Download Portfolio Risk and return

Risk and Return
The relationship between risk and return is a fundamental financial relationship that affects
expected rates of return on every existing asset investment. The Risk-Return relationship is
characterized as being a "positive" or "direct" relationship meaning that if there are expectations
of higher levels of risk associated with a particular investment then greater returns are required
as compensation for that higher expected risk. Alternatively, if an investment has relatively
lower levels of expected risk then investors are satisfied with relatively lower returns.
This risk-return relationship holds for individual investors and business managers. Greater
degrees of risk must be compensated for with greater returns on investment. Since investment
returns reflects the degree of risk involved with the investment, investors need to be able to
determine how much of a return is appropriate for a given level of risk. This process is referred
to as "pricing the risk". In order to price the risk, we must first be able to measure the risk (or
quantify the risk) and then we must be able to decide an appropriate price for the risk we are
being asked to bear.i
Real Rate of Interest
This is the rate that has been adjusted to remove the effects of inflation thereby revealing the real cost of
funds to the borrower, and the real yield to the lender. The real interest rate is the growth rate of
purchasing power derived from an investment. By adjusting the nominal interest rate to compensate for
inflation, the purchasing power of a given level of capital is held constant over time.ii
The real rate of interest is calculated as below:
Exercise
1. Assume the current rate of inflation is 8% and a nominal interest rate of 10% per annum. What is
the real rate of interest for the investor?
[(1 + .10) ÷ (1 + .08)] - 1 = 1%
This means that the investor’s true increase in purchasing power is only 1% after inflation is
taken into account.
2. What would you expect the nominal rate of interest to be if the real rate is 5% and the expected
inflation rate is 3%?
[(1 + x) ÷ (1 + .03)] – 1 = 0.05
(1 + x) ÷ (1 + .03) = 1.05
1 + x = 1.0815
x = 8.15%
Term Structure of Interest Rates
This refers to the relationship between interest rate charged or paid for money and the length of time that
money is held. Therefore, bonds with identical risk, liquidity, and tax characteristics may have different
interest rates because the time remaining to maturity is different. The interest rates can be graphically
illustrated on a yield curve.
The Yield curve is a plot of the yields on bonds with differing terms to maturity but the same risk,
liquidity and tax considerations. Yield curves may be
– Upward-sloping - this occurs when long-term rates are above short-term rates
– Flat – this occurs when short- and long-term rates are the same
– Inverted – this occurs when long-term rates are below short-term rates
– Humped –A number of reasons may cause this scenario because the hump can be above
short and long rates or can be a dip below them.
Sample of Upward sloping and Downward sloping yield curves
Expected Return
This reflects the probabilities and the respective possible outcomes of making an investment decision.
The uncertainty of a decision influences the value the cash flows that accrue from choosing a particular
alternative.
Expected Cash flow
This is the weighted average of the possible cash flow outcomes such that the weights are the
probabilities of the occurrence of the various states of the economy.
For example, JohnRay Co. is planning on investing $1,000,000 in a security. There is a 15% probability
of economic boom during which the investment will yield $60,000. There is a 20% chance of mild growth
in the economy during which the company will earn $40,000. If there is a recession, the company will
earn only $20,000. Compute the Expected return and the Expected rate of return.
State of economy
Boom
Mild growth
Recession
A
B
Probability
15%
20%
65%
Cash flows
60,000
40,000
20,000
120,000
C=AxB
Expected
Return
9,000.00
8,000.00
13,000.00
30,000.00
D = B ÷ 1,000,000
Weight
6.00%
4.00%
2.00%
Out Beach is reviewing an investment that will provide various possible returns as below:
Probability
Return
0.1
-8%
0.2
10%
0.5
15%
0.2
20%
Compute the expected rate of return on this investment.
Answer:
(0.01)
0.02
0.08
0.04
13%
E=AxD
Expected Rate
of Return
0.900%
0.800%
1.300%
3.000%
Standard Deviation
A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the
deviation. Standard deviation (∂) is calculated as the square root of variance.
For example, Hillary Brown Cosmetics is considering investing in a security. The probability and the
respective possible returns are detailed as below:
Probability
5%
25%
30%
35%
Returns
-8%
10%
15%
20%
Compute the expected return and the standard deviation of the returns.
A
B
C=AxB
D = (B-k*)² x A
5%
-8%
-0.4%
0.0022
25%
10%
3%
0.0002
30%
15%
5%
0.0001
35%
20%
7%
0.0017
∑=
k* =
14%
0.004265
∂=
6.53%
This means that given the expected return of 14%, the investor can reasonably anticipate the actual return
to be 14% ± 6.53% two-thirds of the time.
Portfolio Risk and return
The above is relevant when reviewing a single investment. An element of risk can be reduced by
diversification. There are two types of risk


Systematic
Unsystematic
Systematic risk is also called market risk and is the risk inherent in the aggregate market that cannot be
solved by diversification. Some common sources of market risk are recessions, wars, interest rates and
others that cannot be avoided through a diversified portfolio. Though systematic risk cannot be fixed with
diversification, it can be hedged.
Company or industry specific risk that is inherent in each investment. The amount of unsystematic risk
can be reduced through appropriate diversification. For example, news that is specific to a small number
of stocks, such as a sudden strike by the employees of a company you have shares in, is considered to be
unsystematic risk.
Holding period
The total return received from holding an asset or portfolio of assets. Holding period return/yield is
calculated as the sum of all income and capital growth divided by the value at the beginning of the period
being measured.
Holding period return is a very basic way to measure how much return you have obtained on a particular
investment. This calculation is on a per-dollar-invested basis, rather than a time basis, which makes it
difficult to compare returns on different investments with different time frames. When making
comparisons such as this, the annualized calculation shown above should be used.
Exercise
a. Using the following data on the HGN stock price, calculate the holding-period returns for each
month from August 2011
Month
July
Aug
Sept
Oct
Nov
Dec
HGN
1,328
1,350
1,355
1,407
1,455
1,490
Holding
Period returns
1.66%
0.37%
3.84%
3.41%
2.41%
b. Assume you purchase a share of stock last year for $100. One year later the stock price is
$116.91 and it has paid a dividend of $4.50. What is the holding period gain?
= 4.50 + (116.91 – 100) ÷ 100
= 21.41%
The standard deviation of these holding period returns can also be calculated, however, because historical
data is being used, we assume that each observation has an equal probability of occurrence. It is computed
as follows:
Using the data for the HGN stock, the standard deviation is computed as follows:
Month
HGN
July
Aug
Sept
Oct
Nov
Dec
1,328
1,350
1,355
1,407
1,455
1,490
Average =
Holding
period
return
1.66%
0.37%
3.84%
3.41%
2.41%
2.34%
0.001%
0.008%
0.005%
0.002%
0.000%
0.015%
∂ = 1.24%
Coefficient of Variation
This is a statistical measure of the dispersion of data points in a data series around the mean. It is
calculated as follows:
The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful
statistic for comparing the degree of variation from one data series to another, even if the means are
drastically different from each other.
In the investing world, the coefficient of variation allows you to determine how much volatility (risk) you
are assuming in comparison to the amount of return you can expect from your investment. In simple
language, the lower the ratio of standard deviation to mean return, the better your risk-return trade
off.
Note that if the expected return in the denominator of the calculation is negative or zero, the ratio will not
make sense.
Using the data for HGN, the Coefficient of Variation is = 1.24 ÷ 2.34 = .53
Portfolio Beta
Beta measures the relationship between an investment’s return and the market’s return. The higher the
beta, the greater the volatility of the return and the greater the likelihood that it will provide much higher
or much lower returns than the market.
The portfolio beta reveals the relationship between the portfolio’s return and the market’s various returns.
It shows the non-diversifiable risk of the portfolio. It is computed by finding the weighted average of the
individual stock betas. For example, assume a simple portfolio of two stocks –
 $100,000 of ABC stock with a beta of 1.1
 $50,000 of XYZ stock with a beta of 2
The portfolio beta is = (100,000 ÷ 150,000) x 1.1 + (50,000 ÷ 150,000) x 2
= 1.4
The addition of a stock with a relatively low beta will reduce the portfolio’s volatility and the addition of
a stock with a relatively high beta will do the reverse.
Capital Asset Pricing Model (CAPM)
A model that describes the relationship between risk and expected return and that is used in the pricing of
risky securities.
The general idea behind CAPM is that investors need to be
compensated in two ways: time value of money and risk. The time
value of money is represented by the risk-free (rf) rate in the
formula and compensates the investors for placing money in any
investment over a period of time. The other half of the formula
represents risk and calculates the amount of compensation the
investor needs for taking on additional risk. This is calculated by
taking a risk measure (beta) that compares the returns of the asset to
the market over a period of time and to the market premium (Rm-rf).
The CAPM says that the expected return of a security or a portfolio equals the rate on a risk-free security
plus a risk premium. If this expected return does not meet or beat the required return, then the investment
should not be undertaken. The security market line plots the results of the CAPM for all different risks
(betas).
Using the CAPM model and the following assumptions, we can compute the expected return of a stock in
this CAPM example: if the risk-free rate is 3%, the beta (risk measure) of the stock is 2 and the expected
market return over the period is 10%, the stock is expected to return 17% (3%+2(10%-3%)).
Discrete Probability Distributions
If a random variable is a discrete variable, its probability distribution is called a discrete probability
distribution. Assume you flip a coin two times. This simple statistical experiment can have four possible
outcomes: HH, HT, TH, and TT. Now, let the random variable X represent the number of Heads that
result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a
discrete random variable.
Continuous Probability Distributions
If a random variable is a continuous variable, its probability distribution is called a continuous probability
distribution. A continuous probability distribution differs from a discrete probability distribution in
several ways including the following –
 The probability that a continuous random variable will assume a particular value is zero.
 As a result, a continuous probability distribution cannot be expressed in tabular form.
 Instead, an equation or formula is used to describe a continuous probability distribution.
i
http://uwf.edu/rconstand/5994content2003/T1-Overview/T1-OverviewP04.htm
ii
http://www.investopedia.com