Concept of Risk & Return.pptx help understand the risk and return

OUTLINE
• Concept of risk and return
• Relationship between Risk and Return
• Risk and Return of a Single Asset
• Risk and Return of a Portfolio
• Measurement of Market Risk

Introduction
⚫It is important to understand the relation between risk
and return due to uncertainty.
⚫The relation between risk and return is useful for
investors (who buy/sell securities), corporations (that
sell securities to finance themselves), and for financial
intermediaries (that invest, borrow, lend, and price
securities on behalf of their clients).
⚫The relationship between risk & return is fundamental
in finance theory.

Introduction
⚫If given the choice between
❑Investing in a low risk opportunity that says it will pay you a
10% return on your money.
❑Investing in a high risk opportunity that says it will pay you a
10% return on your money.
❑Most people would choose the lower risk opportunity.
• The principle we follow in finance is that investors
need the inducement of higher reward to take on
perceived higher risks

Defining a return on an Investment
• Investment returns measure the financial results of
an investment.
• Returns may be historical or prospective/expected
(anticipated).
• We invest in a stock to have positive return.
• For stocks, we have two components
• May receive Dividend
• Stock price may appreciate

Defining Historical Return
Income received on an investment plus any change in
market price, usually expressed as a percent of the
beginning market price of the investment.
Dt + (Pt - Pt-1 )
Pt-1
R =

RISK AND RETURN OF A SINGLE ASSET
Rate of Return
Rate of Return = Annual income + Ending price-Beginning price
Beginning price Beginning price
Current yield Capital gains yield

Return Example
The stock price for Stock A was Rs 10 per share
1 year ago. The stock is currently trading at Rs
11.50 per share and shareholders just received
a Rs 1 dividend. What return was earned over
the past year?
1.00 + (11.50 - 10.00 )
10.0
0
R = = 25%

Return
⚫ The previous example calculated what actually
happened.
⚫ We call this a “historic return”. (Its history)
⚫ However, prior to making the investment, we may have
had an expected return of 30%?
⚫ In this case, What actually happened fell short of our
expectation.
⚫ Alternatively, may be our expectation was to earn only
10%.
⚫ In this case the actual return exceeded our expectations.

Expected Return
⚫The future is uncertain.
⚫Investors do not know with certainty whether the economy
will be growing rapidly or be in recession.
⚫Investors do not know what rate of return their investments
will yield.
⚫Therefore, they base their decisions on their expectations
concerning the future.
⚫The expected rate of return on a stock represents the
mean of a probability distribution of possible future returns
on the stock.

Expected Return
⚫ The table below provides a probability distribution for the returns on
stocks A and B
State Probability Return On Return On
Stock A Stock B
1 20% 5% 50%
2 30% 10% 30%
3 30% 15% 10%
4 20% 20% -10%
⚫ The state represents the state of the economy one period in the future
i.e. state 1 could represent a recession and state 2 a growth economy.
⚫ The probability reflects how likely it is that the state will occur. The sum
of the probabilities must equal 100%.
⚫ The last two columns present the returns or outcomes for stocks A and
B that will occur in each of the four states.

Expected Return
⚫Given a probability distribution of returns, the expected
return can be calculated using the following equation:
N
E[R] = Σ (piRi)
i=1
⚫Where:
⚪ E[R] = the expected return on the stock
⚪ N = the number of states
⚪ pi = the probability of state i
⚪ Ri = the return on the stock in state i.

Expected Return
⚫In this example, the expected return for stock A
would be calculated as follows:
E[R]A = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%
⚫Now you try calculating the expected return for
stock B!

Expected Return
⚫Did you get 20%? If so, you are correct.
⚫If not, here is how to get the correct answer:
E[R]B = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20%
⚫So we see that Stock B offers a higher expected
return than Stock A.
⚫However, that is only part of the story; we haven't
considered risk.

Risk
⚫The fact that what actually happens (and often does)
differ from what we either expect or would like to
happen is defined as risk.
⚫Its useful to have a mathematical tool to measure
risk.
⚫A common approach is to look at expected returns
and calculate the volatility of the returns.
⚫Volatility is measured either by SD or Variance.
⚫The larger the volatility, the greater the risk.

Defining Risk
What rate of return do you expect on your
investment (savings) this year?
What rate will you actually earn?
Does it matter if it is a bank CD or a share of
stock?
The variability of returns from those
that are expected.

What is Investment risk?
■Typically, investment returns are not known
with certainty.
■The greater the chance of a return far below the
expected return, the greater the risk.
■It is the potential for divergence between the
actual outcome and what is expected.
■In finance, risk is usually related to whether
expected cash flows will materialize, whether
security prices will fluctuate unexpectedly, or
whether returns will be as expected.

Probability distribution
Rate of
return (%)
50
1
5
0
-
20
Stock X
Stock Y
■ Which stock is riskier?
Why?

19
Average Returns and St. Dev. for Asset Classes,
2000-2017
1. Investors who want higher returns have to take more risk
2. The incremental reward from accepting more risk seems
constant
Bills Bonds
Stocks
Average Return (%)
Standard Deviation (%)

Expected Risk & Preference
⚫Risk-averse investor
⚫Risk-neutral investor
⚫Risk-seeking investor

Measurement of Risk
⚫One way to measure risk is to calculate the
variance and standard deviation of the distribution
of returns.
⚫Standard Deviation, σ, is a statistical measure of
the variability of a distribution around its mean.
⚫The larger the standard deviation, the higher the
probability that returns will be far below the
expected return.
⚫The variance is the sum of the squares of deviations
of actual returns from the expected return, weighted
by the associated probabilities.

How to Determine the Expected Return and
Standard Deviation
Stock BW
Ri Pi (Ri)(Pi) (Ri - R )2
(Pi)
-.15 .10 -.015 .00576
-.03 .20 -.006 .00288
.09 .40 .036 .00000
.21 .20 .042 .00288
.33 .10 .033 .00576
Sum 1.00 .090 .01728

Determining Standard Deviation (Risk Measure)
σ = Σ ( Ri - R )2
( Pi )
σ = .01728
σ = .1315 or 13.15%
n
i=1

Measures of Risk
⚫Probability Distribution:
State Probability Return On Return On
Stock A Stock B
1 20% 5% 50%
2 30% 10% 30%
3 30% 15% 10%
4 20% 20% -10%
⚫E[R]A = 12.5%
⚫E[R]B = 20%

Measures of Risk
⚫The variance and standard deviation for stock A is
calculated as follows:
σ2
A = .2(.05 -.125)2
+ .3(.1 -.125)2
+ .3(.15 -.125)2
+ .2(.2 -.125)2
= .002625
σΑ 002625
= (. )0 5
.
0512 5 12
= . = . %
⚫Now you try the variance and standard deviation for stock B!

Measures of Risk
σ2
B = .2(.50 -.20)2
+ .3(.30 -.20)2
+ .3(.10 -.20)2
+ .2(-.10 - .20)2
= .042
σΒ 042
= (. )0 5
.
2049 20 49
= . = . %
⚫Although Stock B offers a higher expected return than
Stock A, it also is riskier since its variance and standard
deviation are greater than Stock A's.

Classification of risk
Unique risk of a security represents that portion of its total
risk which stems from company specific factors & not
explained by general market movements. Also known as
Unsystematic Risk . It is avoidable through diversification.
Examples – Workers declare strike, Company loses a big
contract, Emergence of a new competitor.
Total Risk = Unique Risk +Market Risk

Continued……
⚫Market risk of security represents that portion of
its risk which is attributable to economy –wide
factors. Also known as Systematic Risk . It is
unavoidable.
⚫Examples - Changes in economy, tax reforms,
Changes in interest rate policy, Inflation,
Restrictive credit policy.

Market Risk (Systematic Risk)
Total
Risk
Unsystematic risk
Systematic risk
STD
DEV
OF
PORTFOLIO
RETURN
NUMBER OF SECURITIES IN THE PORTFOLIO

Unique Risk (Unsystematic risk)
Total
Risk
Unsystematic risk
Systematic risk
STD
DEV
OF
PORTFOLIO
RETURN
NUMBER OF SECURITIES IN THE PORTFOLIO

Portfolio Risk and Return
⚫Most investors do not hold stocks in isolation.
⚫Instead, they choose to hold a portfolio of several
stocks.
⚫When this is the case, a portion of an individual stock's
risk can be eliminated, i.e., diversified away.
⚫From our previous calculations, we know that:
⚪ the expected return on Stock A is 12.5%
⚪ the expected return on Stock B is 20%
⚪ the variance on Stock A is .00263
⚪ the variance on Stock B is .04200
⚪ the standard deviation on Stock A is 5.12%
⚪ the standard deviation on Stock B is 20.49%

⚫The Expected Return on a Portfolio is computed as the
weighted average of the expected returns on the stocks which
comprise the portfolio.
⚫The weights reflect the proportion of the portfolio invested in
the stocks.
⚫This can be expressed as follows:
N
E[Rp] = Σ wiE[Ri]
i=1
⚫Where:
⚪ E[Rp] = the expected return on the portfolio
⚪ N = the number of stocks in the portfolio
⚪ wi = the proportion of the portfolio invested in stock i
⚪ E[Ri] = the expected return on stock i

Variance & Standard deviation of a Two-asset
portfolio
⚫For a portfolio consisting of two assets, the above equation
can be expressed as:
E[Rp] = w1E[R1] + w2E[R2]
= w1E[R1] +( 1-w1)E[R2]
⚫If we have an equally weighted portfolio of stock A and stock
B (50% in each stock), then the expected return of the
portfolio is:
E[Rp] = .50(.125) + .50(.20) = 16.25%
The portfolio weight of a particular security is the percentage
of the portfolio’s total value that is invested in that security.

Expected Return of a Portfolio
Example
Portfolio value = 2,000 + 5,000 = 7,000
rA = 14%, rB = 6%,
wA = weight of security A = 2,000 / 7,000 = 28.6%
wB = weight of security B = 5,000 / 7,000 = (1-28.6%)= 71.4%

PORTFOLIO RISK : THE 2-SECURITY CASE
⚫Risk of individual assets is measured by their variance or
standard deviation.
⚫We can use variance or standard deviation to measure the
risk of the portfolio of assets as well.
⚫The risk of portfolio would be less than the risk of
individual securities, and that the risk of a security should
be judged by its contribution to the portfolio risk.

⚫The variance/standard deviation of a portfolio reflects only
the variance/standard deviation of the stocks that make up
the portfolio and not how the returns on the stocks which
comprise the portfolio vary together.
⚫Two measures of how the returns on a pair of stocks vary
together are the covariance and the correlation coefficient.
⚫The portfolio variance or standard deviation depends on the
co-movement of returns on two assets.
⚫Covariance of returns on two assets measures their co-
movement.
⚪ Correlation coefficient, is often used to measure the degree of co-
movement between two variables. The correlation coefficient simply
standardizes the covariance.

⚫The Covariance between the returns on two stocks can be
calculated as follows:
Cov(RA,RB) = σA,B = Σ pi(RA - E[RA])(RB - E[RB])
⚫Where:
⚪ σΑ,Β = the covariance between the returns on stocks A and B
⚪ pi = the probability of state
⚪ RA = the return on stock A
⚪ E[RA] = the expected return on stock A
⚪ RB = the return on stock B
⚪ E[RB] = the expected return on stock B

Covariance of Returns of Securities X and Y

⚫The Correlation Coefficient between the returns on two
stocks can be calculated as follows:
σA,B Cov(RA,RB)
Corr(RA,RB) = ρA,B = σAσB = SD(RA)SD(RB)
⚫Where:
⚪ ρA,B=the correlation coefficient between the returns on stocks A and B
⚪ σA,B=the covariance between the returns on stocks A and B,
⚪ σA=the standard deviation on stock A, and
⚪ σB=the standard deviation on stock B

⚫Using either the correlation coefficient or the covariance, the
Variance on a Two-Asset Portfolio can be calculated as
follows:
σ2
p = (wA)2
σ2
A + (wB)2
σ2
B + 2wAwBρA,B σAσB
OR
σ2
p = (wA)2
σ2
A + (wB)2
σ2
B + 2wAwB σA,B
⚫The Standard Deviation of the Portfolio equals the
positive square root of the variance.

Portfolio Risk and Return: Example
⚫The covariance between stock A and stock B is as follows:
σA,B = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) +
.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105
⚫The correlation coefficient between stock A and stock B is as
follows:
-.0105
ρA,B = (.0512)(.2049) = -1.00

⚫Let’s calculate the variance and standard deviation of a portfolio
comprised of 75% stock A and 25% stock B:
σ2
p =(.75)2
0512
(. )2
+(.25)2
(.2049)2
+2(.75)(.25)(-1)(.0512)(.2049)= .00016
σp = .00016 = .0128 = 1.28%
⚫Notice that the portfolio formed by investing 75% in Stock A and
25% in Stock B has a lower variance and standard deviation than
either Stocks A or B and the portfolio has a higher expected
return than Stock A.
⚫This is the purpose of diversification; by forming portfolios,
some of the risk inherent in the individual stocks can be
eliminated.

MEASUREMENT OF MARKET RISK
THE SENSITIVITY OF A SECURITY TO MARKET MOVEMENTS IS
CALLED BETA .
BETA REFLECTS THE SLOPE OF A THE LINEAR REGRESSION
RELATIONSHIP BETWEEN THE RETURN ON THE SECURITY AND
THE RETURN ON THE PORTFOLIO
Relationship between Security Return and Market Return
Security
Return
Market return

CAPITAL ASSET PRICING MODEL(CAPM)
EXPECTED RATE OF RISK-FREE RATE OF + RISK
RETURN RETURN PREMIUM
RISK PREMIUM = BETA [EXPECTED RETURN ON MARKET
PORTFOLIO – RISK FREE RATE OF RETURN]
Example
Beta = 1.20
Expected return on the market portfolio = 15 percent
Risk free rate = 10 percent
Expected rate of return = 10 +1.2 (15-10)
= 16 percent
= +

Rate of Return
C Risk premium for an aggressive
17.5 B security
15.0 A
12.5 Risk premium for a neutral security
Rf = 10
Risk premium for a defensive security
0.5 1.0 1.5 2.0 Beta
BETA (MARKET RISK) & EXPECTED RATE OF
RETURN

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Concept of Risk & Return.pptx help understand the risk and return