1. Level I Fixed Income: valuing floating rate bonds

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 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.

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

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

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: pricing forward contracts

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!