Under IFRS the permissible cost flow methods are:

• Specific Identification
• FIFO
• Weighted average cost

Liquidity ratios

Cash ratio is the same as Quick without the accounts receivable

Solvency ratios

Long term debt to equity =  total debt / total equity

Debt to equity = total debt / total equity

Total debt ratio = total debt / total assets

Financial leverage ratio = total assets / total equity

LOS 8a

A random variable is an uncertain value determined by chance

An outcome is the realization of a random variable

An event is a set of one or more outcomes.  Two events that cannot both occur are termed “mutually exclusive”
and a set of events that includes all possible outcomes is an “exhaustive” set of events.

LOS 8b

The 2 properties of probabilities are

1. The sum of probabilities of all possible mutually exclusive events is 1
2. The probability of any event cannot be greater than 1 or less than zero

A priori probability measures probabilities based on well defined inputs; empirical probability measures probability from observations or experiments; and subjective probability is an informed guess.

LOS 8c

If the probability of an event is A out of B trials (A/B), the ‘odds for’ are A to (B-A) and the ‘odds against’ are (B-A) to A

LOS 8d

Unconditional probability (marginal probability)  is the probability of an event occurring; conditional probability, P(A|B), is the probability of an event (A) occurring given that another event (B) has occurred.

LOS 8e

The joint probability of 2 events, P(A|B), is the probability that they will both occur.

P(A|B), is the probability of an event (A) occurring given that another event (B) has occurred.

The probability that at least one of two events will occur is P(A or B) = P(A) + P(B) – P(AB).
For a mutually exclusive evens, P(A or B) = P(A) + P(B), since P(AB) = 0

The joint probability of any number of independent events is the product of their individual probabilities.

LOS 8f

The probability of an independent event if unaffected by the occurrence of other events, but the probability of a dependent event is changed by the occurrence of another event.

LOS 8g

Using the total probability rule, the unconditional probability of A is the probability weighted  sum of the conditional probabilities:

LOS 8h

Conditional expectations are used in investments to update expectations when a conditioning event has occurred.

LOS 8i

A tree diagram shows the probabilities of 2 events and the conditional probabilities of 2 subsequent events

LOS 8j

Covariance measures the extent to which 2 random variables tend to be above and below their respective means for each joint realization.  It can be calculated as:

Correlation is a standardised measure of association between 2 random variables; it ranges in value from –1 to +1 and is equal to

Correlation coefficient = Cov A,B / (SD A x SD B)
= -7.2 / (2.450 x 3.098)
= -0.9486

LOS 8K

The expected value of a random variable, E(X) equals

The variance of a random variable Var(X) equals

Standard deviation

The expected returns and variance of a 2 asset portfolio are given by

LOS 8l

Given the joint probablities for Xi & Yi ie  P(XiYi) the covariance is calculated as

where Bi is a set of mutually exclusive and exhaustive events.

LOS 8h

Conditional expectations are used in investments to update expectations when a conditioning event has occurred.

LOS 8i

Bayes formula for updating probabilities based on the occurrence of an event O is:

LOS 8n

The number of ways to order n objects is n factorial n! = n x (n-1) x (n-2) x…. x1.

There are   ways to assign k different labels to N items, where nⁱ 

is the number of items with the label i  

The number of ways to choose a subset of size r from a set of size n when order doesnt matter is  

when order matters, there are   permutations.

Notes

The probability distribution of annual returns from investing in Company A is given below.

Return %  Probability

20              0.1

30              0.6

40              0.3

Press 2nd Data. Enter in the following:
X01: 20 Y01: 10
X02: 30 Y02: 60
X03: 40 Y03: 30
You are entering the probabilities as whole numbers in the 'Y'values e.g. a probability of 0.1 = 10% for Y01.
Now press 2nd STAT. Make sure you have 1-V on the screen.
(If it is not keep presssing, 2nd then ENTER until 1-V appears.)
Scroll down until you find the mean as 32% and the population standard deviation of 6%.

LOS 7a

Descriptive statistics summarize the characteristics of a data set; inferential statistics are used to make probabilistic statements about a population based on a sample.

A population includes all members of a specified group, while a sample is a subset of the population used to draw inferences about the population.

Nominal scale  -- data is put into categories that have no particular order

Ordinal scale – data is put into categories that can be ordered with respect to some characteristic.

Interval scale – differences in data values are meaningful, but ratios, such at twice as much or twice as large are not meaningful

Ratio scale --   ratios of values, such as twice as much or half as large are meaningful, and zero represents the complete absence of the characteristic being measured.

LOS 7b

Any measurable characteristic of a population is called a parameter.

A characteristic of a sample is given by a sample statistic.

An interval is a range of values.

A frequency distribution groups observations into classes or intervals.

LOS 7c

Relative frequency is the percentage of total observations falling within an interval; cumulative relative frequency for an interval is the sum of the relative frequencies for all values less than or equal to a given maximum value.

Relative frequency is found by dividing the frequency of the interval by the total number of frequencies

Histograms and frequency polygons are graphical tools used to illustrate frequency distributions.

LOS 7d

median – midpoint of dataset

mode – most frequent value

LOS 7e

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies.  Examples of quantiles include:

1. Quartiles – distribution is divided into quarters
2. Quintile – distribution is divided into fifths
3. Decile – distribution is divided into tenths
4. Percentile – distribution is divided inot hundreths

LOS 7f

The range is the difference between the largest and smallest values in the dataset

Mean absolute deviation (MAD) is the average of the absolute values of the deviations from the arithmetic mean:

Standard deviation is the positive square root of the variance and is frequently used as a quantitative measure of risk.

LOS 7g

Chebyshev’s inequality states that the proportion of the observations within k standard deviations of the mean is at least 1-1/k² for all k > 1

LOS 7h

The coefficient of variation for sample data, is the ratio of the standard deviation of the sample to its mean (expected value of the underlying distribution)

The Sharpe ratio measures excess return per unit of risk

LOS 7i

Skewness describes the degree to which a distribution is not symmetric about its mean.

• A right skewed distribution has positive sample skewness and has a mean that is greater than its median that is greater than its mode
• A left skewed distribution has a negative skewness and has a mean that is less than its median that is less than its mode.
• Sample skew with an absolute value greater than .5 is considered significantly different from zero

LOS 7j

Kurtosis measures the peakedness of a distribution and the probability of extreme outcomes (thickness of tails)

• Excess kurtosis is measured realtive to a normal distribution, which has a kurtosis of 3.
• Positive values of excess kurtosis indicate a distribution that is leptokurtic (fat tails, more peaked) so that the probability of extreme outcomes is greater than the normal distribution.
• Negative values of excess kurtosis indicate a platykurtic distribution (thin tails, less peaked)
• Excess kurtosis with an absolute value greater that 1 is considered significant

LOS 6a

NPV – The NPV is the present value of a projects future cash flows, discounted at the firms cost of capital, less the projects cost.

NPV = PV of Cash Inflows - PV of Cash Outflows

IRR – Internal rate of return is the discount rate that makes the NPV == 0 (equates the PV of the expected future cash flows to the projects initial cost)

The NPV rule is to accept a project if NPV > 0; the IRR rule is to accept a project if IRR > required rate of return.

For an independent (single) project, these rules produce the exact same decision.

For mutually exclusive projects, IRR rankings and NPV rankings may differ due to differences in project size or in the timing of the cash flows.  Choose the project with the higher NPV as long as it is positive.

A project may have multiple IRRs or no IRR.

The IRR method assumes all cashflows from the project are reinvested at the IRR        

LOS 6b

HPR

– Holding Period Return is the  % change in the value of an investment over the period it is held.

The holding period return (or yield) is calculated as:

HPY

– Holding period yield is the actual return an investor will receive if the money market instrument is held until maturity

LOS 6c

The money weighted rate of return is the IRR calculated using periodic cash flows into and out of an account and is the discount rate that makes the PV of cash inflows equal to the PV of cash outflows.

Time weighted rate of return – is calculated from the accounts periodic holding period returns and is the preferred performance measure.

LOS 6d

BDY -

– Bank Discount Yield is the % discount from face value, annualized by multiplying by 360/days to maturity

Bank discount yields are not true yields because they are based on a percentage of face (maturity) value instead of on the original amount invested.  They are annualized without compounding since the actual discount from face value is simply multiplied by the number of periods in a “year”.  The year used in 360 days.

EAY

– Effective annual yield is the annualized HPY on the basis of a 365 day year and incorporates the effects of compounding.  It converts a t-day holding period to a compound annual yield based on a 365 day year.

MMY

– Money market yield (or rMM) is the annualized yield that is based on price and a 360 day year and does not account for the effects of compounding – it assumes simple interest

           NB:// easy in relation to the above!

Examples

money weighted rate of return

Assume an investor purchases a share of stock for $50 at time t=0

and another share at $65 at time = t1

and at the end of year 1 and year 2, the stock paid a $2.00 dividend

Also at the end of year 2, the investor sold both shares for $70 each

1. [CF]2nd [CLR WORK]
2. 50 [+/-] [ENTER]
3. 63 [+/-] [ENTER]
4. 144 [+/-] [ENTER]
5. [IRR] [CPT]                      -------------- NB:// Watch your minus signs here or it wont work!

time weighted rate of return

HPR1 = (65+2)/50 – 1 = 34%

HPR2 = (140+4)/130 –1 = 10.77%

thus = [(1.34)()1.1077]^{.5 –}1 = 21.83%

Calculator – Texus Instruments BAII Plus Calculator manual

• N      No of compounding periods
• I/Y    Interest rate per compounding period
• PV    Present Value
• FV    Future Value
• PMT  Annuity payments or constant periodic cash flow
• CPT  Compute

Terms

Required rate of return -- The rate of return needed to induce investors or companies to invest in something.

Discount rate -- is an interest rate a central bank charges depository institutions that borrow reserves from it.

Opportunity cost -- or economic opportunity loss is the value of the next best alternative forgone as the result of making a decision.

Types of risk

• Default risk  -- The risk that  a borrower will not make the promised payments in a timely manner.
• Liquidity risk -- is the risk of receiving less than a fair value for an investment if it must be sold for $$$ quickly
• Maturity risk  -- Longer maturity bonds have more maturity risk than the shorter term bonds and require  maturity risk premium

Required interest rate on a security = nominal risk-free rate + default risk premium + liquidity premium + maturity premium

Effective annual rate (EAR)  = (1 + periodic rate)^m – 1

periodic rate = stated annual rate/m

m = number of compounding periods per year

Annuity is a stream of equal cash flows that occurs at equal intervals over a given period

• Ordinary annuity – payment received at the end of each compounding period
• Annuity due – payment received at start – Calc [2nd][BGN][2nd][SET][2nd][QUIT]

Perpetuity  -- a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time

Loan Amortization – is the process of paying off a loan with a series of periodic load payments, whereby a portion of the outstanding loan amount is paid off, or amortized, with each payment.

LOS 5a

An interest rate can be interpreted as the return required in equilibrium, the discount rate for calculating the present value of future cash flows, or as the opportunity cost of consuming now, rather than saving and investing.

LOS 5b

The required rate of return on a security = real risk free rate + expected inflation + default risk premium + liquidity premium + maturity risk premium.

LOS 5c

FV = PV(1+I/Y)^N

PV = FV (1+I/Y)^N

LOS 5d

An annuity is a series of equal cash flows that occurs at evenly spaced intervals over time.

Ordinary annuity cash flows occur at the end of each time period.  Annuity due cash flows occur at the beginning of each time period.

Perpetuities are annuities with infinite lives  (perpetual annuities)

PV perpetuity = PMT

                          I\Y

LOS 5e

The present (future) value of any series of cash flows is equal to the sum of the present (future) values of the individual cash flows.

A mortgage is an amortizing load, repaid in a series of equal payments (an annuity), where each payment consists of the interest for the period (which decreases with each payment) and the amount of principle repaid (which increases with each payment)

Notes

For semi annual compounding – halve the interest rate and halve N periods (4 if this was quarterly)

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