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INTRODUCTION
As with many other subjects, finance is studied with theories and models.
Started out as a branch of applied Micro Economics around the turn of the century. Finance
remained largely descriptive until the late 1950’s.
About half a century later however, finance has transformed into a discipline fall of rigorous
theory and models. In addition to such theories as M+M Theory of Franco Modigliani(1985)
and Merton Miller (1990). The Portfolio Theory of Harry Markowitz (1990). The Capital
Assets Pricing Model (CAPM) Theory of William Sharpe (1990) and Option Pricing
Model of Myron Scholes and Robert Merton other notable finance theories include;
Efficient Market theory
Gordown’s dividend valuation model
Signoring and symmetric information Model
Agency Theory
Arbitrage pricing Theory (APT) etc.
As in many other behavioural science fields, explanations and predictions are the main goals
in finance. Therefore both theoretical analysis and Empirical investigation are necessary.
Theories employ abstract deductive reasoning by which conclusions are drawn from sets of
assumptions whereas purely empirical studies are inductive in nature.
Financial Management involves a wide range of functions in business organization. It must
provide answers to the following questions:-
1. How, where and when is money for investment to be raised. (Finance Needs, Sources).
Financing function.
2. What investment short-Term or Long- Term are worth (investment decisions) making
3. What should be done with the firms profits (Dividends Decision), (liquidity decision)
The focus in finance is between risk and return. The investors make trade-off between risk
and return based on their risk profile.
Some investors are Risk averse i.e. Those who accept risky port folio’s only if they provide
compensation of risk.
Risk Neutral who finds the levels of risks relevant and considers only the expected returns.
Risk lovers i.e. Those who are willing to accept lower expected returns on prospects with
higher amount of risk.
1
RETURNS
Return is what accrues from an investment i.e. what is earned from any investment.
The following are the measures of Returns namely;
1. Holding Period Return (HPR)
HPR=V1-V0 +D1 X100
V0
Where V1-V0 = Capital Gain
D1 = Divided or interest
The assumptions of HPR includes:
i) Any interest or dividends received on the asset which is not distributed to the investor is
re-invested in the portfolio.
Hence the value changes from V0 to V1.
ii) Any distribution occurs at the end of the investment horizon
iii)No capital inflows occur during the investment horizon
2.) Average Return:/Arithmetic Return (AR)
A.R = R1 + R2 + R3 +…………………….+ Rnn
Where n= N0 of observations or periods
R1 Rn = The Returns
3) Expected returns (E(R))
This is based on the arithmetic model but the only difference is that we add a futuristic
element. This is given by:-
E ( R ) = ∑ FR
Where F =Frequency
N= № of observationF/N = Likelihood or Probability = Pi
Ri = Returns
Thus;
E (R ) = ∑ Pi Ri
2
ⁿ
i =1 N
n
i =1
Illustration
Suppose you want to estimate the returns of a certain asset in the economy and you consulted
stock broker who gave you the estimates. The current price for the asset is sh.15 and 5
brokers gave an estimate of sh.12, 3 brokers an estimate of sh.14, 2 brokers an estimate of sh.
20, 6 brokers an estimate of sh.25 and 4 brokers an estimate of sh.10. What would be the
expected returns for the asset?
Frequency Observation Probability Returns Expected Returns
F N Pi = F/N Ri E (R) = PiRi
5 20 0.25 12 3
3 20 0.15 14 2.1
2 20 0.10 20 2
6 20 0.30 25 7.5
4 20 0.20 10 2
20 E (R ) = 16.6%
RISK
Risk is a potential variability in future cash flow.
The wider the range of possible events that can occur, the greater the risk
Risk is seen as a chance or possibility of a loss which may not be correct.
The following are the means how risk is measured.
1. Expected monetary Value
2. Variance, Standard deviation and co-variance
3. Coefficient of variation
4. Sensitivity analysis
5. Decision Tree Analysis
6. Simulation etc
Illustration
You are evaluating an investment in 2 companies whose past 10 years of returns are shown
below.
Companies % age Returns during the years
1 2 3 4 5 6 7 8 9 10
A 37 24 -7 6 18 32 -5 21 18 6
3
B 32 29 -12 1 15 30 0 18 27 10
RQD: Calculate:-
i) The average returns of each security.
ii) Use standard deviation to find the risk of the two.
iii)Which of the two securities would you choose as an investor.
Solution
E (R)A = = 37+24+(-7)+6+18+32+(-5)+21+18+6 = 15%
10
E (R)B = 32+29+(-12)+1+15+30+0+18+27+10 = 15%10
Based on expected returns Go for either security A or B
or
A: B:
Returns (Ri - Returns (Ri -
37 484 32 28924 81 29 196-7 484 -12 7296 81 1 19618 9 15 032 289 30 225-5 400 0 22521 36 18 918 9 27 1446 81 10 25
1954 2038
A
Based risk, the lower percentage of the standard deviation, the lower the risk.
Based on the returns, an investor would choose any security and based on risk of standard
deviation, the investor goes for security.
4
Standard Deviation is an absolute measure of risk i.e it can be expressed using the units of the
variables that were used to calculate it.
In order to get a relative measure, we calculate the co-efficient of variation (CV) given by:
C.V =
CV measures the amount of risk per unit of expected return. The higher the CV, the higher
the relative risk inherent in the project/security.
For the above 2 securities, CV can be calculated as follows:
A B
CV = CV=
Example
A company is planning an advertising campaign in 3 different market areas. The estimates of
probability of success and associated additional profits in each of the three market areas are
provided as below:
Profits(sh.) Market 1 Market 2 Market 3
Probability Profits(sh.) Probability Profits(sh.) Probability
Fair 10,000 0.4 5,000 0.2 16,000 0.5
Normal 18,000 0.5 8,000 0.6 20,000 0.3
Excellent 25,000 0.1 12,000 0.2 25,000 0.2
RQD: i) Computed the expected monetary value and standard Deviation of profits resulting
from the advertising campaign in each of the market areas.
ii) Rank the 3 markets according to the riskness using the coefficient of variation.
Expected Monetary Value
Market 1 Market 2 Market 3
Condition Prob. Profit EMV Prob. Profits EMV Prob. Profits EMV
Fair 0.4 10,000 4,000 0.2 5,000 1000 0.5 16,000 8000
Normal 0.5 18,000 9,000 0.6 8,000 4800 0.3 20,000 6000
Excellent 0.1 25,000 2,500 0.2 12,000 2400 0.2 25,000 5000
5
15,500 8,200 19,000
Standard deviations
Market 1 EV= 15,500
Condition Prob Profits (V-EV) (V-EV)2. Pi
Fair 0.4 10,000 -5,500 12,100,000
Normal 0.5 18,000 2,500 3,125,000
Excellent 0.1 25,000 9500 9,025,000
24,250,000
= 4924.43
Market 2 EV= 8,200
Condition Prob Profits (V-EV) (V-EV)2. Pi
Fair 0.2 5000 -3200 2,048,000
Normal 0.6 8000 -200 24,000
Excellent 0.2 12,000 3,800 2,888,000
4,960,000
= 2227.11
Market 3 EV= 1900
Condition Prob Profits (V-EV) (V-EV)2. Pi
Fair 0.5 16000 -3000 4,500,000
Normal 0.3 20,000 1000 300,000
Excellent 0.2 25,000 6000 7,200,000
12,000,000
= 3,464.10
Coefficient of variation =
6
Markets: Std deviation Expected Value (EV) Coefficient of var. (CV)Market 1 4924.43 15500 0.318 31.8%Market 2 2227.11 8.200 0.272 27.2%Market 3 3.464.10 19,000 0.182 18.2%
Ranking the Markets Market 3Market 2Market 1
Apart from the expected Returns, and the holding returns the following methods can also be
used to calculate returns:
1. Returns Relative
In this method negative returns cannot be used and hence the ending price is compared
with the beginning price.
R.R = C+PE
PB
Where C = Cash payment or receipts during the period
PB = Beginning Price
PE = Ending Price
OR:
R.R = I+ Total Returns %
2. Cummulative Wealth IndexAge in total Returns reflect change in the level of wealth. To measure wealth level you must measure cumulative effect of returns over time given some stated initial amount usuallyKsh.1. The Cummulative wealth index measure the cumulative effects of total Returns measured as;
CWIn = WI0 (1+R1) (1+R2)………………….(1+Rn)
Where:
CWIn = Cummulative wealth index at the end of n years
WI0 = Beginning index value which is typically Ksh.1
Ri = Total Returns for years 1,2…………………n
Illustration
Given R1 =0.14; R2 = 0.12; R3 = -0.08; R4 = 0.25; R5 = 0.02;
7
Calculate cumulative wealth index at the end of 5 years assuming a beginning value of Ksh.1
CWIn = WI0 (1+R1) (1+R2)…………. (1+R5)
I(1+0.14) (1+0.12) (1-0.08) (1+0.25) (1+0.02)
= 1.498 =1.5
Ksh.1 Invested in year 1 will be Ksh. 1.5 at the end of year 5.
2. Geometric Return
This is used to calculate the average compound rate of growth that has actually occurred over
multiple period. It reflects the compound rate of growth overtime. This is computed as.
= [(1+R1) (1+R2)……….. (1+Rn)] ½-1
Illustration
Calculate the Geometric Return for the illustration above
[(1+0.14)(1+0.12)(1-0.08)(1+0.25)(1+0.02)] ½-1
[(1.14)(1.12)(0.92)(1.25)(1.02)]1/2 - 1 = 0.2238 = 22.4%
NB// The above is referred to as nominal or money returns to make it real returns. It will be
adjusted for inflation as follows:
Real Returns = 1+Nominal Returns -1
1+Inflation Rate
Suppose in the above example, inflation rate is 6% what would be the Real Returns:
R.R = 1+0.224 -1 = 1.224 -1 = 0.15471+0.06 1.06
= 15.47= 15.5%
IllustrationThe following represent returns of ABC shares over 6 years period. Find the risk and the expected Returns.
Period Returns1 152 123 204 -105 146 9
8
E(R) = 15+12+20-10+14+9 = 60 = 106 6
E ( R) = 10
E (R) =
Risk =
ER =10
Period Returns(R) R-E(R) [R0-E(R)] 2
1 15 5 25
2 12 2 4
3 20 10 100
4 -10 -20 400
5 14 4 16
6 9 -1 1
546
= =9.539
= 9.54
Risk Measured by standard deviation = 9.54
Coefficient variation: CV = =
= 0.954
PORT FOLIO THEORY
Portfolio Theory originally proposed by Harry Markowitz (1990) was the first formal attempt
of quantifying the risk of a portfolio and develop a methodology of determining optimal
portfolio. The means of controlling portfolio risk is called diversification whereby investment
is made in a variety of assets so that exposure to any particular type of asset is limited.
Port Folio Returns
9
Portfolio means holding many assets of different company. Portfolio Returns is a weighted
average of individual returns of each asset which is in a portfolio.
Suppose one invests in two assets and assume that the proportion invested in asset1=W1 and
that asset is denoted by W2.
RP =W1X1+W2X2
Where X1, X2= Individual assets;
Suppose we are planning to invest in the following Assets x and y.
Suppose the probability and Return on the 2 assets are as follows:-
Probability Returns (x) Returns (y)
% %
0.25 10 -6
0.25 8 23
0.25 20 5
0.25 15 15
RQO: Calculate the expected Returns of the portfolio to be invested in x and y as follows:-
i) 25% in x and 75%in y
ii) 100% in x and 0 in y
iii) 50% in x and 50% in y
Solution:
E(R)x =0.25x10 + 0.25x8 +0.25x20 +0.25x15 = 13.25
E (R)y =0.25x-6 +0.25x23 +0.25x5 +0.25x15 = 9.25
Rp = W1X1 = W2X2
i) 25% in x and 75% in y
(0.25 x 13.25) + (0.75 x 9.25) = 10.25
ii) 100% in x
1x13.25 = 13.25
10
iii) 50%in x and 50% in y
(0.5x13.25)+(0.5x9.25) = 11.25
For n securities /Assets, the returns of the portfolio is given by:-
RP =
Suppose the expected Returns on asset:- ER1= 10%; ER2 = 15%; ER3 = 20%; ER4 = 16%.
If the investor invested in these 4 assets equally, what are the returns of the Portfolio?
RP =
= (0.25x10)+(0.25x15)+(0.25x20)+(0.25x16) = 15.25%.
RISK
For a two (2) asset case. The risk of the portfolio with the proportion of W1and W2 is
Variance p= W12. + W2
2. + 2W1.W2. .cor12
OR
p = W12. + W2
2. + 2W1.W2. cov12
Cor12 =
Cov 12 =
Where = Variance of Portfolio
Wi = Weight/ Portfolio of portfolio i
= Variance of Asset 1
Variance of Asset 2
Covariance
Covariance reflects the degree to which the returns of the two securities vary or change
together.
11
A positive (+ve) covariance means that the returns of the two securities move together while
a negative (-ve) covariance mean that the returns do not move together.
Covariance of two assets is:-
Cov. AB =
Coefficient of Correlation
Covariance and correlation are conceptually analogue as both represent the degree of co-
movement of the two variables. Mathematically they are related as follows:-
Pij = CorRiRj =
Where Pij = Cor RiRj = is the correlation coefficient between security i and j
CovRiRj = Covariance between returns of security i and j
Con. Coefficient (r) is -1 r 1 i.e Ranges between -1 and +1
Negative (-ve) correlation means perfect co-movement in the opposite direction
Positive (+ve) correlation means perfect co-movement in the same direction.
O Correlation Means no correlation co-movement.
Portfolio variance for two securities involves finding the sum of the following two cells as
shown below.
Section 1 Section 2
Section 1 W . W .W2. 1 2 r12
Section 2 W .W2. 1 2 r12 W .
2 P = W . + W . + 2 W .W2. 1 2 r12
Illustration
Suppose you want to invest in two securities x and y in the following proportions and returns
are as follows:
Prob. Rx(%) Ry (%)
0.2 11 -3
0.2 9 15
0.2 25 2
0.2 7 20
0.2 -2 6
Calculate the portfolio variance and risk if
the proportions invested in x and y are:
X Y
100% 4 0%
75% 4 25%
50% 4 50%
12
Solution
ERi =
ERx = 0.2x11 + 0.2x9 +0.2x25+0.2x7+0.2x-2 =10
ERY = 0.2 x-3 +0.2x15+0.2x2+0.2x20 +0.2x6 = 8
Pro. Rx (R-ERx) (R-ERx)2.P RY (R-ERy) (R-ERy)2.P
0.2 11 1 0.2 -3 -11 24.2
0.2 9 -1 0.2 15 7 9.8
0.2 25 15 45 2 -6 7.2
0.2 7 -3 1.8 20 12 28.8
0.2 -2 -12 28.8 6 -2 0.8
76 70.8
= =8.718 8.7 = 0.087 = =8.414 8.4 0.084
Cov.xy =
Prob. (R-ERx) (R-ERy) (R-ERx)(R-ERy).Pi
0.2 1 -11 -2.2
0.2 -1 7 -1.4
0.2 15 -6 -18
0.2 -3 12 -7.2
0.2 -12 -2 4.8
-24
Cov.xy = -24
- 0.0024
Cor.xy = = = -0.3284
= W . + W . + 2. W .W . + rxy
i)100% proportion in x and 0% in y2 + 0+0 = 0.0077
ii) 75% in x and 25% in y
13
= (0.752 X 0.0872) + (0.252X0.0842) + 2 x 0.75 x 0.25 x 0.0087 x 0.084 x-0.32
0.004258 +0.000441-0.0009 =0.003799 0.0038
= 0.570069+0.000441 +-0.00089998 =0.56961002
0.57
PORTFOLIO RISKS FOR N SECURITIES
For n securities the variance of the portfolio is given by:
= Cor. ij
If this can be represented in an n x n matrix; It can be shown as follows:-
Security X1 Security X2 Security X3 Security Xn
Security X1 21X X P X P … Xn P
Security X2 X P X X P … X X P
Security X3 X P X X P X …. X X P
Security Xn X P X X P X X P …. X
A portfolio consist of 4 securities: 1,2,3 and 4. The Proportions of the securities are;
W1= 0.2, W2 = 0.3, W3 = 0.4 and W4 = 0.1
= 4; = 8; =20; =10
The correlation coefficient between these securities is:
P =0.3; P =0.5; P =0.2; P = 0.6; P = 0.8; P =0.4
What is the standard deviation of the Portfolio?
= W +W . +W . + W . +2.W . W Cor 1.2
MINIMUM VARIANCE PORTFOLIO
Most investors and portfolio managers invest in 2 brood categories, namely bonds and stocks.
The portfolio of bonds and stocks that minimizes risk would be determined as follows:
Varp = W . +W . +2.Wb.Ws. .r
Where: Varp= Variance of portfolio consisting of bonds and stocks
Ws = Proportion invested in stocks
Ws = 1-Wb
= Standard deviation of Returns from bonds
14
= Standard deviation of returns from stocks
r = Correlation coefficient between returns from bonds and stocks
BUT = Ws = 1-Wb
Then;
Varp = W . + (1-Wb)2 . + 2. Wb (1-Wb). . rbs
First Derivative with respect to Wb: would:-
= 2Wb. + 2(1-Wb) x-1x + 2(1-Wb) r bs - 2Wbrbs
Then equating to zero;
Wb (Minimum) = =
Illustration
Suppose the expected return of bond is 8% E(r) of stock is 15%
Find the E(r) of portfolio and standard deviation of portfolio consisting of
bonds and stocks. E (R) P= Wb x 8 + Ws x15 = 8 Wb + 150Ws
= W .10 +W .20 +2xW xW x10 x20 X rbs
100W +400 W +400W W .rbs
Find the minimum Variance Portfolio for correlation of -1, 0 and 0.5.
Rbs -1 0 0.5
Wb(min) 0.67 0.8 1
E(R)P 10.31 9.4 8
0.01 80 100
Ws 0.33 0.2 0
Wb (min) = =
Ws = 1Wb i.e 1-0.67 = 0.33
Wb (min) = = = 0.8
Ws = 1-Wb i.e 1-0.8 = 0.2
15
Wb (min) = = = 1
Ws = 1-Wb i.e 1-1=0
Rbs =-1
ERP = (0.67x8) +(0.33x15) = 10.31
= 100x0.67 +400x0.33 + 400x0.67x0.33x-1 =0.01
Rbs =0
ERP = (0.8x8) +(0.2x15) = 9.4
Rbs = 0.5E(R) P =(1x8) +(0x15) = 8
=100
Indifference CurvesWhen choosing the portfolio, the investor would wish to maximize E(R) and minimize the risk. The indifference curve for an investor represents the mixtures of risk and returns which will be acceptable in turns of an investment. Any point along the indifference curves, the risk returns relationship is the same for the investor and will be indifferent to whether the investment is at point A or B as shown below:-
Indifference curves are curved because of the diminishing returns provided as the quantities of risk or return become disproportionate in the mixture towards the extremes of one curve. A will be preferable to D because it offers the same E(r). For a lower risk and to c because it offers a higher E(R ). A is said to dominate C and D. Whether an investor would choose A or B will depend on their attitude to risk. The Markowith Model of investment analysis seeks to
measure investors utilizing function U as. U= F .
Where
.
16
XX
Expected Returns E(R)
CB
C
A D
Indifference Curve
Risk
Different indifference curves for investors can be plotted as follows:-
The further the indifference curves goes to the left, the greater will be the value of utility to
the investor because these portfolios provides higher E (R) for the same level of risk or lower
risk for the same level of E (R).
Portfolio’s to the left of the curves are preferred because those below are Mean Variance
inefficient and those on or above the utility curve are Mean Variance Efficient. Mean
Variance Efficient Portfolio’s are those which give maximum return for a given level of risk
or have the minimum return for a given level of return.
Markowith assumption was mainly on Risk Averse Investors. However investors can be risk
seeking or Risk neutral. Investors who are risk seekers will choose portfolio with higher risks
hence suggesting that they will have negative sloping indifference curve which is concave.
17
Risk
E (R)U3
U2U1
.A
.B.C
.D
Risk
E( R)
I3
I2
I1
Risk Seekers
The investor will choose I1which is in the North East indifference curve for risk neutral is between that of risk seekers and risk averse investors.The risk averse investor does not want to take a fair gamble while risk seeking investor will take the fair gamble and the risk neutral investor does not care whether the gamble is taken i.e. The risk is unimportant to the risk neutral investors in horizontal lines as shown below:
The investor will prefer portfolio B to A.
Efficiency Frontier
The efficient set Theory states that an investor would choose his/her portfolio from set of
portfolio’s that;
1) Offers Maximum E (R) for varrying levels of risks.
2) Offers minimum Risk for Varying levels of E (R).
The set of portfolio’s meeting these 2 conditions is known as efficient set also known as
efficient frontier. Feasible set also known as opportunity set is the set from which the
efficient set can be identified.
Case 1: Two Securities
Suppose an investor is evaluating 2 securities A and B with E (R) A = 12% and E (R) B =
20% and . Correlation coefficient between the Z is rAB = -0.20. These
Securities can be combined as follows:
18
Risk
E (R)
.B
. A
Portfolio WA WB E ( R) P
1 1 0 12 20
2 0.9 0.1 12.8 17.64
3 0.8 0.2 14 16.27
4 0.5 0.5 16 20.49
5 0.25 0.75 18 29.41
6 0 1 20 40
The feasible frontier for different degree’s correlations e.g -1, 0 and 1 can be illustrated as follows:
When rAB = 1 Diversification does not reduce risk while when rAB = -1 diversification can
reduce maximum risk. This relationship can be mathematically determined as follows:
Var (P) = WA
If rAB = -1 Then;
19
4020
B
A
20%
12%
p
E ( R)p
E ( R) P
p
A
P=-1
P=1
WA=1
B =WB=1
P=0
This is the same as;Since (W -W )
Since W = 1-WThen;
If it can be set to zero:
WA =
If rAB =-1 Then;
Case 2: Efficient Frontier For N SecuritiesIn a 2 security cases, a curved line separates all possible portfolios. In a Multi-security case,
the collection of all possible portfolio is represented by a broken egg shape region referred to
as the feasible region. What matters to the investor is the North West boundary of the feasible
region which is represented by the thick dark line referred to as efficient frontier.
Efficient frontier contains all the efficient portfolios. A portfolio is efficient if and only if
there is no alternative with:
1. The same E (R) and lower portfolio Risk
2. The same portfolio Risk and a higher E (R)
3. A higher E (R) and a lower portfolio risk. This is as shown below:-
20
SFeasible set
Efficient Frontier
E(R)p
All possible portfolio’s that could be formed from any securities lie either on or within the
boundary of the feasible set. Only those portfolios’s lying on the North–West boundary
between point E and S are referred to as efficient which the Risk averse investors finds it
optimal.
AssignmentWhy is the feasible region broken Egg shaped?
Selection of optimal Portfolio
The investor selects optimal portfolio by plotting his/her indifference curve on the graph as
the efficient set and then choose the portfolio on the furthest North–West indifference curve;
This portfolio will correspond to the point at which an indifference is just tangent to the
efficient set as shown below.
ARBITRAGE PRICING MODEL (APT)
CAPM made enormous contribution to the field of finance. However, many empirical studies
have pointed out deficiencies in explaining the relationship between risk and return. Sets of
these studies suggested that it is possible to rely on certain firm on security characteristics
21
H
G
E
p (Risk)
S
H
G
O
Feasible Region
Efficient Frontier
E(R)p
p (Risk)
Point O is the optimal portfolio.
*
I3
I2
I1
and earn superior returns even after adjustments for risk as measures by Betas. Among these
studies includes:
Bans found that small cap stock performs larger capstock on risk adjusted basis.
Basu Found out that low priced earning stocks out performed large priced earning stock after
adjustment of risk.
Fama and French: More recently documented that value stock i.e. stock with high Books
market price Ratio generated large returns than growth stocks i.e. stocks with low book to
market price Ratio on a risk adjusted basis.
Some stocks not traded in the S.E. How do we calculate beta.
In an efficient market such differentials should not exist. This made financial economists to
look for other models that help to answer the problem of this variation. Stephen Ross (Mid –
1970) developed an alternative model called Arbitrage Pricing Theory (APT) which is
reasonable initiative requires only limited assumptions and allows for multiple risk factors.
These few assumptions of APT include:
1. Capital markets are perfectly competitive
2. Investors always prefer more wealth to less wealth and certainty i.e. investors are
rationale.
APT assumes that the return on an asset is linearly related to a set of risk factor given by:-
Ri = E (Ri ) + Bi Ij + Bi 2 I2+…………bik Ik +ei
Where Ri = Actual Returns on asset i during a special time period i.e. 1,2…..n
E (Ri) = Expected return on asset i. If all the risto factors have zero changes
Bi, = The sensitivity of asset is return to the common risk factor j
Ii, I2…….Ij Ij = Deviation of systematic Risk factor from its expected Value:
Ei = Random error term unique to asset.
In developing equilibrium risk–return relationship, the key idea that guides the development
of equilibrium relationship is the law of one price which says that 2 identical things cannot
sell at different prices. This means in portfolio that 2 portfolio’s that have the same risk
cannot offer different expected returns. If it happens the Arbitragers will step in and their
action will restore the law of one price.
At equilibrium, APT is given by
22
E (Ri) + bi1 + bi2 + …….+ bij
E(Ri) = Expected Return on asset =i
= Expected Return on asset with zero systematic risk
Bij = Sensitivity of asset it return on the common risk factor j
= The risk premium related to the Jth risk factor.
Arbitrage is the process of earning riskless profit by taking advantage of differential pricing
for the same physical asset or security. It entails the sale of security at relatively high price
and simultaneous price of the same security at relatively low price.
According to APT, an investor would explore the possibility of forming an arbitrage portfolio
in order to increase the E (R) of his/her current portfolio without increasing its risk. The
following are the characteristics of Arbitrage portfolio.
Characteristics of Arbitrage Portfolio
1. It does require any additional fund from the investor E.G if Xi denotes the change in the
investors holding of security Y and hence the weight of security in the Arbitrage
portfolio, then for three security Arbitrage is given as;
a) = X1+X2+X3 = 0
2. It has no sensitivity to any factor i.e. the sensitivity of a portfolio to that factor is the
weighted average of the sensitivity or the securities to that factor. For 3 security
Arbitrage; This is given by:
a) = b1 x1+b2x2+ B3X3 =0
3. It has a positive Expected Retun i.e. For 3 security case it is given by:
b) X1
Example: Suppose an investor owns 3 stocks with amount of holding in each to be Ksh. 4M.
The three studies have the following E(R) and sensitivities.
Stock bi
1 1.5% 0.9
2 21% 3.0
3 12% 1.8
B1x1 +b2x2+b3x3=0
(0.9x0.1) + (3.0x0.075) + (1.8x-0.175) =0
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What would happen to stock prices to restore equilibrium through arbitrage?
Solution
There is an finite number of portfolio combinations of values for x1, x2 & x3
Considering the characteristic number one x1 +x2 + x3 = 0 and characteristic number two:
b1X1+b2X2 +b3 X3 =0 ie.
0.9 X1 +3.0 X2 +1.8 X3 =0.
This gives 2 equations with 3 unknowns
Arbitray, assign a Value to X1 or X2 or X3, So as to find one Combination E.G
Assign 0.1 to X1:
X1 +X2 +X3= 0 0.9X1 +3.0X2 +1.8X3 =00.1+X2 + X3=0 0.9(0.1) +3X2 +1.8X3=0X2+X3 =-0.1 0.09+3X2 +1.8X3=0
3X2+1.8X3= -0.09
Solve Simultaneously:X2+X3=-0.1 X2=-0.1 –X33X2 +1.8X3 = -0.09 Substitute X2:3(-0.1-X3)+ 1.8 X3= -0.09
-1.2X3 = -0.09+0.3-1.2X3 = 0.21
X3 = 0.21 = -0.175 X3 = -0.175 -1.2
X2 +X3 = -0.1 but X3 = -0.175X2 = -0.1-X3
X2 = -0.1-(-0.175)X2 =0.075 X2 = 0.075
In order to establish that this is an arbitrage portfolio its E(R) must be determined,as follows:
E (R)i=X1R1 +X2 R2 +X3 R3 >0:
15% x 0.1 +21% x 0.075+ 12% x-0.175= 0.975 >0
This is an arbitrage portfolio because E (R)i >0 1> 0.975.
These changes of the holding are as follows:Stocks:
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1 0.1 x 12,000,000 =1.2M2 0.075x 12M=0.9M3 -0.175x 12M= -2.1M
This means that stock 3 was disposed of 2.1m and acquired stocks:1> 1.2m and stock 2>0.9M.
The net effect as zero i.e
Characteristic 1: X1 + X2+ X3=0 1.2M +0.9M +(-2.1M)= 0
The Arbitrage portfolio effect on investor’s position is as shown below:
Weight Old Portfolio + Arbitrage Portfolio = New PortfolioX1 0.33 + 0.1 = 0.433X2 0.33 + 0.075 = 0.408X3 0.33 + -0.175 = 0.158RP 16% 0.975 = 16.975Bp 1.9 000 = 1.9Dp 11 small = 11
The consequecies of buying pressure of stock 1 and 2 will make the stock prices to rise and hence E(R) to fall.
The selling pressure of pot on stock 3 will make stock price to fall but E (R) to rise.
This buying and selling activity will continue until all arbitrage possibilities are significantly reduced or eliminated.
This causes approximately a linear relationship to exist because E(R) and sensitivities as shown below.
Ri = Lo +L1bi
Where ho and h1 are constant and the equation is the Asset pricing equation of APT when returns are generated by one factor.
Example: Suppose one possible equilibrium seting have 10 =8 and 11 =4 giving a pricing equation of :
Ri = 8 +4bi
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What would be the equilibrium levels of E(R) for asset 1, 2 and 3 from the previous examples.
Initially Process of Arbitration
R1= 8+4 x 0.9 =11.6% 15% Buying Falling E(R) ed Prices
R2 = 8+4 x3 = 20% 21% Buying Falling E(R) ed Price R3 = 8+4 x1.8 = 15.2% 12% Selling Rising E(R) ed Price
APT can be shown graphically as follows.
APT line will present the investor’s with Arbitrage opportunity. Investors who buy security B and sell security 5 in equal shillings amounts will have formed an arbitrage portfolio in the equation:
Ri = Lo + Lihi
If bi= 0 ; Then Ri =ho
Ho is equivalent of risk free rate (Rƒ) making the equation to be:
Ri =Rƒ + hibi
And hi is the expected excess return over and above risk free rate also referred to as factor risk premium or factor Expected Return premium.
OPTIONS
An option is a contract that confers to its holder or, owner the right but not the obligation to buy or sell a specified security at a specified price on or before a given date.
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Bb = bS Beta=(b)
S
BrB
rs
λ0
APT Asset Pricing Line
Definition of Terms
Call Option- Is an option to buy the underlying asset but not an obligation.Put option:-Is an option to sell to sell the underlying asset.Exercise Price/ Strike Price:- It is the fixed price at which the option holder can buy or sell the underlying asset.Expiration/ Maturity Date: The date when the option expires after the expiration date, the option is worthiless
The act of buying or selling the underlying asset as per the option contract is referred to as exercising the option.
Options are essentially of 2 types namely: (Call and Put Options).
American Option: This is a contract that gives the holder, the right but not the obligation to buy specified securities at a specified price on or before a specified exercise date.
European Option: This is one which can be exercised only on the maturity date.Options traded on an exchange is known as Exchange Traded options and options not traded on an exchange are referred to as over- the-counter options.
Options are said to be-At-the- Money (ATM)In- the- money (ITM)Out-of-the- Money (OTM)
At-the Money Option is an option that would lead to zero cash loss i.e No profit or Loss to the holder if it were exercised immediately.
In- the- Money Option: - Is an option that would lead to a positive cash flow to the holder if it were exercised immediately.
Out –of-the-Many-option: One which would lead to a negative cash-flow to the holder if it were exercised immediately.
This is as shown below:
Call Option Put Option
ATM Exercise Price = Market Price Exercise Price= Market Price
ITM Exercise Price < Market Price Exercise Price> Market Price
OTM Exercise Price> Market Price Exercise Price< Market Price
Options and their Pay
Options and their Pay-offs Just before Expiration
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Call Option: The call option gives the holder the right to buy an asset at a fixed price during a certain period.
There are no restrictions on the kind of assets to be used but the most popular type of call option is the option on stocks.
To provide protection to option traders, the option contract generally specifies that the exercise price and the number of shares will be adjusted for stock splits and stock dividends.
The option buyer normally pays to the option seller, a price referred to as option price or premium.
Pay- off of a call Option From Buyers Point of View
The pay off of call option denoted as which just because expiration depends on the relationship because the stock prices S1 and the
Exercise Price E as show below:
C= S1-E if S1> EC= O if S1 < E
This means that C= Max (S1- E,0)
Example: Suppose the market price of Equity shares of Company A on the expiration date is Ksh. 140 and the Exercise price is Ksh. 125. Illustrate diagrammatically this position of buyer.
Market stock Price S1= Ksh. 140Exercise Price E = Ksh.125
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60
Value of option
Share price Exercise Value of Call Price
50 125 S1<E 50<125 0100 125 S1<E 100>125 0125 125 S1=E 100=125 0150 125 S1>E 150>125 150 – 125 = 25200 125 S1>E 200>125 200 – 125 = 75
Incase of Premium
What would be the gain or Loss to call option buyer if a premium of Ksh. 5 was paid.
C= S1-E - Premium:Share Price Premium Value (Gain loss)50 5 -5100 5 -5125 5 -5150 5 20200 5 70
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At Kshs.130, therefore the value will be zero.
10
0
50 100 150 200
20
30
40
50
Price of shares
S1< EPotential profit area S1≥ E
450
Value of cell option
30
40
Pay-offs of call option from the seller’s Point of view
A call writer or seller of a call option collects the option premium from the buyer or holder of the option and in return is obliged to deliver the shares should the option buyer exercise the option.
If the stock Price S1 is less than Exercise Price E i.e S1 < E the option holder will not exercise in this case, the options writer liability is zero.On the other hand, If the stock Price S1 > E (Exercise Price), the option holder will exercise the option and the option writer Losses S1-E.
Using the Previous Example C= SI – E - P C = S1-(E+P)
Share Price Value of call Gain /Loss Buyer Gain/loss seller 50 0 -5 5100 0 -5 5125 0 -5 5150 25 20 -20200 75 70 -70
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50 100 150 200 Price of sharePremium paid
S1 < E + PS1 > E + P
Break E+PEven Point
X
-20
-10
-5
0
10
20
10
5
0
Premium receivedBreak even point
Price of share25 50 75 100 125 150 175 200
130
Value of option
PUT OPTION:While the call option gives the holders the right to buy a stock at a fixed price, the put option gives the holders a right to sell the stock at a fixed price.
The pay off of a put–off Just before expiration depends on the relationship between the Exercise price (E) and the price of the underlying stock.
-If S1< E the option has a value of E - S1 and is said to be in-the-Money.
-On the other hand, if S1 >/E, put option is worthless and is said to be out of-the-Money. Thus, the pay-off of a put option just before expiration will be:-
S1 <E S1>/ E
Value of put option S-S1 0
Example:Suppose an investor wants a right to sell the above shares (Previous Example) of Company A at Shs.135 after two months. Illustrate this diagrammatically at different prices of the shares in the market.
Share Price Exercise Price Pay off100 135 35125 135 10135 135 0145 135 0160 135 0
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S1≤E+P
Potential loss area
S1≤E+P
10
20
30
40
Value of option
0
E ≤ S1
E > S1
Profit potential area
X
Incase of PremiumSuppose the premium for the option is Ksh.5: Show the gain/losses from the sellers point of view:Share Price Exercise Price Gain/loss (Seller) C = S1 – E + P 100 135 -30125 135 -5135 135 5145 135 5160 135 5130 135
Factors that determine Option Value:
The following are the factors that affect the value of the option.
1.) Exercise Price: -The exercise price on the date of expiration has a negative influence on the value of a call option i.e.The Value of C (pay off) is negatively related to E. The higher the value of E, the lower the value of C and vice versa.
2.)Expiration Date:-The larger the time to expiration date, the more valuable is the call option.
3.) Stock Market Price:- The value of a call option, increases with the stock prices. It has a positive relationship on the call value.
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100 120 140 160Share price
Share price
Value of option
Premium received
Potential loss areas
S1 ≥ E + P
100 120 140 160-5
-10
-15
-20
-25
-30
4.) Variability of the Stock Price:- A call option has a value when there is a possibility that the stock price exceeds the exercise price before the expiration date.
The higher the variability of the stock price, the greater the likelihood that the stock price will exceed the exercise price.
5.) Risk-free Rate:-When you buy a call option, you do not pay the exercise price until you decide to exercise the call option. The higher the interest rate, the greater the interest will be from the delayed payment and vice versa.Generally, the value of the call option is a function of:-
C = F (S0,E,0-2,rf,t)
BINOMIAL MODEL FOR VALUING OPTIONS
The standard discounted cash-flows that are used to appraise or value other investments may
not be used to value options.
This is because the option prices cannot be certainly estimated.
Therefore option values can be determined by use of either:-
1) Binomial Model
2) Block and Scholes model
The basic idea of binomial model which was developed by Fisher Black and Myron Scholes
(1973) is to set up a portfolio which imitates the call option in its pay off.
According to them, the value of the call can be determined by use of a single period binomial.
The following are the assumptions made about the model:
1) The stock currently selling for S can take 2 possible values next year i.e. US and DS where
Us>ds
2) An amount of b can be borrowed or lend at a rate of r the risk free rate. The interest factor
1+r may be represented as R i.e 1+r =R
3) The value of R is greater than d but smaller than U i.e d<R<u
This condition ensures that there is no risk free arbitrage opportunity.
4.) The exercise Price is E
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The value of the call option just before expiration if the price goes up to US is given by:
Cu = Max (Us-E, 0)
Likewise, the value of the call option just before expiration if the stock price goes down to ds, the value is:
Cd= Max (ds - E,0) Cu=Max (Us - E,0)
Suppose, there are portfolio consisting of D shares of the stock and B shillings of borrowing.
Since this portfolio is set-up in such a way that it has a pay off identical to that of a call option at time 1 and the following equations will be satisfied:
Stock Price Rise: CU= D!US - RBStock Price Fall: CD= Dds - RB
Solving for D and B. Simultaneously using the above equations, Then;
D = Cu-Cd = Spread of Possible option Price S(u-d) Spread of Possible share Price
Since the portfolio consisting of D shares and B debt has the same pay off as that of call option then the value of the call option is given by:
C= -DS - B
B= dCu- Ucd(u-d) R
Illustration 1
The following information relates to pioneers stock:
S = Sh.200 280= dsU = 1.4 180=dsD = 0.9E = Sh.220r = 0.1R = 1+r = 1+0.1 = 1.1Cu = Max of (Us-E,0)Cd = Max of (ds- E,0)Us = 1.4 x 200= 280
Cu= Max (Us- E,0) E=220
280-220=60
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Cu =60, 0= 60
Cd = Max (ds-E,0) d=0.9.5 = 200
ds = 0.9 x 200 = 180
Cd= Mo x (180 - 220,0) Cd = 0
D = Cu - Cd =60 - 0 =60 = 0.6 S (u- d) 200(1.4 - 0.9) 100
B = d (u-Ucd = 0.9 (60) -1.4 (0) (u – d)R (1.4 - 0.9) 1.1
= 54 – 0 = 98.18 0.55
C= - DS - B
C = DS - B = 0.6 (200) - 98.18 =21.82Value of call option = 21.8
Examples 2:
Given E = Sh.50 d = 0.8 S =Sh.60 r = 0.12 R = 1+r =1+0.12=R=1.12 U =1.4 12 = 1+0.12 = 1.12
Find the value of the call option C:
Cd = Max (ds - E,0) and Cu = Max (Us - E,O)Ds = 0.8 x 60 = 48 Us = 1.4 x 60 = 84
Cd= Max (48 - 50,0) Cu = Max (84 - 50,0) Cd = 0 Cu = 34
D = Cu-cd =34 – 0 = 34 = 0.944 S(u – d) 60(1.4-0.8) 36
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B = dCu- Ucd = 0.8 (34) – 1.4 (0) = 27.2 (U –d) R (1.4 -0.8) 1.12 0.672 B = 40.476
C= DS- B = 0.944 (60) - 40.476=
56.64 - 40.476 =16.164
Value of call option is 16.164
Assignment
A stock is currently selling for sh.60, the call option on the stock is exerasable a year from now at an Exercise Price of Sh.55 is currently selling for sh.15. The Risk free interest rate is 12% ( r= R= 1+0.12= 0.12),The stock can either rise or fall after a year.It can fall by 30%=. By what Percent can it rise?
d = 0.3S = 60E = 55Co = 15R = 1.12d = 0.7
Cu = Max (Us-E,O) = 60u -55
Cu= Max (Us- E,O ) = 60U- SSCd = Max (ds- E,0).0.7 x 60 - Ss= -0
D = Cu-cd S (u- d)
=Cu- O60 (u-07)
B= dcu - Ucd = 0.3 Cu- 0 (U-d) R (U-0.7) 1.12
Cu = Max (Us- E)Cu = 60U- SS
C = 60 Cu - 0.3Cu 60u-42 1.12u-0.784
55 = 60Cu - 0.3Cu
36
60u - 42 1.12u- 0.784
60(60u - 55) - 0.3 (60u-55) 60u-42 1.124u-0.784
BLACK AND SCHOLES MODEL OF VALUING OPTIONS:
The above analysis was based on the assumption that there were 2 possible values of the stock price at the end of are year.If we assume that there are to possible stock prices at the end of each six month period. The number of possible end of year price increases.
As the period is further shortened from 6 months to 3 months or to 1 month, we get more frequent changes in stock price and a wider range of possible end of year prices.
Eventually, we would reach a situation where prices change more or less continuously leading to a continuam of possible prices at the end of the year.
Calculating the value of such a portfolio, would be done by use of a model developed by Black and Scholes as follows:
Co = SoN (d1) –E Nd2
ert
Where; Co= Equilibrium value of a call option now So = Price of the stock now E = Exercise Price E = Bases of natural logarithm. R = Annualised continuously compounded risk free interest rate t = Length of time in years to the expiration date Nd = The value of cumulative normal density function d1 =
d1 = Ln So + r +1/2 2 t C = M In 1+r/m E
d2 = d1 -
Where Ln = Natural Logarithm U = Standard deviation of the annualised continuously compounded return on stock.
Assumptions of the Model1.) The call option is European.2.) The stock price is continuous is distributed normally
37
3.) There are no transaction costs and taxed4.) There are no restrictions on or Penalties for short selling5.) The stock pays no dividends.6.) The risk free interest rate is known and constant.
IllustrationsFrom the following information available to a market participant determine the value of an European option as per Black and Schole. Model:
- Spot Price of the share Ksh.1120- Exercise Price of call option Ksh.1.100- Short-Term Risk free interest rate (Continuously Compounded) 10% p.a - Time remaining for expiration= 1 month.- Volatility of the share/standard deviation (0) = 0.2
SolutionStep 1: Calculate d1 and d2.
d1 = Ln So + (r + 1/2 o2) t
E
So = 1120E = 1100r = 10%t =11/
12
U = 0.2
d2 = d1-
d1 = In (1120) + 0.1+0.2 2 1/2
1100 2
0.2
d1 = 0.018 +0.01 = 0.483 0.058
d2 = 0.483 - 0.2
0.483 - 0.058= 0.425
Step 2: Find N(d1) and N(d2)
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This represents the probabilities that a random variable has standardized Normal distribution which will assume values less than d1 and d2.
N(d1) = N(0.483) = 0.6844N(d2) = N(0.425) = 0.6628
Step 3: Estimate the present value of the Exercise price using the discounting principle given by:
E = 1100 = 1090.871ert e (0.1 x 1/12
STEP 4: Plug the numbers obtained in the formula to get the value of the Call option
Co= So N(d1) – E N(d2)ert
= 1120 x 0.6844 – 1090.87 x 0.6628 = 43.5
Value of Call option is Ksh.43.5
Illustration 26You are provide with the following information:
Exercise Price Ksh 100Spot Price of the share 80Risk Free Rate 12%Standard deviation of the stock 0.3Time to Maturity 1 year.
RQD: Calculate the value of the call option as per Black & Scholes Model:
SolutionSo = 80E =100r =12% =0.12O = 0.3t = 1
d1 = In So + r + ½ 2 t E
d1 = In (80) + 0.12 + 0.3 2 1
100 2
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d2 = d1 -
d1 = -0.2231 + 0.165 =-0.19370.3
d2 = -0.1937 - 0.3 = -0.4937
N(d1) = N(-01937) = 0.4207N(d2) = N(-0.4937) = 03985
E/ert = 100 = 88.692 e 0.12 x 1
WARRANTS - MAKING CONVERSIONS
Co = So N (d1) – E N(d2) ert
Co = 80 x 0.4207 – 88.692 x 0.3085= 6.295 = 63
Value of Call option = Co = 6.3
PUT- CALL PARITY
The following information is available for the call and Put options on the stocks of A Ltd.
Call (Co) Put (Po)Time to Expiration in months (t) 3 3Risk free Rate (r) 10% 10%Exercise Price (E) sh.50 Sh.50Spot Price so (So) Sh. 60 Sh.60Price/ Value of Option Sh.16 Sh.2
Determine if the Put-Call Parity is working:
Co = So + Po- E/ert Po = Price of Put Co = Price of call
CO = 60+ 2- 50e 0.1 x 0.25 Co = 13.235
But calculated Co = 16Hence the Put-Call Parity is not working:
Po = Co - So + E Put- call Parity is not working ert
16 - 60 + 50 = 4.77 but the calculated is Po = 2 e 0.1 x 0.25
40
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