performance analysis of an investment requires
investors to measure returns over time
how are return and risk related
intricately
return measurement helps
in the understanding of investment risk
holding period return (HPR)
measures the cumulative return
rHPR =
Pend + distributions - Pbeg / Pbeg
dollar profit (loss) on an investment =
pi = Pend + distributions - Pbeg
percentage return
pi / pbeg
with varying holding periods, holding period returns are
not good for comparison
converting holding period return to annual returns
necessary state an investment's performance in terms of annual percentage rate (APR) and an effective annual rate of return (EAR) by using two formulas: simple annual return, effective annual return
simple annual return
APR = HPR / n
effective annual rate
EAR = (1 + HPR)^1/n -1
where n is the number of years or proportion of a year that the holding period consists of
extrapolating short term HPRs into APRs and EARs is
mathematically correct, but often unrealistic and infeasible
extrapolating holding period returns implies
earning the same period rate over and over again in one year
a short holding period with fairly high HPR would lead to
huge numbers if returns is extrapolated
future performance of more investments is
uncertain
risky implies
not only the potential for loss but also for uncertain gain (risk does not take sides)
risk can be defined as
a measure of the uncertainty in a set of potential outcomes for an event in which there is a chance of some loss
why is it important to measure and analyze the risk potential of an investment
to make an informed decision
variance and standard deviation are measures of
dispersion
variance and standard deviation help researchers
determine how spread out or clustered together a set of numbers or outcomes is around their mean or average value
the larger the variance,
the great is the variability and hence the riskiness of a set of values
variance (x) =
the sum of (Xi - fixedX)^2 / n-1 = o^2
standard deviation =
square root of variance
over the past 5 decades, riskier investment groups have earned
higher returns (and vice versa)
history shows that the higher the return once expects the
great would be the risk (variability of return) that one would have to tolerate
for future investments we need
expected or ex-ante rather than ex-post return and risk measures
for ex-ante we use
probability distributions, and then the expected return and risk measures are estimated using an equation
expected payoff =
the sum of pay offi x probabilityi
when setting up probability distributions the following 2 rules must be followed
1. the sum of all probabilities must always add up to 1 or 100%
2. each individual probability estimate must be positive
investments must be analyzed in terms of both
their return potential as well as their riskiness or variability
historically, it has been proved that high returns are accompanied by
higher risk
investment rule number 1
if faced with 2 investment choices having the same expected returns, select the one with the lower expected risk
investment rule number 2
if two investment choices have similar risk profiles select the one with the higher expected return
to maximize return and minimize risk, it would be ideal to select an investment that has
a higher expected return and a lower expected risk than the other alternatives
realistically, higher expected returns are accompanied by
greater variances and the choice is not that clear cut, the investor's tolerance for and attitude towards risk matters
in a world fraught with uncertainty and risk
diversification is key!
diversification
the spreading of wealth over a variety of investment opportunities so as to eliminate some risk
by dividing up one's investments across many relatively low-correlated assets, companies, industries, and countries it is possible to
considerably reduce one's exposure to risk
the portfolio's expected return E(rp), return can be measured in 2 ways:
1. weighted average of each stock's expected return
2. expected return of the portfolio's conditional returns
weighted average of each stocks expect return =
weight in Zig
E(rZIG) + Weight in Zag
E(rZAG)
expected return of the portfolio's conditional returns =
sum of prob of econ state X portfolio return in econ state
total risk is made up of two parts
1. unsystematic or diversifiable risk
2. systematic or non-diversifiable risk
unsystematic risk, co-specific, diversifiable risk
product or labor problems
systematic risk, market, non-diversifiable risk
recession or inflation
well diversified portfolio
one whose unsystematic risk has been completely eliminated (large mutual fund companies)
beta
measures volatility of an individual security against the market as a whole
average beta
1.0 - market beta
Beta < 1.0
less risky than market
ex. utility socks
Beta > 1.0
more risky than the market
ex. high-tech socks
Beta = 0
independent of the market
ex. t-bill
betas are estimated by
running a regression of stock returns against market returns (independent variable)
what measures beta or the systematic risk estimate of the stock
the slope of regression line (coefficient of the independent variable)
once individual stock betas are determined, the portfolio beta
is easily calculated as the weighted average
2 different measures of risk related to financial assets
standard deviation (or variance) and beta
standard deviation
measure of total risk of an asset, both its systematic and unsystematic risk
beta
measure of asset's systematic risk
if an asset is part of a well-diversified portfolio
use beta as the measure of risk
if we do not have a well-diversified portfolio, it is more prudent to use
standard deviation as the measure of risk for our asset
the Security Market Line (SML)
shows the relationship between an assets required rate of return an its systematic risk measure
SML is based on 3 assumptions
1. there is a basic reward for waiting: the risk-free rate. consumption.
2. the greater the risk, the greater the expect reward. Investors expect to be proportionately compensated for bearing risk.
3. there is consistent trade-off between risk and reward
These three assumptions imply that SML is
upward sloping, has a constant slope (linear) and has the risk-free rate as its y intercept
The Capital Asset pricing model (CAPM)
equation form of the SML, used to quantify the relationship between expected rate of return and systematic risk
CAMP states that the
expected return of an investment is a function of:
1. the time value of money (the reward for waiting)
2. a reward for taking on risk
3. the amount of risk
the slope of the SML is the
market risk premium (E(rm)-rf) and not beeta
CAMP equations comes from basic y = a + bx where substituting
E(ri) - y variable
rf - intercept a
(E(rm)-rf) the slop b
beta - random variable on the x-axis
therefore we get
E(ri) = rf + B(E(rm)-rf)
the SML has many practical applications such as
1. determining the prevailing market or risk premium
2. determining the investment attractiveness of stock
3. determining portfolio allocation weights and expected return
performance analysis of an investment requires
investors to measure returns over time
how are return and risk related
intricately
return measurement helps
in the understanding of investment risk
holding period return (HPR)
measures the cumulative return
rHPR =
Pend + distributions - Pbeg / Pbeg
dollar profit (loss) on an investment =
pi = Pend + distributions - Pbeg
percentage return
pi / pbeg
with varying holding periods, holding period returns are
not good for comparison
converting holding period return to annual returns
necessary state an investment's performance in terms of annual percentage rate (APR) and an effective annual rate of return (EAR) by using two formulas: simple annual return, effective annual return
simple annual return
APR = HPR / n
effective annual rate
EAR = (1 + HPR)^1/n -1
where n is the number of years or proportion of a year that the holding period consists of
extrapolating short term HPRs into APRs and EARs is
mathematically correct, but often unrealistic and infeasible
extrapolating holding period returns implies
earning the same period rate over and over again in one year
a short holding period with fairly high HPR would lead to
huge numbers if returns is extrapolated
future performance of more investments is
uncertain
risky implies
not only the potential for loss but also for uncertain gain (risk does not take sides)
risk can be defined as
a measure of the uncertainty in a set of potential outcomes for an event in which there is a chance of some loss
why is it important to measure and analyze the risk potential of an investment
to make an informed decision
variance and standard deviation are measures of
dispersion
variance and standard deviation help researchers
determine how spread out or clustered together a set of numbers or outcomes is around their mean or average value
the larger the variance,
the great is the variability and hence the riskiness of a set of values
variance (x) =
the sum of (Xi - fixedX)^2 / n-1 = o^2
standard deviation =
square root of variance
over the past 5 decades, riskier investment groups have earned
higher returns (and vice versa)
history shows that the higher the return once expects the
great would be the risk (variability of return) that one would have to tolerate
for future investments we need
expected or ex-ante rather than ex-post return and risk measures
for ex-ante we use
probability distributions, and then the expected return and risk measures are estimated using an equation
expected payoff =
the sum of pay offi x probabilityi
when setting up probability distributions the following 2 rules must be followed
1. the sum of all probabilities must always add up to 1 or 100%
2. each individual probability estimate must be positive
investments must be analyzed in terms of both
their return potential as well as their riskiness or variability
historically, it has been proved that high returns are accompanied by
higher risk
investment rule number 1
if faced with 2 investment choices having the same expected returns, select the one with the lower expected risk
investment rule number 2
if two investment choices have similar risk profiles select the one with the higher expected return
to maximize return and minimize risk, it would be ideal to select an investment that has
a higher expected return and a lower expected risk than the other alternatives
realistically, higher expected returns are accompanied by
greater variances and the choice is not that clear cut, the investor's tolerance for and attitude towards risk matters
in a world fraught with uncertainty and risk
diversification is key!
diversification
the spreading of wealth over a variety of investment opportunities so as to eliminate some risk
by dividing up one's investments across many relatively low-correlated assets, companies, industries, and countries it is possible to
considerably reduce one's exposure to risk
the portfolio's expected return E(rp), return can be measured in 2 ways:
1. weighted average of each stock's expected return
2. expected return of the portfolio's conditional returns
weighted average of each stocks expect return =
weight in Zig
E(rZIG) + Weight in Zag
E(rZAG)
expected return of the portfolio's conditional returns =
sum of prob of econ state X portfolio return in econ state
total risk is made up of two parts
1. unsystematic or diversifiable risk
2. systematic or non-diversifiable risk
unsystematic risk, co-specific, diversifiable risk
product or labor problems
systematic risk, market, non-diversifiable risk
recession or inflation
well diversified portfolio
one whose unsystematic risk has been completely eliminated (large mutual fund companies)
beta
measures volatility of an individual security against the market as a whole
average beta
1.0 - market beta
Beta < 1.0
less risky than market
ex. utility socks
Beta > 1.0
more risky than the market
ex. high-tech socks
Beta = 0
independent of the market
ex. t-bill
betas are estimated by
running a regression of stock returns against market returns (independent variable)
what measures beta or the systematic risk estimate of the stock
the slope of regression line (coefficient of the independent variable)
once individual stock betas are determined, the portfolio beta
is easily calculated as the weighted average
2 different measures of risk related to financial assets
standard deviation (or variance) and beta
standard deviation
measure of total risk of an asset, both its systematic and unsystematic risk
beta
measure of asset's systematic risk
if an asset is part of a well-diversified portfolio
use beta as the measure of risk
if we do not have a well-diversified portfolio, it is more prudent to use
standard deviation as the measure of risk for our asset
the Security Market Line (SML)
shows the relationship between an assets required rate of return an its systematic risk measure
SML is based on 3 assumptions
1. there is a basic reward for waiting: the risk-free rate. consumption.
2. the greater the risk, the greater the expect reward. Investors expect to be proportionately compensated for bearing risk.
3. there is consistent trade-off between risk and reward
These three assumptions imply that SML is
upward sloping, has a constant slope (linear) and has the risk-free rate as its y intercept
The Capital Asset pricing model (CAPM)
equation form of the SML, used to quantify the relationship between expected rate of return and systematic risk
CAMP states that the
expected return of an investment is a function of:
1. the time value of money (the reward for waiting)
2. a reward for taking on risk
3. the amount of risk
the slope of the SML is the
market risk premium (E(rm)-rf) and not beeta
CAMP equations comes from basic y = a + bx where substituting
E(ri) - y variable
rf - intercept a
(E(rm)-rf) the slop b
beta - random variable on the x-axis
therefore we get
E(ri) = rf + B(E(rm)-rf)
the SML has many practical applications such as
1. determining the prevailing market or risk premium
2. determining the investment attractiveness of stock
3. determining portfolio allocation weights and expected return