Investments Ch 3 Flashcards
RISK AND RETURN
RISK AND RETURN
- Risk - Risk can be defined as the uncertainty of future outcomes.
Involves the uncertainty of future outcomes or the possibility of an
adverse result
- Returns
Individuals forgo consumption for the opportunity of future returns.
Return is the reward for investing
METHODS OF MEASURING RETURNS
METHODS OF MEASURING RETURNS
The two sources of return from an investment are the yield and the
appreciation.
Yield + Capital Gain = Total Investment Return
- Yield - is the amount of cash the investment generates and the amount the investor receives or reinvests during a period. Yield is expressed as a percentage of the amount invested.
- Capital Gain - is the percentage change in the investment’s value from the beginning to the end of the period
EXPECTED AND REQUIRED RATE OF RETURN
EXPECTED AND REQUIRED RATE OF RETURN
Expected
* The level of return an investor thinks an investment will earn in
the future
* The expected return depends on internal and external forces.
Required
* The rate of return an investor must earn on an investment to be
fully compensated for its risk
MEASURES OF RETURNS
MEASURES OF RETURNS
- Holding Period Return * Annualized Return
- Arithmetic Mean Return * Effective Annual Rate of Return
- Geometric Mean Return * Bond Equivalent Yield
- Internal Rate of Return * Weighted Average Return
- Dollar-Weighted * After-tax Return
- Time-Weighted * Real Return
HOLDING PERIOD RETURN (HPR)
HOLDING PERIOD RETURN (HPR)
A measure of return over a specific time and includes:
- Capital gain or loss
- Current income
- HPR is most meaningful if holding period is one year
[Sale Price – Purchase Price +/- cash flows] HPR = --------------------------------------------------------------- Purchase Price
HOLDING PERIOD RETURN:
EXAMPLE
Enrique buys a stock for $60 and receives dividends of $10.
He then sells the stock for $100.
What is his holding period return?
Was this a good return?
HOLDING PERIOD RETURN:
EXAMPLE
Enrique buys a stock for $60 and receives dividends of $10.
He then sells the stock for $100.
What is his holding period return?
$100 - $ 60 + $10
—————————— = 83.33 %
$60
Was this a good return? yes
HOLDING PERIOD RETURN:
EXAMPLE
Enrique buys a stock for $60 and receives dividends of $10.
He then sells the stock for $100.
What is his holding period return?
Was this a good return?
ARITHMETIC MEAN RETURN
Arithmetic return is a simple average return, but it does not account
for compounding.
Year 1 2%
Year 2 4%
Year 3 6%
Year 4 8%
What is the Arithmetic Mean Return?
2 + 4 + 6 + 8
——————— = 5%
4
ARITHMETIC MEAN RETURN: EXAMPLE 1
Consider this example:
Year 1 45%
Year 2 (10%)
Year 3 (25%)
Year 4 30%
What is the Arithmetic Mean Return?
ARITHMETIC MEAN RETURN: EXAMPLE 1
Consider this example:
Year 1 45%
Year 2 (10%)
Year 3 (25%)
Year 4 30%
What is the Arithmetic Mean Return?
45 + (10) + (25) + 30
—————————– = 10%
4
ARITHMETIC MEAN RETURN: EXAMPLE 2
The disadvantage to the Arithmetic Mean is that it doesn’t take
compounding into account.
ARITHMETIC MEAN RETURN: EXAMPLE 2
The disadvantage to the Arithmetic Mean is that it doesn’t take
compounding into account.
Example:
Destiny buys a stock for $100. The stock is worth $200 at
the end of the year. Next year the stock falls back to $100.
What is Destiny’s average return
GEOMETRIC RETURN
GEOMETRIC RETURN
The geometric (mean) is a compounded rate of return.
Better measure of an investor’s effective return over several periods than the arithmetic mean.
The geometric mean of a set of holding period returns for a security is used to calculate: – The time weighted return.
GM = 1 + rଵ 1 + rଶ … (1 + r୬) − 1 x 100
GM = Geometric Mean
n = number of returns
rn = actual return for period n
- The geometric mean formula calculates the average return over time
assuming all earnings remain invested. In other words, the geometric
mean calculates the compound annual return, not the simple average return.
-The greater the volatility of investment returns, the lower the geometric return over time as compared with the arithmetic average return.
-The arithmetic average tends to overstate the average return, especially when both positive and negative returns are experienced.
GEOMETRIC RETURN : EXAMPLE
Year 1 45%
Year 2 (10%)
Year 3 (25%)
Year 4 30%
GEOMETRIC RETURN : EXAMPLE
Year 1 45%
Year 2 (10%)
Year 3 (25%)
Year 4 30%
GM = 1.45 0.90 (0.75)(1.3) − 1 𝑥 100
𝐺M = ర 1.272 − 1 𝑥 100 = 6.2
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Dalton Instructor formula method:
-You cannot use negative so instead of -10% = .90 and -25% = .75
1 Step 1 solve for FV
1.45 x .90 x .75 x 1.30 = 1.27
- Solve for I
1.2724 FV
-1 PV ( its negative one because a dollar was invested , outflow)
4 N
I = 6.20%
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INTERNAL RATE OF RETURN
INTERNAL RATE OF RETURN
The internal rate of return (IRR) is the earnings rate of a series of
cash inflows and outflows over a period of time assuming all earnings
are reinvested. It equates the present value of the future cash flows to the price of the investment.
CF0 = the price of the investment
CFn = the cash flow that occurs at period n
IRR = the rate that forces the price of the investment to be equal to the present value of the cash flows
INTERNAL RATE OF RETURN METHODS
INTERNAL RATE OF RETURN METHODS
________________________________________________________________
Dolla R Weighted - “ from the InvestoR perspective”
- Measures the effect of all cash flows an investor controls
-Combines the result of the timing and dollar volume of investor
trades during a period and the performance of the investment
security
– Used to determine the investor’s combined result of the timing and dollar volume of the transactions during the period, as well as the performance of the investment security. It is also appropriate for gauging the performance of an investment manager with full discretion over an investor’s account.
__________________________________________________________________
T ime Weighted Return - “ from the investmenT perspective”
- Measures the effect of cash flows associated with an investment
- It ignores the dollar volume and timing of investor-driven trades
- It assumes a buy and hold approach
- Mutual funds report their returns using a time weighted return
–More appropriate for assessing the performance of a fund manager or a security.
RR EXAMPLE
Example: Christine purchased 100 shares of Fly Cheap Airlines stock
for $12 per share. One year later the stock paid a dividend of $1 per
share. Christine buys 100 additional shares for $11 per share. At the
end of the second year Christine sells all 200 shares for $20 per share.
What was the time weighted return on Fly Cheap Airline stock?
What was Christine’s dollar weighted return?
RR EXAMPLE
Example: Christine purchased 100 shares of Fly Cheap Airlines stock
for $12 per share. One year later the stock paid a dividend of $1 per
share. Christine buys 100 additional shares for $11 per share. At the
end of the second year Christine sells all 200 shares for $20 per share.
What was the TIME WEIGHTED return on Fly Cheap Airline stock?
0 1 2
I——————-I——————–I-
-12 1 20
-12 CF
1 CF
20 CF
IRR = 33.33%
What was Christine’s DOLLAR WEIGHTED return?
0 1 2
I——————–I——————–I-
-1200 100 4,000
-1,100
-1200 CF
-1000 CF
4000 CF
IRR = 45.60%
ANNUALIZED RETURN
ANNUALIZED RETURN
The annualized return is equivalent to a compound rate of return.
- It assumes that cash flows (i.e., interest, dividends) are reinvested
at the same rate. - If this assumption is not met, then the actual return will be different
than the calculated annualized return. - When investments are held for less than one year, it is useful to
calculate the annualized yield on the investment.
NNUALIZED RETURN: EXAMPLE
Annualized return equation:
Annualized Return = 1 + Rp N − 1
Rp = return for the period being measured
N = number of periods in a year
Assume Jack earned a quarterly return of 3%
.
The annualized rate of return would be:
ANNUALIZED RETURN: EXAMPLE
Annualized return equation:
Annualized Return = 1 + Rp N − 1
Rp = return for the period being measured
N = number of periods in a year
Assume Jack earned a quarterly return of 3%
.
The annualized rate of return would be:
Annualized return = (1.03)4 - 1 = 0.1255 = 12.55%
EFFECTIVE ANNUAL RATE OF RETURN
EFFECTIVE ANNUAL RATE OF RETURN
Takes the impact of compounding into account. When compounding occurs more frequently than annually, the effective annual rate of interest is higher than the nominal rate. The more frequent the compounding, the higher the effective rate
Chelsea invests in a bank CD with a nominal yield of 7% compounded
weekly. Chelsea’s effective annual rate of return is:
i 2 EAR = [ 1 + ------- ] - 1
n
52
.07 [ 1 + -------- ] - 1 = .0725 = 7.25 %
52
BOND EQUIVALENT YIELD (BEY)
BOND EQUIVALENT YIELD (BEY)
BEY determines the yield for a bond sold at a discount based on the
current price and the remaining days until maturity.
Par Value − Price 365
BEY = x , where
Price d
Par value = the face value of the bond, generally $1,000
Price = the price of the bond or the price paid for the bond
d = the number of days until maturity
BOND EQUIVALENT YIELD: EXAMPLE
Example 1: Betty buys a bond with 192 days until maturity for $950.
The BEY is computed as follows:
BOND EQUIVALENT YIELD: BEY
Used to compare yields for bonds with different maturities.
Does NOT work for selling at a premium
Par Value - Price 365 BEY = ------------------------------------------ x -----------
Price d
Par value = the face value of the bond, generally $1,000 Price = the price of the bond in the market or the price paid for the bond 365 = the number of days in a year d = the number of days until maturity
EXAMPLE
Example 1: Betty buys a bond with 192 days until maturity for $950.
The BEY is computed as follows:
$1000 - $950 365 BEY = -------------------------- x -------- = 10 %
$950 192
Example 2: Bobby buys a bond with 289 days until maturity for $931.
The BEY is computed as follows:
$1000 - $931 365 BEY = -------------------------- x -------- = 9.36 %
$931 289
The BEY is often used to compare yields for bonds with different
maturities.
WEIGHTED AVERAGE RETURN
WEIGHTED AVERAGE RETURN
A return that is “weighted” based on the dollar amount of the individual investments.
Rw =
Rw = weighted average return
Ri = return for investment i
Wi = weight for investment i as a percentage of the portfolio
N = number of investments
The overall weighted average return is computed by: 1. summing the market value of the investments, 2. multiplying the percent return of each security by the individual market value to obtain the individual dollar returns, which are then added together, and 3. dividing the total dollar return by the total market value.
Example:
Riley has a portfolio worth $10,000 consisting of three
stocks. The expected return for each stock is given, along with its
proportionate value of the portfolio. What is the expected return for the portfolio?
Stock Expected Return Value
Apple 20% $3,000
Google 25% $3,000
Amazon 50% $4,000
WEIGHTED AVERAGE RETURN: EXAMPLE
Stock Expected Return % of Portfolio Weighted Return
__________________________________________________________________
Apple 20% 30% 6.0
Google 25% 30% 7.5
Amazon 50% 40% 20
WEIGHTED AVERAGE RETURN: EXAMPLE
Stock Expected Return % of Portfolio Weighted Return
__________________________________________________________________
Apple 20% 30% 6.0
Google 25% 30% 7.5
Amazon 50% 40% 20
= 33.5%
(20%)(0.30) + (25%)(0.30) + (50%)(0.40) = 33.5%
AFTER-TAX RETURNS & CAPITAL GAINS
AFTER-TAX RETURNS & CAPITAL GAINS
After-Tax Return = Taxable Return x (1 – Marginal Tax Rate)
- Realized taxable return = the return from the investment without
considering income tax
investors in high marginal income tax brackets should evaluate investments based on their after-tax returns to determine whether any of the investment options have some sort of tax advantage or penalty
- Marginal income tax rate = the tax rate percentage applied to
incremental taxable income
EXAMPLE:
Hank compares a corporate bond with a municipal bond. The corporate offers a yield to maturity of 5% while the muni has a yield of just 4%.
At first glance, it appears the corporate offers a superior investment. Hank, however, pays taxes at a marginal rate of 30% and will receive tax benefits from the muni bond, but not from the corporate. In fact, the after-tax return of the corporate will be only 3.5% or [5% x (1 - 30%)], making it less attractive than the muni.
EXAMPLE
The equation can be rearranged or solved for the tax rate that equates two bonds subject to different tax treatment. Consider the following example.
Jesse is considering a corporate bond yielding 8.0% and a municipal bond yielding 5.6%. The tax rate at which Jesse would be indifferent between the two bonds is 30%, as follows:
8.0% x (1 - Tax rate) = 5.6%
1 - Tax rate = 70%
Tax rate = 30%
-If Jesse’s marginal rate is less than 30%, then he would be better off from an after-tax perspective with the corporate bond.
i.e. 8.0% x ( 1 - .25 ) = 6.00 ( so if in 25 % tax bracket, then the corporate is better.
-If his tax rate is in excess of 30%, then he would be better off with the municipal bond.
REAL RETURN
Example: Benji has a retirement fund that is expected to earn 9%
annually, but inflation is 3%. Benji’s real annual return is =
REAL RETURN
Inflation erodes purchasing power. When attempting to estimate the
future purchasing power (investment returns net of inflation), the
inflation-adjusted, or real rate of return is appropriate
1 + Rn Real Rate of Return = ( ----------- - 1 ) x 100
1 + i
Rn = nominal rate or Return ,
i = inflation
Example:
Benji has a retirement fund that is expected to earn 9%
annually, but inflation is 3%. Benji’s real annual return is =
1.09% [ ( -------------- -1 ) x 100 ] = 5.83%
1.03 %
RISK
RISK
- Risk involves the uncertainty associated with returns.
- The chance that the actual return from an investment may differ from what is expected
- The more risk, the higher the return required.
- Risk in the context of Investments, has two components:
—Systematic or Market Driven
—Unsystematic or Unique (specific to the security)
TOTAL RISK
TOTAL RISK
- Total Risk = Systematic + Unsystematic Risk
- Total risk is measured by standard deviation.
- Systematic risk is measured by beta.
SYSTEMATIC RISK
SYSTEMATIC RISK “prime “
- Purchasing Power
- Reinvestment Rate
- Interest Rate
- Market Risk
- Exchange Rate
- Systematic (Non-Diversifiable Risk)
- Systematic risks are those risks that are inherent in the “system.”
- These risks are non-diversifiable, regardless of how many securities and industries are combined in a portfolio
—–No amount of diversification can reduce or eliminate systematic risk.
UNSYSTEMATIC RISK
UNSYSTEMATIC RISK
Unsystematic risk can be diversified away by combining multiple asset classes and industries in a portfolio.
Unsystematic (Diversifiable Risk) Risk includes:
* Business
* Country
* Credit or Default
* Financial
* Government / Regulation
* Others include: Call Risk, Investment Manager, Event Risk, Liquidity
and Marketability Risk
STANDARD DEVIATION
STANDARD DEVIATION
- Measure of TOTAL RISK = Systematic + Unsystematic Risk
- The greater the standard deviation, the greater the risk
- Measure of total return variability
- Measures variation of returns around an average
The variance of a distribution is its standard deviation squared.
Measures variability of returns in relation to the mean return.
A larger standard deviation implies greater uncertainty of outcomes.
With a normal distribution,_________________________________________
approximately 68% of the outcomes fall within ± 1 standard deviation from the mean,
approximately 95% of the outcomes fall within ± 2 standard deviations from the mean,
and approximately 99% of the outcomes fall within ± 3 standard deviations from the mean.
POPULATION STANDARD DEVIATION
POPULATION STANDARD DEVIATION
Measure the risk for a population
The variation of the # of returns within the entire population
𝑟௧ = past returns
𝑟̅ = arithmetic mean of past returns
n = number of returns used
𝜎 = population standard deviation
SAMPLE STANDARD DEVIATION
SAMPLE STANDARD DEVIATION
The calculation will be a little larger since its from a sample within the population.
𝑟௧ = past returns
𝑟̅ = arithmetic mean of past returns
n = number of returns used
𝜎 = sample standard deviation
CALCULATE STANDARD DEVIATION
Calculate the standard deviation of the following annual returns:
Year 1 5%
Year 2 9%
Year 3 1%
Year 4 (2%)
Year 5 7%
CALCULATE STANDARD DEVIATION
example
Calculate the standard deviation of the following annual returns:
Year 1 5%
Year 2 9%
Year 3 1%
Year 4 (2%)
Year 5 7%
5 +
9
1
-2
7
orange shift Sx.Sy = 4.47%
POPULATION STANDARD DEVIATION: EXAMPLE
POPULATION STANDARD DEVIATION: EXAMPLE
SAMPLE STANDARD DEVIATION: EXAMPLE
SAMPLE STANDARD DEVIATION: EXAMPLE
PROBABILITY OF RETURNS ASSUMING A
NORMAL DISTRIBUTION
PROBABILITY OF RETURNS ASSUMING A
NORMAL DISTRIBUTION
The following exhibit helps to explain the relationship between standard deviation and probabilities of outcomes
The curve in Exhibit 3.5 shows the bell-shaped curve of a normal distribution.
The vertical axis represents the probability of observing an outcome and the horizontal axis represents the possible outcomes.
The curve peaks at the mean of the distribution, as the mean is the most likely outcome to be observed.
Observing values further and further away from the mean becomes less and less likely, hence the curve slopes down towards the tails. The likelihood of seeing observations far out in the tails becomes extremely low, but never quite hits zero
DISTRIBUTION OF RETURNS
- Normal Distribution
- Classic Bell Curve Distribution of Data
- Distributions that are NOT normal. May be caused by:
- Skewness
- Kurtosis
DISTRIBUTION OF RETURNS
- Normal Distribution
- Classic Bell Curve Distribution of Data
- Distributions that are NOT normal. May be caused by:
- Skewness
- Kurtosis
DISTRIBUTION OF RETURNS: SKEWNESS
DISTRIBUTION OF RETURNS: SKEWNESS
Skewness: Measures the lack of symmetry in a Bell Curve
- Skewness for a normal curve is zero.
- Data skewed to the left has negative skewness.
- Data skewed to the right has positive skewness.
- Example: T-Bill returns are positively skewed since returns cannot
be negative
KURTOSIS
DISTRIBUTION OF RETURNS: KURTOSIS
Degree to which Bell Curves are peaked or flat at the top.
High Kurtosis data is peaked at the mean;
low Kurtosis data is rounded at the mean.
The Kurtosis for normal data is 3.
BETA: A MEASURE OF SYSTEMATIC RISK
BETA: A MEASURE OF “SYSTEMATIC RISK”
Beta is used to measure the volatility of a portfolio as compared to the market. Beta measures systematic risk only
Beta is an indication of a security’s volatility relative to the market. The market is defined as having a beta of 1.0.
Beta can be used as a measure of risk for portfolios that are sufficiently diversified to eliminate unsystematic risk.
- ABC Stock Beta = 1.5
- ABC returns should be 50% more volatile than market returns. If
the market increases by 10%, then ABC returns are expected to
increase by 1.5 x 10% = 15%. - What if the market decreases by 5%?
- XYZ Stock Beta = 0.7
- XYZ returns should be 30% less volatile than market returns. If
the market increases by 10%, then XYZ returns are expected to
increase by 0.7 x 10% = 7%.
BETA: THE FORMULA
BETA: THE FORMULA
(Std Dev of investment) ( Correlation between asset & market) BETA = -------------------------------------------------------------------------------------
Std Dev of Market
Cov im (P im ) ( σ i ) βi = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ = \_\_\_\_\_\_\_\_\_\_\_\_\_\_
2 σ m
σ m
βi = beta of asset i
Cov im = covariance between asset i and the market
Rim = correlation between asset i and the market
σ i = standard deviation of asset i
σ m = standard deviation of the market
P im = Correlation coefficient between the individual security and the market
