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Chapter 2
Measuring
Returns and Risk
Measures of Investment Returns
• Holding Period Return (HPR) and Return Relative
(HPRR)
• Per-Period Return (PPR) and Return Relative
(PPRR)
• Compounding
• Expected Return
• Annualized Return
• Geometric Mean (GMR) and Arithmetic Mean
Returns
Ex Ante Returns
• Returns that are derived from a probability
distribution
Ex Post Returns
• Returns that come from a time series of
historical data
Components of Return
• Periodic payments: dividends, interest,
rents, or royalties
• Changes in market value: price appreciation
or decline
Holding Period Return (HPR)
• Rate of return for the period held
Holding Period Return Relative (HPRR)
• End-period value relative to beginning-of-period
value for specific holding period
• holding period return plus one (1)
HPR = (Income Received + Change in Value) 
Beginning Value
HPRR = (Income Received + Ending Value) 
Beginning Value
HPR = HPRR – 1
or
1 + HPR = HPRR
Per-Period Return (PPR)
• Return earned for particular period (for example,
annual return)
Per-Period Return = (Period’s Income
+ Price Change)
 Beginning Period Value
Per-Period Return Relative (PPRR)
Per-Period Return Relative=
(Period’s Income + End of Period Value)
 Beginning Period Value
Impact of Compounding
Ending Value = Beginning Value x
(1 + rate of return)t
where t = the number of time periods
Example:
beginning value = $100
rate of return = 10%
t = 10 years
Ending Value = $100 x (1+.10) 10 = $259.37
Annualized Rate of Return
•Converting Returns for time periods other than one year
into annualized returns
Rear = (1 + HPR)#
where Rear = effective annual interest rate
HPR = holding period rate of return
# = adjustment factor determined by whether
holding period is measured in days, weeks,
months, quarters, etc.
Expected Return
• Probability Distribution
• E(return)=P1xR1 + P2xR2 + … + PnxRn
A Portfolio’s Expected Rate of Return
• Same formula as HPR for security, or
• Weighted average rate of return = W1 x E(R1) +
W2 x E(R2) + . . . + Wn x E(Rn)
where Wi = % of portfolio invested in security i,
E(Ri) = the per-period return on security i, and
W1 + … + Wn = 1
Geometric Mean Return
• Value of compounded per-period average rate of
return of financial asset determined during
specified time
HPRR = PPRR1 x PPRR2 x …x PPRRn
GMR = HPRR(1  n) – 1
Arithmetic Mean Return
• Simple average return found by dividing sum of
separate per-period returns by number of periods
over which they were earned
Why Arithmetic Mean Is a Really Bad
Measure of Returns Over Time
R1 = 100%
R2 = –50%
Arithmetic average return = +25%
Geometric mean return = 0%
Relationship between GM and
Arithmetic Mean Returns
1. Only when all PPRs are identical will
GMR and arithmetic mean be equal
2. If PPRs are not identical, then GMR will
always be less than arithmetic mean return
3. Difference increases as variability among
PPRs increases
Predicting Future Performance Based
on Past Performance
• If predicting one period in future, use
arithmetic mean
• If predicting n periods in future, where n =
number of historical periods, use GMR
Risk
• Pure Risk versus Speculative Risk
• Types of Risk
– Purchasing power (or inflation) risk
– Interest rate risk
– Market risk
– Business (and default) risk
– Liquidity risk
– Political (sovereign) risk
– Exchange rate risk
– Tax risk
– Additional commitment Risk
Pure Risk
• Involves only chance of loss but no chance of gain
Speculative Risk
• Associated with speculation in which there is
some chance of gain and some chance of loss
Purchasing Power Risk
• Loss of purchasing power of investment
asset’s future cash flows
1 + real rate = (1 + nominal rate) 
(1 + inflation rate)
real rate ~ nominal rate – inflation rate
Interest Rate Risk
• for debt securities, risk associated with changes in interest
rates; consists of price risk and reinvestment rate risk
Price Risk
• a change in market interest rates produces an opposite
change in the value of investments
Reinvestment Rate Risk
• risk as to what interest rate will be when income and/or
principal from investments are reinvested
Market Risk
• Degree to which asset’s return is affected by
events affecting entire market
• Also called systematic risk
• Risk that is nondiversifiable
Business Risk
• Unique for each enterprise
• Also called nonsystematic risk
• Risk that can be reduced or eliminated
through diversification
Default Risk
•Risk that contractual payments will
not be honored
Financial Risk
•Risk that companies with heavy use
of debt financing will have more
volatile rates of return
Liquidity Risk
• Relative inability to convert an asset to cash
quickly, at any time, and without any loss of
principal
Political (Sovereign) Risk
• Degree to which investment asset subject to events in
foreign markets that can cause the value of these
investments to drop precipitously
• Includes:
– effects of trade disputes
– wars
– political unrest
– tariffs
– corruption
– expropriation
Exchange Rate Risk
• Degree to which investment asset affected by
movements in currency exchange rates in country
where investment is located
• Affects investments in some U.S. companies
because of overseas markets, production facilities,
and raw materials
Tax Risk
• Extent to which investment’s returns
are exposed to changes in tax laws
• Income that is not currently taxable
may be taxable later
Investment Manager Risk
• Risk that managed fund will perform
below average due to poor
investment decisions
• Can minimize risk through
diversification or use of index funds
Additional Commitment Risk
• Degree to which investment asset may
require the buyer to put additional money
into that investment
• Inability to meet commitment might create
loss of value
Measures of Risk
• Range
• Semivariance
n
 Pi xmin R i  ER ,0
2
i 1
• Standard Deviation
• Coefficient of Variation
• Beta
CV  σ R
Why Is Risk Important?
• It is the driver for expected return:
Expected
Return
Expected return
increases with risk
Risk
Standard Deviation
• Measure of degree of dispersion of distribution
- Standard deviation is square root of the variance
• Normal distribution
- Two times out of three actual value will be within one
standard deviation on either side of mean value
- 19 out of 20 times will be within two standard
deviations
- Well diversified portfolios with a large number of
stocks have rates of return that approximate a normal
distribution
Computing Variance and Standard
Deviation
Using Historical (ex post) data
n
1 n
R  Rt n  Rt
n t 1
t 1
1
2
R t  R 
Variance [σ ] 

n  1 t 1
n
2
Skewness
• Distribution of returns have one tail which is
longer than the other
• Investors prefer returns that are skewed to the right
Kurtosis
• Measures the “peakedness” of a distribution
- leptokurtic refers to a distribution that has a
very high center with thick tails
- platykurtic is a distribution with short center
and negligible tails
Monte Carlo Simulation
• Technique using repeated samplings from a
probability distribution to determine the
distribution of the dollar value of a portfolio
- Assumes returns are generated by a random distribution
process
- Results are dependent upon the mean and standard
distribution of the hypothesized distribution
- Useful tool to simulate both the accumulation and
decumulation of a portfolio
Buying on Margin
• Margin rate: percentage of securities
purchase that must come from investor’s
funds rather than from borrowing
• Initial margin rate: used when determining
cash needed for new purchase
• Maintenance margin rate: used when
determining if margin call is needed
Margin Rates
• Federal Reserve Board vs. In-house rule
• Regulation T
– 50% initial margin rate
• NYSE's Rule 431 & FINRA's Rule 2520
– 25% maintenance margin rate
– 30% on short positions
• In-house, only higher, never lower
Buying Power
• Dollar value of additional securities that can
be purchased on margin with current equity
in margin account
E
BP =
 MV
IMR
Net Equity
• Total value of account minus amount of
debt outstanding (with respect to a margin
account)
E = MV – LOAN
where MV = market value of portfolio
LOAN = current loan balance
Maintenance Margin
• Minimum percentage of equity that ongoing
margin account is required to maintain at all
times
MV x (1 – MMR) ≥ LOAN
where MMR = maintenance margin rate
Ways to Satisfy Margin Call
• Deposit cash in account
• Add more collateral (marginable securities)
to account.
• Sell stock from account and use proceeds
to reduce the margin debt
In each case, result must raise equity
percentage to margin maintenance
minimum to satisfy margin call.
Cash Necessary to Meet a Margin Call
Cash added = LOAN – [MV x (1 – MMR)]
Example:
Loan = $50,000
MV = $60,000
MMR = 30%
Cash added = $50,000 – [$60,000 x (1 - .30)]
= $8,000
Permitted Cash Withdrawals
Maximum Cash Withdrawal=
MV x (1 – IMR) – LOAN
Example:
MV = $100,000,
IMR = 60%,
LOAN = $15,000
Max. Cash W/D = $100,000 x (1 – .60)–$15,000
= $25,000
The Impact of Leverage
ROA = (Ending Value – Beginning Value) 
Beginning Value
ROE = {(Ending Value – Beginning Value) –
Interest Charges}  Initial Investment
Broker Call-Loan Rate
• Interest rate charged by banks to brokers for
loans that brokers use to support their
margin loans to customers
• Usually scaled up for margin loan rate