September 2015 Professional Examination
Paper 2.3: Questions & Solutions
Derivatives Valuation Analysis
Portfolio Management
Commodity Trading and Futures
Level 2
Page 1 of 11
SECTION B:
Question 2 – Derivative Valuation and Analysis
Over the next three months, you expect a significant drop in the price of a particular stock. Call
and put options are traded on the stock. You plan to profit from your expectation by shortselling the stock. Your compliance department has however reminded you that short-selling of
stock is not allowed in the stock market – (short-selling of other securities are however
allowed).
Explain, using put-call parity, how you can profit from your expectation without violating market
regulation.
(3 marks)
Solution to Question 2
I will need to create a synthetic short stock. From put-call parity:
C + Ee – r.T = P + S
S = C + Ee – r.T – P
The right-hand side is a long synthetic stock. If we multiply both sides –1, we have:
–S = P – C – Ee – r.T
The right-hand side is now the synthetic short stock, which means:
 long the put

short the call

short (i.e. borrow) the riskless bond
1½ marks
1½ marks
The net pay-off from this will be the same as that of a short stock.
Question 3 – Portfolio Management
An investor owns the following portfolio today.
Stock
R
S
T
Market Value
(N)
2,000
3,200
2,800
Expected
Annual Return
17%
8%
13%
Calculate the expected market value of the portfolio after three years and the total expected
return over the same period.
(4 marks)
Solution to Question 3

Portfolio total value = 2,000 + 3,200 + 2,800 = 8,000

Portfolio weights:
R
S
T
= 2,000 / 8,000
= 3,200 / 8,000
= 2,800 / 8,000
= 0.25
= 0.40
= 0.35
(1/2 mark)
(1/2 mark)
Page 2 of 11

Portfolio expected annual return
(0.25 × 17) + (0.40 × 8) + (0.35 × 13) = 12%

Expected portfolio value after 3 years is:
8,000(1.12)3

(1 mark)
= N11,239.42
(1 mark)
Expected total return:
11,239.42 – 1
8,000
Or (1.12)3 – 1
= 40.49%
= 40.49%
(1 mark)
4 marks
Question 4 – Commodity Trading and Futures
Briefly indicate the factors distinguishing a forward contract from a futures contract? What do
forward and futures contracts have in common? What advantages does each have over the
other?
(3 marks)
Solution to Question 4
While both forward and futures contracts are agreements to purchase a good at a future
date, a futures contract provides liquidity by having a central marketplace and standardized
contract terms. This allows holders of futures contracts to sell them in the market at any
time prior to expiration. Futures trading is governed by the formal regulations of the futures
exchange. Most important, the losses incurred by futures traders are guaranteed by the
clearinghouse, which requires the daily settlement of gains and losses. That is, the holders of
profitable contracts do not have to worry about whether their gains will be paid by the
holders of losing contracts. Forward contracts, however, are subject to default risk. Forward
contracts can be tailored to the unique needs of firms. For example, a firm may need to
execute a hedge in which the expiration is a specific date. Futures contracts expire only on
certain dates, which may not fit the needs of the firm.
(3 marks)
Page 3 of 11
SECTION C:
Question 5 – Derivative Valuation and Analysis
Kola Iwelabi is a senior quantitative analyst in Eko Asset Management (EAM). Kola is going
through market data concerning options on the stock of KB plc. Kola has the following
information presented in Table 1.
Table 1 : Stock and Options Data for KB plc and Risk-Free Interest Rate
Current Call Price
Current Put Price
Exercise Price
Days to Expiration*
Current Stock Price
N(d1)
Up Move on Stock
Down Move on Stock
Risk-Free Interest Rate
N2.30
N4.70
N130.00
60
N128.55
0.64
15%
10%
3%
* Note: Assume a 365-day year. Assume continuous compounding.
5(a1) Using the information in Table 1, calculate the price of a synthetic 60-day call option with
a N130.00 strike price.
(2 marks)
(5a2)
5(b)
A new trainee is interested in knowing why it is useful to construct and value synthetic
calls and puts. One of your assistants responds, “Deriving synthetic values enables us to
determine whether it is possible to earn arbitrage profits.
Based on the information in Table 1 and using a one-period binomial model, calculate the
value of 60-day KB plc call option with a strike of N130.00. What is the value of a
corresponding put option?
(5 marks)
5(c) Now assume you are currently short 104,000 units of KB’s stock and long 50,000 call
options on the stock.
Calculate what position you need to take on the stock’s put option for delta-hedging.
(2 marks)
5(d)
Using the Black-Schole option-pricing model, another assistant of Kola, Kevy, calculates
the price of a 3-month call option and notices the option’s calculated value is different
from its market price. With reference to Kevy’s use of the Black-Scholes option-pricing
model:
5d1)
5d2)
Discuss why the calculated value of an out-of-the-money European option may differ
from its market price.
Discuss why the calculated value of an American option may differ from its market
value.
(4 marks)
Solution to Question 5 – Derivative Valuation and Analysis
5(a1)
To create synthetic call option, make ‘C’ (i.e. call) the subject of the formula (in the put-call
parity equation)
Page 4 of 11
= P + S – Ee – rt
= 4.7 + 128.55 – 130e – 0.03 × 60/365
= ₦3.89
C
(2 marks)
(5a2)
The assistant is correct about the reason for calculating synthetic option values; it allows one
to determine if it is possible to earn arbitrage profits. However, the assistant is
incorrect about the set of transactions that can be used to earn an arbitrage profit if the
current price of the call option is greater than the synthetic value. The correct strategy is to
sell the call option and then take long positions in the put and the stock and a short position
in the bond (purchase the synthetic call). He incorrectly states that a short position should be
taken in the stock.
(2 marks)
5(b)
 Stock and call prices
Su = 128.55 × 1.15 = 147.8325
Cu = Max(0,147.8325 – 130) = 17.8325
(1 mark)
So = 128.55
Sd = 128.55 × 0.90 = 115.695
Cd = Max(0,115.695 – 130) = 0

(1 mark)
Risk-neutral probability
Let π = probability of ‘up’
= 0.4198
π=
1–
𝑒 𝑟𝑇 − 𝑑
𝑈 −𝑑
=
𝑒 0.03 ×(60/365 ) − 0.90
1.15 − 0.90
π = 1 – 0.4198 = 0.5802
(1 mark)
π = erT – d = e(0.03 x 60/365) – 0.90
U–d

1.15 – 0.90
Value of call (Co)
Co = (π Cu + (1 –
π)Cd)e – rT
(1 mark)
[(0.4198 x 17.8325) – (0.5802 x 0)] x (e-0.03x60/360)
= ₦7.45
The corresponding value of put can be calculated using put-call parity
P = C + So – Ee – rt
= 7.45 + 128.55 – 130e – 0.03 × 60/365
= ₦6.64
5(c)
Let
x = number of put needed
Delta of total holding is
Page 5 of 11
Security
Stock
Call
Put
Qty
Delta/unit
Total Delta
–104,000
1
– 104,000
50,000
x
0.64
32,000
– 0.36
– 0.36x
Total portfolio delta
= – 72,000 – 0.36x
For delta hedging:
– 72,000 – 0.36x = 0
x = – 200,000 put options
For delta-hedging, we need to sell 200,000 put options
(2 marks)
5d1)
When European options are out of the money, investors are essentially saying that they are
willing to pay a premium for the right, but not the obligation, to buy or sell the underlying
asset. The out-of-the-money option has no intrinsic value, but, since options require
little
capital (just the premium paid) to obtain a relatively large potential payoff, investors are
willing to pay that premium even if the option may expire worthless. The Black-Scholes
model does not reflect investors' demand for any premium above the time value of the
option. Hence, if investors are willing to pay a premium for an out-of-the-money option
above its time value, the Black-Scholes model does not value that excess premium.
(2 marks)
5d2)
With American options, investors have the right, but not the obligation, to exercise the
option prior to expiration, even if they exercise for non-economic reasons. This increased
flexibility associated with American options has some value but is not considered in the
Black-Scholes model because the model only values options to their expiration date
(European options)
(2 marks)
Question 6 – Portfolio Management
Tony John, FCS, manages the Big Bank plc’s (BBP) pension fund. He recently presented his
quarterly report to the fund’s board of trustees and is recommending a change in the fund’s
asset allocation. Based on his market expectations, he recommends allocating 30% of assets
to value stocks, 50% to growth stocks, and 20% to bonds. Tony’s market expectations are
shown in Table 1.
Table 1: Tony’s Market Expectations
Expected annual return
Expected standard deviation of
annual returns
Return correlations:
Value stock portfolio
Growth stock portfolio
Bond portfolio
Value stock
Portfolio
12%
Growth Stock
Portfolio
14%
Bond
Portfolio
8%
16%
22%
8%
1.0
-----
0.9
1.0
---
0.3
0.2
1.0
Page 6 of 11
6(a)
Using Tony’s recommended asset allocation and market expectations from Table 1,
calculate the expected standard deviation of annual returns for the pension fund’s
portfolio.
(3 marks)
6(b) If Tony wants to achieve an expected annual return of 12.5% while maintaining the pension
fund’s current 20% allocation to bonds, calculate the proportion of the fund’s assets that
should be allocated to value stocks.
(5 marks)
6(c)
6(d)
In describing the risk-return characteristics of his recommended portfolio, Tony states.
“I believe the recommended asset allocation will produce a portfolio that is the global
minimum-variance portfolio. The global minimum-variance portfolio has the lowest level
of risk compared with all other portfolios on the efficient frontier, and thus it also
dominates all other portfolios on the efficient frontier.”
Is Tony Correct? Explain.
(2 marks)
One of the board members, Ben Ema, suggests that Tony should consider broadening the
diversification of the fund’s portfolio into a “fully diversified portfolio” by adding real estate
and international stocks. He states that these additions will improve the efficiency of the
fund. Ben estimates the fully diversified portfolio would have an expected return of 13%
and a standard deviation of 15%. He would then further expand the investment
opportunity set by combining the proposed fully diversified portfolio with either risk-free
borrowing or lending. He notes that the appropriate risk-free rate of return is 4%.
6d1)
6d2)
If Ben wants to construct an optimal portfolio that has an expected standard deviation
of annual returns of 12%, what proportion of total assets should be invested in riskfree assets and what proportion in risky portfolio?
(2½ marks)
If Ben uses his proposed “fully diversified portfolio” to construct an optimal portfolio
that has an expected standard deviation of annual return of 12%, what will be expected
annual return for the resulting portfolio?
(1½ marks)
6(e) Another board member, Victor Dan, is not too sure he understands the concepts of ‘growth
investing’ and ‘value investing’.
You are required to explain very briefly the two concepts to victor.
(4 marks)
Solution Question 6 – Portfolio Management
6(a)
Standard deviation = (0.32 × 16%2 + 0.52 × 22%2 + 0.22 × 8%2 + 2 × 0.3 × 0.5 × 16% ×
22% × 0.9 + 2 × 0.3 × 0.2 × 16% × 8% × 0.3 + 2 × 0.5 × 0.2 × 22% × 8% × 0.2) 0.5
= 15.9%.
(3 marks)
6(b)
Total portfolio weight
= 100%
% allocated to bond
= 20
% allocated to stocks (value + Growth)
= 80
(1 mark)
Let w = % in growth stock
0.8 – w = % in value stock
(1 mark)
The following equation holds:
12w + 14(0.8 – w) + 8(0.20) = 12.5
12w + 11.20 – 14w = 10.90
–2w = –0.30
(2 marks)
w = 0.15 or 15%
Page 7 of 11
% in value stock = 0.80 – 0.15 = 0.65
The final allocation is:
Growth stock 15%
6(c)
Value stock
65%
Bond
20%
(1 mark)
100%
(5 marks)
Although he is correct that the global minimum-variance portfolio has the lowest level of risk
compared with other portfolios on the efficient frontier, he is incorrect with regard to
dominance. The global minimum-variance portfolio does not dominate portfolios on the
efficient frontier.
(2 marks)
6d1)
Let x = weight of the fully diversified portfolio
1 - x = weight of the risky asset.
σp = (x) (standard deviation of fully diversified portfolio)
12 =(x)(15
x = 0.80
1 – x = 0.20
Thus, 20% should be invested in the risk-free asset and 80% in the fully diversified
portfolio.
(2½ marks)
6d2)
E(R) = RF + (E(R) – RF) x δ
δp
= 4 + (13 – 4) 12 = 11.2%
15
Or:
E(R) = (1 – Wp) × RF + Wp × RP
= (1 – 0.8)(4) + (0.8 × 13) = 11.2%
(1½ marks)
6(e)
When you invest for growth, you are typically seeking capital appreciation over the long
term. You will likely choose investments that you believe will exhibit a faster-than-average
increase in share price over the coming years. Growth investors purchase stocks at
relatively high price-to-earnings, price-to-book, and price-to-sales ratio, and low dividend
yields.
On the other hand, value investors seek out stocks that are undervalued relative to their
fundamentals.
They therefore look for stocks at relatively low prices, as indicated by low price-toearnings, price-to-book, and price-to-sales ratios, and high dividend yields (greater
proportion of total return is from dividend income)
In summary, value investing focuses on the present and letting the future sort itself out. Growth
investing keeps the focus on the future.
(4 marks)
Question 7 – Commodity Trading and Futures
7(a) Explain why futures and forward prices might differ. Assume that platinum prices are
positively correlated with interest rates. What should be the relationship between platinum
forward and futures prices? Explain.
(4 marks)
Page 8 of 11
7(b) Foodful Ltd is a large producer ‘Golden Chocolate’ (GC) whose main ingredient is cocoa
beans. The demand for GC is seasonal with the largest demand occurring mid-November
through the end of December. Production schedules require acquisition of 25,000 bushels
of cocoa seeds in late September to meet the holiday season demand. Foodful
management is concerned about the possibility that a rise in price of cocoa seeds between
now and September could hurt profitability. Cocoa seeds must be acquired at a price of
N450 per bushel or less to ensure profitability. The September cocoa seeds futures
contract (5,000 – bushels quantity) is selling for N422 per bushel.
What can Foodful do to ensure its profitability?
7b1)
(2 marks)
What risk does Foodful face in acquiring cocoa beans by taking delivery of the futures
contract? How should Foodful acquire the cocoa beans it needs?
(4 marks)
7b2)
If the September spot price turns out to be N630 per bushel, show Foodful’s
transaction in the cocoa beans cash and futures markets and calculate its net wealth
change.
(6 marks)
Solution to Question 7 – Commodity Trading and Futures
7(a) Futures are subject to daily settlement cash flows, while forwards are not. If the price of the
underlying good is not correlated with interest rates, futures and forward prices will be equal.
If the price of the underlying good is positively correlated with interest rates, a long trader in
futures will receive daily settlement cash inflows when interest rates are high and the trader
can invest that cash flow at the higher rate from the time of receipt to the expiration of the
futures. Because forwards have no daily settlement cash flows, they are unable to reap this
benefit. Therefore, if a commodity's price is positively correlated with interest rates, there will
be an advantage to a futures over a forward. Thus, for platinum in the question, the futures
price of platinum should exceed the forward price. The opposite price relationship can occur if
there is negative correlation. Generally, this price relationship is not sufficiently strong to be
observed in the market.
(4 marks)
7(b) Foodful can acquire cocoa beans today and store it until September or Foodful can acquire
cocoa beans using the September cocoa beans futures contract, it would buy 5(25,000/5,000)
September contracts at ₦422 per bushel.
(2 marks)
7b1)
When September arrives, Foodful can acquire the cocoa beans in one of two ways. First, it
can take delivery of the beans via the futures contract. Unfortunately, the short side of the
contract chooses the delivery location. This location may or may not be convenient for
Foodful. The second alternative for acquiring cocoa beans eliminates this risk. In this
method, Foodful acquires cocoa beans in the spot market and enters a reversing trade in the
futures market. If the futures price has moved since the initiation of the hedge, any gains
(losses) on the futures contract offset any losses (gains) in the cash market so that the
effective price Foodful pays for cocoa beans is N422.
(4 marks)
7b2)
Cash Market
 Today: Foodfull anticipates need for 25,000 bushels of cocoa beans in September, wants
to pay N422/ bushel or N10,550,000 total.

September: Spot price of cocoa beans is N630. Foodful buys 25,000 bushels for
N15,750,000
Loss over target = N10,550,000 – 15,750,000 = – N5,200,00
Futures Market
Today: Buy five 5,000 – bushel September cocoa beans futures at ₦422/bushel.
September: At maturity, the futures price equals the spot price. Sell 5 futures contracts at
N630/bushel.
Futures profit = 25,000 (630 – 422)
Page 9 of 11
= N5,200,000
Net wealth change:
Profit on futures
Loss in cash market
N
5,200,000
(5,200,000)
0 __
(6 marks)
Page 10 of 11
FORMULAE
1)
Black and Scholes Options pricing model:
;
2)
2)
;
General cost of carry relationship:
)
3)
Continuous time cost of carry relationship:
4)
Determinants of Options Price:
5)
Correlation/Covariance:
6)
Static portfolio insurance using put option:
7)
Hedging with Stock Index Futures:
8)
Risk adjusted performance measures:
9)
Binomial Option Valuation Model:
10)
Note: The variance of 3-asset portfolio is given by:
11)
Page 11 of 11