	– 0.858%	(- 4.29% × 0.15) =	– 0.644%	– 0.214%
Total		4.014%		4.941%	– 0.927%

 

The question has already hinted that the fund underperformed the benchmark.  The calculations above show that the fund’s return was 4.014% and the benchmark’s return was 4.941% equating to underperformance of 0.927%.

In order to determine how each sector contributed to this underperformance, we look at the last column of the table.  It seems that even though manufacturing, pharmaceuticals and information technology performed well in their own right, their contribution to the fund’s performance was negative.  But why has this happened? Look carefully at the fund’s weightings compared to the benchmark’s weightings.  It appears that the fund allocated a lower proportion of the assets to these sectors than the benchmark. Underweighting a sector that performs well has a negative effect on a fund’s performance. Equally, overweighting a sector that performs poorly will also have a negative effect on a fund’s performance.

In a nutshell, the relative weightings of each sector lead to over or under performance of the fund. With this in mind, we can now look at a faster way to get the answer.

Quartic shortcut:  Simply calculate the difference in the weightings for each sector i.e. compare the fund’s weighting to the benchmark’s weighting, and then multiply by the sector’s return.  For example, the fund’s weighting for manufacturing was 22% and the benchmarks’ weighting was 25%.  This equates to a 3% underweighting.  This negative 3% is then multiplied by 15.2% (return on manufacturing sector) to give of -0.456%.  This result represents the manufacturing sector’s contribution to the underperformance of the fund. Repeat the same steps for the other sectors and select the one with the highest result as our answer.

Underperformance attributable to:

i.	Manufacturing (- 0.03 × 15.2%)	= – 0.456%
ii.	Pharmaceuticals (- 0.01 × 10.15%)	= – 0.102%
iii.	IT (- 0.05 × 3.10%)	= – 0.155 %
iv.	Financials (0.05 × – 4.29%)	= – 0.215%

Level III Fixed Income: expected fixed income returns

Question

(Excerpt from structured response question) 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.

Worked solution

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%.

 

2. 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%

 

3. 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 yield²] = [-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.

4. 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.
5. 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%