1
DSC2604
ADDITIONAL STUDY NOTES AND EXERCISES
INDEX
A. Assignment comments and hints p.1
B. Review of discrete random variables and joint distributions p.3
C. Matrix Algebra p.11
D. Supplementary exercises (including example of exam paper) p.14
E. Solutions to supplementary exercises p.23
A. ASSIGNMENT COMMENTS AND HINTS
1. Do not just discard zeroes in calculations! You may want to save time but may end up
making costly mistakes. This is particularly true when working with squares and square
roots. For example: You want to calculate (640 000) 2 and try to save time by writing 642 =
4096. But in your final answer you have to add the correct number of zeroes, so that the
correct answer is 409 600 000 000. Not 40 960 000! Even worse errors occur when trying
to calculate something like 16 000 by using the fact that 16 = 4. The answer is not
4 000 or 400; in fact it is 126,49 (to two decimals). But it is true that 160 000 = 400! Do
you understand the difference?
2. When calculating future or present values, give all steps and draw a time-line to help you.
If there are cash inflows of R2 000 at the end of each of 4 years with an initial investment
of R6 000, the situation is as follows:
t = 0---------l---------2--------3--------4
↓ ↑ ↑ ↑ ↑
6 000 2 000 2 000 2 000 2 000
Assume a discount rate r. The present value (PV) is the t = 0 value of inflows, i.e.
PV = 2 000/(1+r) + 2 000/(1+r)2 + 2 000/(1+r)3 + 2 000/(1+r)4.
The net present value (NPV) is the t = 0 value of inflows minus the initial outflow, i.e.
, 2
NPV = 2 000/(1+r) + 2 000/(1+r)2 + 2 000/(1+r)3 + 2 000/(1+r)4 – 6 000.
The future value (FV) (at t = 4) of inflows is:
FV = 2 000(1+r)3 + 2 000(1+r)2 + 2 000(1+r) + 2 000.
3. The notation σ(R) is the standard deviation of the rate of return, NOT the standard
deviation σ multiplied by rate of return R.
4. Know the difference between probability of an asset return and weight of an asset in a
portfolio. For example, if asset A has rate of return 30% with probability 0,4 and rate of
return 25% with probability 0,6 then the expected rate of return for A is
E(RA ) = 0,4 (0,3) + 0,6 (0,25) = 0,27
and the variance of rate of return is
σ2 (RA) = 0,4 [0,3 - E(RA)]2 + 0,6 [0,25 - E(RA)]2 = 0,0006.
The probabilities are not squared.
But suppose there is another uncorrelated asset B with E(RB) = 22% and σ2 (RB) = 0,009
and portfolio P consists of 0,4 of funds in A and 0,6 of funds in B.
Then for P we have:
E(RP) = 0,4 E(RA ) + 0,6 E(RB) and the variance of rate of return is
σ2 (RP) = (0,4)2σ2 (RA) + (0,6)2σ2 (RB)
= (0,4)2 0,0006 + (0,6)2 0,009.
The weights are squared.
If the returns of A and B are correlated, you will also need to know the covariance or
correlation coefficient. In this case:
σ2 (RP) = (0,4)2σ2 (RA) + (0,6)2σ2 (RB) + 2 (0,4)(0,6)cov(RA,RB).
5. Note that if X is a normal random variable with probability density function f , then
a
P( X ≤ a) =∫ f ( x)dx.
−∞
, 3
This is the area under the graph of f , left of point a. Also note that
∞
) µ=
E( X = ∫−∞
xf ( x)dx;
P(μ – σ ≤ X ≤ μ + σ) = 0,68
P(μ – 2σ ≤ X ≤ μ + 2σ) = 0,95
P(μ – 3σ ≤ X ≤ μ + 3σ) = 0,99
Random normal variables are important: We often assume that the NPV is normally distributed
and that the logarithms of returns on assets are normally distributed.
B. REVIEW OF DISCRETE RANDOM VARIABLES AND JOINT
DISTRIBUTIONS
1. Random variables
In this subsection we give a basic definition of a random variable and examples.
Definition 1 (random variable)
A random variable X is a variable that can take on a given set of values, called the sample space
and denoted by Ω, where the likelihood of the values in Ω is determined by X’s probability
distribution function.
Example 1
Consider the price of Microsoft stock next month. Since the price of Microsoft stock next month
is not known with certainty today, we consider it a random variable. The price next month must
be positive and realistically it can’t get too large. Therefore, the sample space is the set of
positive real numbers bounded above by some large number.
In Financial Modelling, it remained an open question as to what is the best characterisation of
the probability distribution of stock prices. The log-normal distribution is one possibility. In this
regard, if S is a positive random variable such that lnS is normally distributed then S has a log-
normal distribution. You will come across the two distributions very often if you do Honours in
Financial Modelling.
, 4
Example 2
Consider a one month investment in Microsoft stock. That is, we buy 1 share of Microsoft stock
today and plan to sell it next month. Then the rate of return on this investment is a random
variable since we do not know its value today with certainty. In contrast to prices, the rate of
return can be positive or negative and is bounded below by -100%. The normal distribution is
often a good approximation to the distribution of simple monthly rates of return and is a better
approximation to the distribution of continuously compounded monthly rates of return.
Example 3
As a final example under this subsection, consider a random variable X defined to be equal to
one if the monthly price change on Microsoft stock is positive and is equal to zero if the price
change is zero or negative. Hence the sample space is trivially the set {0, 1}. If it is equally
likely that the monthly price change is positive or negative (including zero) then the probability
that X = 1 or X = 0 is 0,5.
2. Discrete random variables
Consider a random variable generically denoted by X and its set of possible values or sample
space denoted by Ω.
Definition 2 (discrete random variable)
A discrete random variable is one that can take on a finite number of n different values x1,
x2,...,xn or, at most, a countably infinite number of different values x1, x2, ....
Definition 3 (probability mass function)
The probability mass function or frequency function of the discrete random variable X is defined
as:
p(x) = P(X=x)
[Read P(X=x)as “the probability that random variable X assumes the value x”]
Both names, probability mass function and frequency function, can be used and we will use them
interchangeably.
A probability mass function has three properties:
1. p(x) ≥ 0 for all x ∈ Ω;
2. p(x) = 0 for all x ∉ Ω;
∑ p( x) = 1
3. x∈Ω
DSC2604
ADDITIONAL STUDY NOTES AND EXERCISES
INDEX
A. Assignment comments and hints p.1
B. Review of discrete random variables and joint distributions p.3
C. Matrix Algebra p.11
D. Supplementary exercises (including example of exam paper) p.14
E. Solutions to supplementary exercises p.23
A. ASSIGNMENT COMMENTS AND HINTS
1. Do not just discard zeroes in calculations! You may want to save time but may end up
making costly mistakes. This is particularly true when working with squares and square
roots. For example: You want to calculate (640 000) 2 and try to save time by writing 642 =
4096. But in your final answer you have to add the correct number of zeroes, so that the
correct answer is 409 600 000 000. Not 40 960 000! Even worse errors occur when trying
to calculate something like 16 000 by using the fact that 16 = 4. The answer is not
4 000 or 400; in fact it is 126,49 (to two decimals). But it is true that 160 000 = 400! Do
you understand the difference?
2. When calculating future or present values, give all steps and draw a time-line to help you.
If there are cash inflows of R2 000 at the end of each of 4 years with an initial investment
of R6 000, the situation is as follows:
t = 0---------l---------2--------3--------4
↓ ↑ ↑ ↑ ↑
6 000 2 000 2 000 2 000 2 000
Assume a discount rate r. The present value (PV) is the t = 0 value of inflows, i.e.
PV = 2 000/(1+r) + 2 000/(1+r)2 + 2 000/(1+r)3 + 2 000/(1+r)4.
The net present value (NPV) is the t = 0 value of inflows minus the initial outflow, i.e.
, 2
NPV = 2 000/(1+r) + 2 000/(1+r)2 + 2 000/(1+r)3 + 2 000/(1+r)4 – 6 000.
The future value (FV) (at t = 4) of inflows is:
FV = 2 000(1+r)3 + 2 000(1+r)2 + 2 000(1+r) + 2 000.
3. The notation σ(R) is the standard deviation of the rate of return, NOT the standard
deviation σ multiplied by rate of return R.
4. Know the difference between probability of an asset return and weight of an asset in a
portfolio. For example, if asset A has rate of return 30% with probability 0,4 and rate of
return 25% with probability 0,6 then the expected rate of return for A is
E(RA ) = 0,4 (0,3) + 0,6 (0,25) = 0,27
and the variance of rate of return is
σ2 (RA) = 0,4 [0,3 - E(RA)]2 + 0,6 [0,25 - E(RA)]2 = 0,0006.
The probabilities are not squared.
But suppose there is another uncorrelated asset B with E(RB) = 22% and σ2 (RB) = 0,009
and portfolio P consists of 0,4 of funds in A and 0,6 of funds in B.
Then for P we have:
E(RP) = 0,4 E(RA ) + 0,6 E(RB) and the variance of rate of return is
σ2 (RP) = (0,4)2σ2 (RA) + (0,6)2σ2 (RB)
= (0,4)2 0,0006 + (0,6)2 0,009.
The weights are squared.
If the returns of A and B are correlated, you will also need to know the covariance or
correlation coefficient. In this case:
σ2 (RP) = (0,4)2σ2 (RA) + (0,6)2σ2 (RB) + 2 (0,4)(0,6)cov(RA,RB).
5. Note that if X is a normal random variable with probability density function f , then
a
P( X ≤ a) =∫ f ( x)dx.
−∞
, 3
This is the area under the graph of f , left of point a. Also note that
∞
) µ=
E( X = ∫−∞
xf ( x)dx;
P(μ – σ ≤ X ≤ μ + σ) = 0,68
P(μ – 2σ ≤ X ≤ μ + 2σ) = 0,95
P(μ – 3σ ≤ X ≤ μ + 3σ) = 0,99
Random normal variables are important: We often assume that the NPV is normally distributed
and that the logarithms of returns on assets are normally distributed.
B. REVIEW OF DISCRETE RANDOM VARIABLES AND JOINT
DISTRIBUTIONS
1. Random variables
In this subsection we give a basic definition of a random variable and examples.
Definition 1 (random variable)
A random variable X is a variable that can take on a given set of values, called the sample space
and denoted by Ω, where the likelihood of the values in Ω is determined by X’s probability
distribution function.
Example 1
Consider the price of Microsoft stock next month. Since the price of Microsoft stock next month
is not known with certainty today, we consider it a random variable. The price next month must
be positive and realistically it can’t get too large. Therefore, the sample space is the set of
positive real numbers bounded above by some large number.
In Financial Modelling, it remained an open question as to what is the best characterisation of
the probability distribution of stock prices. The log-normal distribution is one possibility. In this
regard, if S is a positive random variable such that lnS is normally distributed then S has a log-
normal distribution. You will come across the two distributions very often if you do Honours in
Financial Modelling.
, 4
Example 2
Consider a one month investment in Microsoft stock. That is, we buy 1 share of Microsoft stock
today and plan to sell it next month. Then the rate of return on this investment is a random
variable since we do not know its value today with certainty. In contrast to prices, the rate of
return can be positive or negative and is bounded below by -100%. The normal distribution is
often a good approximation to the distribution of simple monthly rates of return and is a better
approximation to the distribution of continuously compounded monthly rates of return.
Example 3
As a final example under this subsection, consider a random variable X defined to be equal to
one if the monthly price change on Microsoft stock is positive and is equal to zero if the price
change is zero or negative. Hence the sample space is trivially the set {0, 1}. If it is equally
likely that the monthly price change is positive or negative (including zero) then the probability
that X = 1 or X = 0 is 0,5.
2. Discrete random variables
Consider a random variable generically denoted by X and its set of possible values or sample
space denoted by Ω.
Definition 2 (discrete random variable)
A discrete random variable is one that can take on a finite number of n different values x1,
x2,...,xn or, at most, a countably infinite number of different values x1, x2, ....
Definition 3 (probability mass function)
The probability mass function or frequency function of the discrete random variable X is defined
as:
p(x) = P(X=x)
[Read P(X=x)as “the probability that random variable X assumes the value x”]
Both names, probability mass function and frequency function, can be used and we will use them
interchangeably.
A probability mass function has three properties:
1. p(x) ≥ 0 for all x ∈ Ω;
2. p(x) = 0 for all x ∉ Ω;
∑ p( x) = 1
3. x∈Ω
