Risk, return, and portfolio theory

RISK & RETURN

10/12/12
 The concept and measurement of Return:
Realized and Expected return.
 Ex-ante and ex-post returns

Risk, Return and Portfolio Theory
 The concept of Risk:
 Sources and types of risk.
 Measurement of risk :
 Range,
 Std Deviation and

 Co-Efficient of Variation.

 Risk-return trade-off

Risk, Return &Portfolio Theory

Friday, October 12, 2012

LEARNING OBJECTIVES

 The difference among the most important types of

returns
 How to estimate expected returns and risk for individual
securities
 What happens to risk and return when securities are
combined in a portfolio

Friday, October 12, 2012 Risk, Return and Portfolio Theory
INTRODUCTION TO RISK AND RETURN

INTRODUCTION TO RISK AND
RETURN
Risk and return are the two most
important attributes of an
investment.

Research has shown that the two
are linked in the capital markets Return
and that generally, higher %
returns can only be achieved by
taking on greater risk.
Risk Premium

Risk isn’t just the potential loss of
return, it is the potential loss of
the entire investment itself (loss RF Real Return
of both principal and interest).
Expected Inflation Rate

Consequently, taking on additional Risk
risk in search of higher returns is
a decision that should not be
taking lightly.

RISK RETURN TRADE OFF

 The concept of investment return is widely
understood. For example, a 10% per annum return on
a capital sum of $100,000 would result in $10,000
increase in value for the year. However, what exactly
is ‘risk?’.
 Risk is for the most part unavoidable – in life
generally as much as in investing!
 In investments, the term ‘risk’ is often expressed as
‘volatility’ or variations in returns.
 In investment terms, the concept of ‘volatility’ is the
measurement of fluctuation in the market values of
various asset classes as they rise and fall over time.

 The greater the volatility the more rises and falls are
recorded by an individual asset class.

 The reward for accepting greater volatility is the
likely hood of higher investment returns over mid to
longer term.
 The disadvantage can mean lower returns in the
shorter term. It must also be remembered that it can
mean an increase or decrease in capital.
 All investments involve some risk. In general terms
the higher the risk, the higher the potential return, or
loss. Conversely the lower the risk the lower the
potential return, or loss.
 The long-term risk/return trade off between
different asset classes is illustrated in the
following graph:

10/12/12

10/12/12


MEASURING RETURNS

MANY DIFFERENT MATHEMATICAL DEFINITIONS OF
"RETURNS"...

10/12/12 Risk, Return and Portfolio Theory

CHAPTER 9 LECTURE.
QUANTIFYING & MEASURING INVESTMENT
PERFORMANCE: "RETURNS"

RETURNS
 RETURNS = PROFITS (IN THE
INVESTMENT GAME)
 RETURNS = OBJECTIVE TO MAXIMIZE
(CET.PAR.)
 RETURNS = WHAT YOU'VE GOT
 WHAT
YOU HAD TO BEGIN WITH, AS A
PROPORTION OF WHAT YOU HAD TO BEGIN WITH.

QUANTITATIVE RETURN MEASURES
NECESSARY TO:
 MEASURE PAST PERFORMANCE
=> "EX POST" OR HISTORICAL RETURNS;
 MEASURE EXPECTED FUTURE PERFORMANCE
=> "EX ANTE" OR EXPECTED RETURNS.

TYPE 1: PERIOD-BY-PERIOD RETURNS .
. .
"PERIODIC" RETURNS
SIMPLE "HOLDING PERIOD RETURN"
(HPR)
MEASURES WHAT THE INVESTMENT
GROWS TO WITHIN EACH SINGLE
PERIOD OF TIME,
ASSUMING ALL CASH FLOW (OR
VALUATION) IS ONLY AT BEGINNING
AND END OF THE PERIOD OF TIME
(NO INTERMEDIATE CASH FLOWS).

TYPE 1: PERIOD-BY-PERIOD RETURNS
(CONT’D)
RETURNS MEASURED SEPARATELY OVER
EACH OF A SEQUENCE OF REGULAR AND
CONSECUTIVE (RELATIVELY SHORT)
PERIODS OF TIME.
SUCH AS: DAILY, MONTHLY, QUARTERLY, OR
ANNUAL RETURNS SERIES.
E.G.: RETURN TO IBM STOCK IN: 1990, 1991,
1992, ...
PERIODIC RETURNS CAN BE AVERAGED
ACROSS TIME TO DETERMINE THE "TIME-
WEIGHTED" MULTI-PERIOD RETURN.

TYPE 1: PERIOD-BY-PERIOD
RETURNS (CONT’D)
THE PERIODS USED TO DEFINE PERIODIC
RETURNS SHOULD BE SHORT ENOUGH THAT
THE ASSUMPTION OF NO INTERMEDIATE CASH
FLOWS DOES NOT MATTER.

TYPE 2: MULTIPERIOD RETURN
MEASURES
 PROBLEM: WHEN CASH FLOWS OCCUR AT
MORE THAN TWO POINTS IN TIME, THERE IS
NO SINGLE NUMBER WHICH UNAMBIGUOUSLY
MEASURES THE RETURN ON THE
INVESTMENT.

MEASURES (CONT’D)
 NEVERTHELESS,MULTI-PERIOD
RETURN MEASURES GIVE A SINGLE
RETURN NUMBER (TYPICALLY QUOTED
PER ANNUM) MEASURING THE
INVESTMENT PERFORMANCE OF A
LONG-TERM (MULTI-YEAR) INVESTMENT
WHICH MAY HAVE CASH FLOWS AT
INTERMEDIATE POINTS IN TIME
THROUGHOUT THE "LIFE" OF THE
INVESTMENT.

MEASURES (CONT’D)
 THERE ARE MANY DIFFERENT MULTI-PERIOD
RETURN MEASURES, BUT THE MOST FAMOUS
AND WIDELY USED (BY FAR) IS:

THE "INTERNAL RATE OF
RETURN" (IRR).
 THE IRR IS A "DOLLAR-WEIGHTED" RETURN
BECAUSE IT REFLECTS THE EFFECT OF
HAVING DIFFERENT AMOUNTS OF DOLLARS
INVESTED AT DIFFERENT PERIODS IN TIME
DURING THE OVERALL LIFETIME OF THE
INVESTMENT.

ADVANTAGES OF PERIOD-BY-PERIOD
(TIME-WEIGHTED) RETURNS:
1)ALLOW YOU TO TRACK PERFORMANCE OVER
TIME, SEEING WHEN INVESTMENT IS DOING
WELL AND WHEN POORLY.

(TIME-WEIGHTED) RETURNS (CONT’D)

2)ALLOW YOU TO QUANTIFY RISK (VOLATILITY)
AND CORRELATION (CO-MOVEMENT) WITH
OTHER INVESTMENTS AND OTHER PHENOMENA.

(TIME-WEIGHTED) RETURNS (CONT’D)

3) ARE FAIRER FOR JUDGING
INVESTMENT PERFORMANCE WHEN THE
INVESTMENT MANAGER DOES NOT HAVE
CONTROL OVER THE TIMING OF CASH
FLOW INTO OR OUT OF THE INVESTMENT
FUND (E.G., A PENSION FUND).

ADVANTAGES OF MULTI-PERIOD
RETURNS:
1)DO NOT REQUIRE KNOWLEDGE OF MARKET
VALUES OF THE INVESTMENT ASSET AT
INTERMEDIATE POINTS IN TIME (MAY BE
DIFFICULT TO KNOW FOR REAL ESTATE).

RETURNS (CONT’D)
2) GIVES A FAIRER (MORE COMPLETE)
MEASURE OF INVESTMENT
PERFORMANCE WHEN THE
INVESTMENT MANAGER HAS CONTROL
OVER THE TIMING AND AMOUNTS OF
CASH FLOW INTO AND OUT OF THE
INVESTMENT VEHICLE (E.G., PERHAPS
SOME "SEPARATE ACCOUNTS" WHERE
MGR HAS CONTROL OVER CAPITAL
FLOW TIMING, OR A STAGED
DEVELOPMENT PROJECT).

RETURNS (CONT’D)
 NOTE: BOTH HPRs AND IRRs ARE WIDELY
USED IN REAL ESTATE INVESTMENT
ANALYSIS

PERIOD-BY-PERIOD RETURNS...
 "TOTAL RETURN" ("r"):
rt=(CFt+Vt-Vt-1)/ Vt-1=((CFt+Vt)/Vt-1) -1
where: CFt= Cash Flow (net) in period "t"; Vt=Asset
Value ("ex dividend") at end of period "t".
 "INCOME RETURN" ("y", AKA "CURRENT
YIELD", OR JUST "YIELD"):
yt = CFt / Vt-1
 "APPRECIATION RETURN" ("g", AKA
"CAPITAL GAIN", OR "CAPITAL RETURN",
OR "GROWTH"):
gt = ( Vt-Vt-1 ) / Vt-1 = Vt / Vt-1 - 1
NOTE: rt = yt + gt

TOTAL RETURN IS MOST
IMPORTANT:
 To convert y into g, reinvest the cash flow back into
the asset.
 To convert g into y, sell part of the holding in the
asset.
NOTE: This type of conversion is not so easy to do
with most real estate investments as it is with
investments in stocks and bonds.

EXAMPLE:
 PROPERTY VALUE AT END OF 1994: =
$100,000
 PROPERTY NET RENT DURING 1995: =
$10,000
 PROPERTY VALUE AT END OF 1995: =
$101,000

WHAT IS 1995 R, G, Y ?...
 y1995 = $10,000/$100,000 = 10%
 g1995 = ($101,000 - $100,000)/$100,000 = 1%
 r1995 = 10% + 1% = 11%

A NOTE ON RETURN
TERMINOLOGY
 "INCOME RETURN" - YIELD, CURRENT
YIELD, DIVIDEND YIELD.
 IS IT CASH FLOW BASED OR ACCRUAL
INCOME BASED?
 SIMILAR TO "CAP RATE".
 IS A RESERVE FOR CAPITAL EXPENDITURES
TAKEN OUT?
 CI TYPICALLY 1% - 2% /YR OF V.
 EXAMPLE: V=1000, NOI=100, CI=10:
 yt = (100-10)/1000 = 9%, “cap rate” = 100/1000 = 10%

"YIELD"
 CAN ALSO MEAN: "TOTAL YIELD", "YIELD TO
MATURITY"
 THESE ARE IRRs, WHICH ARE TOTAL RETURNS,
NOT JUST INCOME.
 "BASIS POINT" = 1 / 100th PERCENT = .0001

CONTINUOUSLY COMPOUNDED
RETURNS:
 THE PER ANNUM CONTINUOUSLY
COMPOUNDED TOTAL RETURN IS:

WHERE "Y" IS THE NUMBER (OR FRACTION) OF
= ( LN( BETWEEN TIME )) Y ⇔ V + CF
r tYEARSV t + CF t ) - LN( V t - 1"t-1" AND t"t". t = V t - 1 * EXP( Yr t )

EXAMPLE:
01/01/98 V = 1000
03/31/99 V = 1100 & CF = 50

PER ANNUM
r = (LN(1150) – LN(1000)) / 1.25
= 7.04752 – 6.90776
= 11.18%

"REAL" VS. "NOMINAL" RETURNS
NOMINAL RETURNS ARE THE "ORDINARY"
RETURNS YOU NORMALLY SEE QUOTED
OR EMPIRICALLY MEASURED. UNLESS IT
IS EXPLICITLY STATED OTHERWISE,
RETURNS ARE ALWAYS QUOTED AND
MEASURED IN NOMINAL TERMS. The
NOMINAL Return is the Return in
Current Dollars (dollars of the time when the
return is generated).
REAL RETURNS ARE NET OF INFLATION.
The REAL Return is the Return measured
in constant purchasing power dollars
("constant dollars").

EXAMPLE:
Suppose INFLATION=5% in 1992 (i.e., need
$1.05 in 1992 to buy what $1.00 purchased in
1991).
So: $1.00 in "1992$" = 1.00/1.05 = $0.95 in
"1991$“
Ifrt = Nominal Total Return, year t
it = Inflation, year t
Rt = Real Total Return, year t
Then: Rt = (1+rt)/(1+it) - 1 = rt - (it + it Rt )
 rt - it ,
Thus: NOMINAL Return = REAL Return +
Inflation Premium
Inflation Premium = it + it Rt  It

IN THE CASE OF THE CURRENT
YIELD
(Real yt)=(Nominal yt)/(1+it)  (Nominal yt)

EXAMPLE:
1991 PROPERTY VALUE = $100,000
1992 NET RENT = $10,000
1992 PROPERTY VALUE = $101,000
1992 INFLATION = 5%

WHAT IS THE REAL r, y, and g for 1992?

ANSWER:
Real g = (101,000/1.05)/100,000-1= -3.81%
 -4%
(versus Nominal g=+1%)
Real y = (10,000/1.05)/100,000 = +9.52%
 10%
(versus Nominal y=10% exactly)
Real r = (111,000/1.05)/100,000-1=+5.71%
 6%
(versus Nominal r = 11%)
= g + y =+9.52%+(-3.81%)  10%
- 4%

RISK
INTUITIVE MEANING...
THE POSSIBILITY OF NOT MAKING THE
EXPECTED RETURN:

rt ≠ Et-j[rt]

MEASURED BY THE RANGE OR
STD.DEV. IN THE EX ANTE
PROBABILITY DISTRIBUTION OF
THE EX POST RETURN . . .

100%

A
75%
Probability

50%

B
25%
C
0%
-10% -5% 0% 5% 10% 15% 20% 25% 30%
Returns
Figure 1
C RISKER THAN B.
B RISKIER THAN A.
A RISKLESS.

WHAT IS THE EXPECTED
RETURN? . . .

EXAMPLE OF RETURN RISK
QUANTIFICATION:
SUPPOSE 2 POSSIBLE FUTURE RETURN
SCENARIOS. THE RETURN WILL EITHER
BE:
+20%, WITH 50% PROBABILITY
OR:
-10%, WITH 50% PROBABILITY

"EXPECTED" (EX ANTE) RETURN
= (50% CHANCE)(+20%) + (50% CHANCE)(-10%)
= +5%

RISK (STD.DEV.) IN THE RETURN
= SQRT{(0.5)(20-5)2 + (0.5)(-10-5)2}
= 15%

THE RISK/RETURN TRADEOFF..
INVESTORS DON'T LIKE RISK!

COMPENSATE THEM BY PROVIDING
HIGHER RETURNS (EX ANTE) ON
MORE RISKY ASSETS . . .

Expected
Return

rf

Risk

RISK & RETURN:
TOTAL RETURN = RISKFREE RATE +
RISK PREMIUM

rt = rf,t + RPt

RISK FREE RATE
RISKFREE RATE (rf,t)
= Compensation for TIME
= "Time Value of Money"
 US Treasury Bill Return (For Real Estate, usually
use Long Bond)

RISK PREMIUM
RISK PREMIUM (RPt):
EX ANTE: E[RPt]
= E[rt] - rf,t
= Compensation for RISK
EX POST: RPt
= rt - rf,t
= Realization of Risk ("Throw of Dice")

RELATION BETWEEN RISK &
RETURN:
GREATER RISK <===> GREATER RISK
PREMIUM

(THIS IS EX ANTE, OR ON AVG. EX POST,
BUT NOT NECESSARILY IN ANY GIVEN
YEAR OR ANY GIVEN INVESTMENT EX
POST)

EXAMPLE OF RISK IN REAL
ESTATE:
PROPERTY "A" (OFFICE):
VALUE END 1998 = $100,000
POSSIBLE VALUES END 1999
$110,000 (50% PROB.)
$90,000 (50% PROB.)
STD.DEV. OF g99 = 10%

EXAMPLE (CONT’D)
PROPERTY "B" (BOWLING ALLEY):
VALUE END 1998 = $100,000
POSSIBLE VALUES END 1999
$120,000 (50% PROB.)
$80,000 (50% PROB.)
STD.DEV. OF g99 = 20%

EXAMPLE (CONT’D)
B IS MORE RISKY THAN A.

T-BILL RETURN = 7%

EXAMPLE (CONT’D)
A: Office Building B: Bowling Alley
Known as of end 1998 Known as of end 1998
 Value = $100,000  Value = $100,000
 Expected value end 99  Expected value end 99
= $100,000 = $100,000
 Expected net rent 99  Expected net rent 99
= $11,000 = $15,000
 Ex ante risk premium  Ex ante risk premium
= 11% - 7% = 4% = 15% - 7% = 8%

EXAMPLE (CONT’D) – SUPPOSE THE
FOLLOWING OCCURRED IN 1999
A: Office Building B: Bowling Alley
Not known until end Not known until end
1999 1999
 End 99 Value =  End 99 Value =
$110,000 $80,000
 99 net rent = $11,000  99 net rent = $15,000

 99 Ex post risk  99 Ex post risk
premium = 21% - 7% = premium = -5% - 7% =
14% -12%
 (“The Dice Rolled  (“The Dice Rolled
Favorably”) Unfavorably”)

SUMMARY:
THREE USEFUL WAYS TO BREAK
TOTAL RETURN INTO TWO
COMPONENTS...
1) TOTAL RETURN = CURRENT YIELD
+ GROWTH
r=y+g
2) TOTAL RETURN = RISKFREE RATE +
RISK PREMIUM
r = rf + RP
3) TOTAL RETURN = REAL RETURN +
INFLATION PREMIUM
r = R + (i+iR)  R + I

"TIME-WEIGHTED
INVESTMENT". . .
SUPPOSE THERE ARE CFs AT INTERMEDIATE
POINTS IN TIME WITHIN EACH “PERIOD”
(E.G., MONTHLY CFs WITHIN QUARTERLY
RETURN PERIODS).
THEN THE SIMPLE HPR FORMULAS ARE NO
LONGER EXACTLY ACCURATE.

"TIME-WEIGHTED
INVESTMENT". . .
A WIDELY USED SIMPLE ADJUSTMENT IS
TO APPROXIMATE THE IRR OF THE
PERIOD ASSUMING THE ASSET WAS
BOUGHT AT THE BEGINNING OF THE
PERIOD AND SOLD AT THE END, WITH
OTHER CFs OCCURRING AT
INTERMEDIATE POINTS WITHIN THE
PERIOD.
THIS APPROXIMATION IS DONE BY
SUBSTITUTING A “TIME-WEIGHTED”
INVESTMENT IN THE DENOMINATOR
INSTEAD OF THE SIMPLE BEGINNING-OF-
PERIOD ASSET VALUE IN THE
DENOMINATOR.

"TIME-WEIGHTED
INVESTMENT". . .

EndVal − BegVal + ∑ CFi
r=
where: BegVal − ∑ wi CFi
= sum of all net cash flows occurring
inCF
∑ period t,
i

wi = proportion of period t remaining at
the time when net cash flow "i" was
received by the investor.
(Note: cash flow from the investor to the
investment is negative; cash flow from the
investment to the investor is positive.)

EXAMPLE . . .
CF: Date:
- 100 12/31/98
+ 10 01/31/99
+100 12/31/99
Simple HPR: (10 + 100-100) / 100
= 10 / 100
= 10.00%
TWD HPR: (10 + 100-100) / (100 –
(11/12)10)
= 10 / 90.83
= 11.01%

IRR: = 11.00% . . .

EXAMPLE (CONT’D)

11
10 0 100
0 = − 100 + +∑ + ⇒ IRR / mo = 0.87387%
1 + IRR / mo j = 2 (1 + IRR / mo ) j (1 + IRR / mo ) 12
⇒ IRR / yr = (1.0087387)12 − 1 = 11.00%

THE DEFINITION OF THE "NCREIF"
PERIODIC RETURN FORMULA . . .
 THE MOST WIDELY USED INDEX OF
PERIODIC RETURNS IN COMMERCIAL
REAL ESTATE IN THE US IS THE
"NCREIF PROPERTY INDEX" (NPI).
 NCREIF = "NATIONAL COUNCIL
OF REAL ESTATE INVESTMENT
FIDUCIARIES“
 “INSTITUTIONAL QUALITY R.E.”
 QUARTERLY INDEX OF TOTAL
RETURNS
 PROPERTY-LEVEL
 APPRAISAL-BASED

NCREIF FORMULA A TIME-WEIGHTED
FORMULA INCLUDES
INVESTMENT DENOMINATOR,
ASSUMING:
ONE-THIRD OF THE QUARTERLY
PROPERTY NOI IS RECEIVED AT THE END
OF EACH CALENDAR MONTH;
PARTIAL SALES RECEIPTS MINUS CAPITAL
IMPROVEMENT EXPENDITURES ARE
RECEIVED MIDWAY THROUGH THE
QUARTER...

EndVal − BegVal + ( PS − CI ) + NOI
rNPI =
[Note: (1/3)NOI =−(2/3)(1/3)NOI+(1/3) 3) NOI
BegVal (1 2 )( PS − CI ) − (1
(1/3)NOI+(0)(1/3)NOI ]

MULTI-PERIOD RETURNS…
SUPPOSE YOU WANT TO KNOW WHAT
IS THE RETURN EARNED OVER A
MULTI-PERIOD SPAN OF TIME,
EXPRESSED AS A SINGLE AVERAGE
ANNUAL RATE?...

YOU COULD COMPUTE THE
AVERAGE OF THE HPRs ACROSS
THAT SPAN OF TIME.

THIS WOULD BE A "TIME-
WEIGHTED" AVERAGE RETURN.

MULTI-PERIOD RETURNS
(CONT’D)
IT WILL:
=>Weight a given rate of return more if it
occurs over a longer interval or more frequently
in the time sample.

=>Be independent of the magnitude of
capital invested at each point in time; Not
affected by the timing of capital flows into or out
of the investment.

MULTI-PERIOD RETURNS
(CONT’D)
YOU CAN COMPUTE THIS AVERAGE USING
EITHER THE ARITHMETIC OR
GEOMETRIC MEAN...

Arithmetic average return over 1992-94:
= (r 92+ r93+ r94)/3

Geometric average return over 1992-94:
= [(1+r 92)(1+r93)(1+r94)](1/3) – 1

ARITHMETIC VS. GEOMETRIC
MEAN…
Arithmetic Mean:
=> Always greater than geometric mean.
=> Superior statistical properties:
* Best "estimator" or "forecast" of "true"
return.
=> Mean return components sum to the mean
total return
=> Most widely used in forecasts & portfolio
analysis.

MEAN (CONT’D)
Geometric Mean:
=> Reflects compounding ("chain-linking") of
returns:
* Earning of "return on return".
=> Mean return components do not sum to
mean total return
* Cross-product is left out.
=> Most widely used in performance evaluation.

MEAN (CONT’D)
The two are more similar:
- The less volatility in returns
across time
- The more frequent the return
interval
Note: "continuously compounded" returns
(log differences) side-steps around this
issue. (There is only one continuously-
compounded mean annual rate: arithmetic
& geometric distinctions do not exist).

TIME-WEIGHTED RETURNS:
NUMERICAL EXAMPLES
An asset that pays no dividends . . .

Year: End of year asset
value: HPR:

1992 $100,000

1993 $110,000 (110,000 - 100,000) / 100,000 = 10.00%

1994 $121,000 (121,000 - 110,000) / 110,000 = 10.00%

1995 $136,730 (136,730 - 121,000) / 121,000 = 13.00%

THREE-YEAR AVERAGE ANNUAL
RETURN (1993-95):

Arithmetic mean:
= (10.00 + 10.00 + 13.00) / 3
= 11.00%

Geometric mean:
= (136,730 / 100,000)(1/3) - 1
= ((1.1000)(1.1000)(1.1300))(1/3) - 1
= 10.99%

Continuously compounded:
= LN(136,730 / 100,000) / 3
= (LN(1.1)+LN(1.1)+LN(1.13)) / 3
= 10.47%

ANOTHER EXAMPLE
End of year
asset value: HPR:

Year:

1992 $100,000

1993 $110,000 (110,000 - 100,000) / 100,000 = 10.00%

1994 $124,300 (124,300 - 110,000) / 110,000 = 13.00%

1995 $140,459 (140,459 - 124,300) / 124,300 = 13.00%

RETURN (1993-95):
Arithmetic mean:
= (10.00 + 13.00 + 13.00) / 3
= 12.00%

Geometric mean:
= (140,459 / 100,000)(1/3) - 1
= ((1.1000)(1.1300)(1.1300))(1/3) - 1
= 11.99%

Continuously compounded:
= LN(140,459 / 100,000) / 3
= (LN(1.1)+LN(1.13)+LN(1.13)) / 3
= 11.32%

ANOTHER EXAMPLE

Year: End of year
asset value: HPR:

1992 $100,000

1993 $110,000 (110,000 - 100,000) / 100,000 = 10.00%

1994 $121,000 (121,000 - 110,000) / 110,000 = 10.00%

1995 $133,100 (133,100 - 121,000) / 121,000 = 10.00%

RETURN (1993-95):
Arithmetic mean:
= (10.00 + 10.00 + 10.00) / 3
= 10.00%

Geometric mean:
= (133,100 / 100,000)(1/3) - 1
= ((1.1000)(1.1000)(1.1000))(1/3) - 1
= 10.00%

Continuously comp'd:
= LN(133,100 / 100,000) / 3
= (LN(1.1)+LN(1.1)+LN(1.1)) / 3 = 9.53%

ANOTHER MULTI-PERIOD
RETURN MEASURE: THE IRR...
CAN’T COMPUTE HPRs IF YOU DON’T
KNOW ASSET VALUE AT
INTERMEDIATE POINTS IN TIME (AS
IN REAL ESTATE WITHOUT REGULAR
APPRAISALS)

SO YOU CAN’T COMPUTE TIME-
WEIGHTED AVERAGE RETURNS.

 You need the “IRR”.

IRR
SUPPOSE YOU WANT A RETURN
MEASURE THAT REFLECTS THE
EFFECT OF THE TIMING OF WHEN
(INSIDE OF THE OVERALL TIME SPAN
COVERED) THE INVESTOR HAS
DECIDED TO PUT MORE CAPITAL
INTO THE INVESTMENT AND/OR
TAKE CAPITAL OUT OF THE
INVESTMENT.

 You need the “IRR”.

IRR
FORMAL DEFINITION OF IRR

"IRR" (INTERNAL RATE OF RETURN) IS
THAT SINGLE RATE THAT DISCOUNTS ALL
THE NET CASH FLOWS OBTAINED FROM
THE INVESTMENT TO A PRESENT VALUE
EQUAL TO WHAT YOU PAID FOR THE
INVESTMENT AT THE BEGINNING:

IRR
CF 1 CF 2 CF N
0 = CF 0 + + + ... +
(1 + IRR) (1 + IRR )2 (1 + IRR )N
CFt = Net Cash Flow to Investor in Period "t“
CF0 is usually negative (capital outlay).
Note: CFt is signed according to the convention:
 cash flow from investor to investment is
negative,
 cash flow from investment to investor is
positive.
Note also: Last cash flow (CFN) includes two
components:
 The last operating cash flow plus
 The (ex dividend) terminal value of the asset
("reversion").

WHAT IS THE IRR?...
(TRYING TO GET SOME INTUITION
HERE . . .)

A SINGLE ("BLENDED") INTEREST
RATE, WHICH IF ALL THE CASH IN
THE INVESTMENT EARNED THAT
RATE ALL THE TIME IT IS IN THE
INVESTMENT, THEN THE INVESTOR
WOULD END UP WITH THE
TERMINAL VALUE OF THE
INVESTMENT (AFTER REMOVAL OF
CASH TAKEN OUT DURING THE
INVESTMENT):

WHAT IS THE IRR? (CONT’D)
PV (1 + IRR ) − CF1 (1 + IRR )
−  − CFN (1 + IRR ) = CF
N ( N −1)
where PV = -CF0, the initial cash−1"deposit" N
in the "account" (outlay to purchase the
investment).
IRR is "internal" because it includes only
the returns earned on capital while it is
invested in the project.
Once capital (i.e., cash) is withdrawn from
the investment, it no longer influences the
IRR.
This makes the IRR a "dollar-weighted"
average return across time for the
investment, because returns earned when

THE IRR INCLUDES THE EFFECT
OF:
1. THE INITIAL CASH YIELD RATE
(INITIAL LEVEL OF CASH PAYOUT AS
A FRACTION OF THE INITIAL
INVESTMENT;
2. THE EFFECT OF CHANGE OVER
TIME IN THE NET CASH FLOW
LEVELS (E.G., GROWTH IN THE
OPERATING CASH FLOW);
3. THE TERMINAL VALUE OF THE
ASSET AT THE END OF THE
INVESTMENT HORIZON (INCLUDING
ANY NET CHANGE IN CAPITAL
VALUE SINCE THE INITIAL

IRR
THE IRR IS THUS A TOTAL RETURN
MEASURE (CURRENT YIELD PLUS GROWTH
& GAIN).

NOTE ALSO:
THE IRR IS A CASH FLOW BASED
RETURN MEASURE...
•       DOES NOT DIFFERENTIATE
BETWEEN "INVESTMENT" AND
"RETURN ON OR RETURN OF
INVESTMENT".
•       INCLUDES THE EFFECT OF
CAPITAL INVESTMENTS AFTER THE
INITIAL OUTLAY.
•       DISTINGUISHES CASH FLOWS
ONLY BY THEIR DIRECTION: POSITIVE
IF FROM INVESTMENT TO INVESTOR,
NEGATIVE IF FROM INVESTOR TO

IRR
In general, it is not possible to algebraically
determine the IRR for any given set of cash
flows. It is necessary to solve numerically for the
IRR, in effect, solving the IRR equation by "trial
& error". Calculators and computers do this
automatically.

ADDITIONAL NOTES ON THE
IRR . . .
TECHNICAL PROBLEMS:
 IRR MAY NOT EXIST OR NOT BE
UNIQUE (OR GIVE MISLEADING RESULTS)
WHEN CASH FLOW PATTERNS INCLUDE
NEGATIVE CFs AFTER POSITIVE CFs.
 BEST TO USE NPV IN THESE
CASES. (SOMETIMES “FMRR” IS USED.)

ADDITIONAL NOTES ON THE IRR
(CONT’D)
THE IRR AND TIME-WEIGHTED RETURNS:
 IRR = TIME-WTD GEOMEAN HPR IF
(AND ONLY IF) THERE ARE NO
INTERMEDIATE CASH FLOWS (NO CASH
PUT IN OR TAKEN OUT BETWEEN THE
BEGINNING AND END OF THE
INVESTMENT).

(CONT’D)
THE IRR AND RETURN
COMPONENTS:
             IRR IS A "TOTAL RETURN"
             IRR DOES NOT GENERALLY
BREAK OUT EXACTLY INTO A SUM
OF: y + g: INITIAL CASH YIELD +
CAPITAL VALUE GROWTH
COMPONENTS.
             DIFFERENCE BETWEEN
THE IRR AND THE INITIAL CASH
YIELD IS DUE TO A COMBINATION OF
GROWTH IN THE OPERATING CASH
FLOWS AND/OR GROWTH IN THE

THE IRR AND RETURN
COMPONENTS (CONT’D)
 IF THE OPERATING CASH
FLOWS GROW AT A CONSTANT RATE,
AND IF THE ASSET VALUE REMAINS A
CONSTANT MULTIPLE OF THE
CURRENT OPERATING CASH FLOWS,
THEN THE IRR WILL INDEED EXACTLY
EQUAL THE SUM OF THE INITIAL
CASH YIELD RATE PLUS THE GROWTH
RATE (IN BOTH THE CASH FLOWS AND
THE ASSET CAPITAL VALUE), AND IN
THIS CASE THE IRR WILL ALSO
EXACTLY EQUAL BOTH THE
ARITHMETIC AND GEOMETRIC TIME-
WEIGHTED MEAN (CONSTANT

(CONT’D)
THE IRR AND TERMINOLOGY:
 IRR OFTEN CALLED "TOTAL YIELD"
(APPRAISAL)
 "YIELD TO MATURITY" (BONDS)
 EX-ANTE IRR = "GOING-IN IRR".

DOLLAR-WEIGHTED & TIME-WEIGHTED
RETURNS:A NUMERICAL EXAMPLE . . .

"OPEN-END" (PUT) OR (CREF).

INVESTORS BUY AND SELL "UNITS" ON
THE BASIS OF THE APPRAISED VALUE
OF THE PROPERTIES IN THE FUND AT
THE END OF EACH PERIOD.

SUPPOSE THE FUND DOESN'T PAY OUT
ANY CASH, BUT REINVESTS ALL
PROPERTY INCOME. CONSIDER 3
CONSECUTIVE PERIODS. . .

INVESTMENT PERIODIC
RETURNS: HIGH, LOW, HIGH . . .

1996 1997 1998 1999

YR END UNIT VALUE $1000 $1100 $990 $1089

GEOM MEAN TIME-WTD RETURN = (1.089)+10.00%
PERIODIC RETURN +10.00% -10.00% (1/3)
-
1 = 2.88%

INVESTOR #1, "MR. SMART" (OR
LUCKY): GOOD TIMING . . .

END OF YEAR: 1996 1997 1998 1999

UNITS BOUGHT 2

UNITS SOLD 1 1

IRR = IRR(-2000,1100,0,1089) = 4.68%
CASH FLOW -$2000 +$1100 0 $1089

INVESTOR #2, "MR. DUMB" (OR
UNLUCKY): BAD TIMING . . .

END OF YEAR: 1996 1997 1998 1999

UNITS BOUGHT 1 1

UNITS SOLD 1 1

IRR = IRR(-1000,-1100,990,1089) = -0.50%
CASH FLOW -$1000 -$1100 +$990 $1089

EXAMPLE (CONT’D)
DOLLAR-WTD RETURN BEST FOR
MEASURING INVESTOR PERFORMANCE
IF INVESTOR CONTROLLED TIMING OF
CAP. FLOW.

TIME-WTD RETURN BEST FOR
MEASURING PERFORMANCE OF THE
UNDERLYING INVESTMENT (IN THIS
CASE THE PUT OR CREF), AND
THEREFORE FOR MEASURING
INVESTOR PERFORMANCE IF INVESTOR
ONLY CONTROLS WHAT TO INVEST IN
BUT NOT WHEN.

MEASURING RETURNS
INTRODUCTION

Ex Ante Returns
 Return calculations may be done ‘before-the-fact,’

in which case, assumptions must be made about
the future

Ex Post Returns
 Return calculations done ‘after-the-fact,’ in order
to analyze what rate of return was earned.

MEASURING RETURNS
INTRODUCTION

According to the constant growth DDM can be decomposed into the
two forms of income that equity investors may receive, dividends
and capital gains.

 D1 
kc =   + [ g ] = [ Income / Dividend Yield] + [ Capital Gain (or loss) Yield]
 P0 

WHEREAS

Fixed-income investors (bond investors for example) can expect to
earn interest income as well as (depending on the movement of
interest rates) either capital gains or capital losses.


MEASURING RETURNS
INCOME YIELD

 Income yield is the return earned in the form of a
periodic cash flow received by investors.

 The income yield return is calculated by the
periodic cash flow divided by the purchase price.

CF1
[8-1] Income yield =
P0

Where CF1 = the expected cash flow to be received
P0 = the purchase price

INCOME YIELD
STOCKS VERSUS BONDS
Figure 8-1 illustrates the income yields for both bonds and stock in Canada from
the 1950s to 2005

The dividend yield is calculated using trailing rather than


forecast earns (because next year’s dividends cannot be
predicted in aggregate), nevertheless dividend yields have
exceeded income yields on bonds.
 Reason – risk
 The risk of earning bond income is much less than the risk
incurred in earning dividend income.

(Remember, bond investors, as secured creditors of the first have a
legally-enforceable contractual claim to interest.)

(See Figure 8 -1 on the following slide)

EX POST VERSUS EX ANTE
RETURNS
MARKET INCOME YIELDS
8-1 FIGURE

Insert Figure 8 - 1

MEASURING RETURNS
COMMON SHARE AND LONG CANADA BOND YIELD
GAP
 Table 8 – 1 illustrates the income yield gap between stocks and bonds over
recent decades
 The main reason that this yield gap has varied so much over time is that the

return to investors is not just the income yield but also the capital gain (or
loss) yield as well.

Table 8-1 Average Yield Gap

Average Yield Gap (%)
1950s 0.82
1960s 2.35
1970s 4.54
1980s 8.14
1990s 5.51
2000s 3.55
Overall 4.58

MEASURING RETURNS
DOLLAR RETURNS

Investors in market-traded securities (bonds or stock)
receive investment returns in two different form:
Income yield



 Capital gain (or loss) yield

The investor will receive dollar returns, for example:
 $1.00 of dividends
 Share price rise of $2.00

To be useful, dollar returns must be converted to percentage returns as a
function of the original investment. (Because a $3.00 return on a $30
investment might be good, but a $3.00 return on a $300 investment
would be unsatisfactory!)

MEASURING RETURNS
CONVERTING DOLLAR RETURNS TO PERCENTAGE
RETURNS

An investor receives the following dollar returns a stock
investment of $25:
$1.00 of dividends



 Share price rise of $2.00

The capital gain (or loss) return component of total return is calculated:
ending price – minus beginning price, divided by beginning price

P − P0 $27 - $25
[8-2] Capital gain (loss) return = 1 = = .08 = 8%
P0 $25

MEASURING RETURNS
TOTAL PERCENTAGE RETURN

 The investor’s total return (holding period
return) is:

Total return = Income yield + Capital gain (or loss) yield
CF + P − P
= 1 1 0
[8-3] P0
 CF   P − P 
= 1+ 1 0
 P0   P0 
 $1.00   $27 − $25 
=  +  $25  = 0.04 + 0.08 = 0.12 = 12%
 $25   

MEASURING RETURNS
TOTAL PERCENTAGE RETURN – GENERAL FORMULA

 The general formula for holding period return is:

Total return = Income yield + Capital gain (or loss) yield
CF1 + P − P0
= 1

[8-3] P0
 CF1   P − P0 
= +
1

 P0   P0 

MEASURING AVERAGE RETURNS
EX POST RETURNS

 Measurement of historical rates of return that have
been earned on a security or a class of securities allows

us to identify trends or tendencies that may be useful
in predicting the future.
 There are two different types of ex post mean or
average returns used:
 Arithmeticaverage
 Geometric mean

ARITHMETIC AVERAGE

n

∑r

[8-4] i
Arithmetic Average (AM) = i =1
n

Where:
ri = the individual returns
n = the total number of observations

 Most commonly used value in statistics
 Sum of all returns divided by the total number of
observations

GEOMETRIC MEAN

1

[8-5]
Geometric Mean (GM) = [( 1 + r1 )( 1 + r2 )( 1 + r3 )...( 1 + rn )] -1
n

 Measures the average or compound growth rate
over multiple periods.

GEOMETRIC MEAN VERSUS ARITHMETIC AVERAGE

If all returns (values) are identical the geometric mean =
arithmetic average.

If the return values are volatile the geometric mean <
arithmetic average

The greater the volatility of returns, the greater the
difference between geometric mean and arithmetic average.

(Table 8 – 2 illustrates this principle on major asset classes 1938 – 2005)

AVERAGE INVESTMENT RETURNS AND STANDARD
DEVIATIONS

Table 8 - 2 Average Investment Returns and Standard Deviations, 1938-2005

Annual Annual Standard Deviation
Arithmetic Geometric of Annual Returns
Average (%) Mean (%) (%)

Government of Canada treasury bills 5.20 5.11 4.32
Government of Canada bonds 6.62 6.24 9.32
Canadian stocks 11.79 10.60 16.22
U.S. stocks 13.15 11.76 17.54

So urce: Data are fro m the Canadian Institute o f A ctuaries

The greater the difference,
the greater the volatility of
annual returns.

MEASURING EXPECTED (EX ANTE)
RETURNS

 While past returns might be interesting, investor’s are
most concerned with future returns.

 Sometimes, historical average returns will not be
realized in the future.
 Developing an independent estimate of ex ante returns
usually involves use of forecasting discrete scenarios
with outcomes and probabilities of occurrence.

ESTIMATING EXPECTED RETURNS
ESTIMATING EX ANTE (FORECAST) RETURNS

 The general formula

n
[8-6] Expected Return (ER) = ∑ (ri × Prob i )
i =1

Where:
ER = the expected return on an investment
Ri = the estimated return in scenario i
Probi = the probability of state i occurring

ESTIMATING EX ANTE (FORECAST) RETURNS

Example:
This is type of forecast data that are required to make
an ex ante estimate of expected return.

Possible
Returns on
Probability of Stock A in that
State of the Economy Occurrence State
Economic Expansion 25.0% 30%
Normal Economy 50.0% 12%
Recession 25.0% -25%


ESTIMATING EX ANTE (FORECAST) RETURNS USING A
SPREADSHEET APPROACH

Example Solution:
Sum the products of the probabilities and possible
returns in each state of the economy.

(1) (2) (3) (4)=(2)×(1)
Possible Weighted
Returns on Possible
Probability of Stock A in that Returns on
State of the Economy Occurrence State the Stock
Economic Expansion 25.0% 30% 7.50%
Normal Economy 50.0% 12% 6.00%
Recession 25.0% -25% -6.25%
Expected Return on the Stock = 7.25%


ESTIMATING EX ANTE (FORECAST) RETURNS USING A FORMULA
APPROACH

Example Solution:
Sum the products of the probabilities and possible
returns in each state of the economy.

n
Expected Return (ER) = ∑ (ri × Prob i )
i =1

= (r1 × Prob1 ) + (r2 × Prob 2 ) + (r3 × Prob 3 )
= (30% × 0.25) + (12% × 0.5) + (-25% × 0.25)
= 7.25%


MEASURING RISK

RISK

 Probability of incurring harm

 For investors, risk is the probability of earning an
inadequate return.
 Ifinvestors require a 10% rate of return on a given
investment, then any return less than 10% is considered
harmful.

RISK
ILLUSTRATED

The range of total possible returns
on the stock A runs from -30% to
Probability
more than +40%. If the required

return on the stock is 10%, then
those outcomes less than 10%
Outcomes that produce harm represent risk to the investor.

A

-30% -20% -10% 0% 10% 20% 30% 40%
Possible Returns on the Stock

RANGE

 The difference between the maximum and minimum

values is called the range
 Canadian common stocks have had a range of annual
returns of 74.36 % over the 1938-2005 period
 Treasury bills had a range of 21.07% over the same period.
 As a rough measure of risk, range tells us that
common stock is more risky than treasury bills.

DIFFERENCES IN LEVELS OF RISK
ILLUSTRATED

Outcomes that produce harm The wider the range of probable
outcomes the greater the risk of the
Probability
investment.

B A is a much riskier investment than B

A

-30% -20% -10% 0% 10% 20% 30% 40%
Possible Returns on the Stock

HISTORICAL RETURNS ON
DIFFERENT ASSET CLASSES

 Figure 8-2 illustrates the volatility in annual returns on

three different assets classes from 1938 – 2005.
 Note:
 Treasury bills always yielded returns greater than 0%
 Long Canadian bond returns have been less than 0% in some
years (when prices fall because of rising interest rates), and
the range of returns has been greater than T-bills but less
than stocks
 Common stock returns have experienced the greatest range of
returns
(See Figure 8-2 on the following slide)

MEASURING RISK
ANNUAL RETURNS BY ASSET CLASS, 1938 - 2005
FIGURE 8-2


REFINING THE MEASUREMENT OF
RISK
STANDARD DEVIATION (Σ)

 Range measures risk based on only two observations

(minimum and maximum value)
 Standard deviation uses all observations.
 Standard deviation can be calculated on forecast or possible
returns as well as historical or ex post returns.

(The following two slides show the two different formula used for Standard
Deviation)

MEASURING RISK
EX POST STANDARD DEVIATION

n _

∑ ( ri − r ) 2
[8-7] Ex post σ = i =1

n −1

Where :
σ = the standard deviation
_
r = the average return
ri = the return in year i
n = the number of observations


MEASURING RISK
EXAMPLE USING THE EX POST STANDARD DEVIATION

Problem
Estimate the standard deviation of the historical returns on investment
A that were: 10%, 24%, -12%, 8% and 10%.
Step 1 – Calculate the Historical Average Return

n

∑r i
10 + 24 - 12 + 8 + 10 40
Arithmetic Average (AM) = i =1
= = = 8.0%
n 5 5

Step 2 – Calculate the Standard Deviation
n _

∑ (r − r ) i
2
(10 - 8) 2 + (24 − 8) 2 + (−12 − 8) 2 + (8 − 8) 2 + (14 − 8) 2
Ex post σ = i =1
=
n −1 5 −1
2 2 + 16 2 − 20 2 + 0 2 + 2 2 4 + 256 + 400 + 0 + 4 664
= = = = 166 = 12.88%
4 4 4


EX POST RISK
STABILITY OF RISK OVER TIME

Figure 8-3 (on the next slide) demonstrates that the relative riskiness of
equities and bonds has changed over time.

Until the 1960s, the annual returns on common shares were about four
times more variable than those on bonds.

Over the past 20 years, they have only been twice as variable.

Consequently, scenario-based estimates of risk (standard deviation) is
required when seeking to measure risk in the future. (We cannot safely
assume the future is going to be like the past!)

Scenario-based estimates of risk is done through ex ante estimates and
calculations.


RELATIVE UNCERTAINTY
EQUITIES VERSUS BONDS
FIGURE 8-3


MEASURING RISK
EX ANTE STANDARD DEVIATION

A Scenario-Based Estimate of Risk

n
[8-8] Ex ante σ = ∑ (Prob i ) × (ri − ERi ) 2
i =1

SCENARIO-BASED ESTIMATE OF RISK
EXAMPLE USING THE EX ANTE STANDARD DEVIATION –
RAW DATA

GIVEN INFORMATION INCLUDES:
- Possible returns on the investment for different
discrete states
- Associated probabilities for those possible returns

Possible
State of the Returns on
Economy Probability Security A

Recession 25.0% -22.0%
Normal 50.0% 14.0%
Economic Boom 25.0% 35.0%


SCENARIO-BASED ESTIMATE OF
RISK
EX ANTE STANDARD DEVIATION – SPREADSHEET
APPROACH

 The following two slides illustrate an approach to
solving for standard deviation using a spreadsheet
model.

FIRST STEP – CALCULATE THE EXPECTED RETURN

Determined by multiplying
the probability times the
possible return.

Possible Weighted
State of the Returns on Possible
Economy Probability Security A Returns

Recession 25.0% -22.0% -5.5%
Normal 50.0% 14.0% 7.0%
Economic Boom 25.0% 35.0% 8.8%
Expected Return = 10.3%

Expected return equals the sum of
the weighted possible returns.


SECOND STEP – MEASURE THE WEIGHTED AND SQUARED
DEVIATIONS

Now multiply the square deviations by
First calculate the deviation of
their probability of occurrence.
possible returns from the expected.

Deviation of Weighted
Possible Weighted Possible and
State of the Returns on Possible Return from Squared Squared
Economy Probability Security A Returns Expected Deviations Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600
Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070
Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531
Expected Return = 10.3% Variance = 0.0420
Standard Deviation = 20.50%

Second, square those deviations
The sum of the weighted and square deviations
The standardthe mean. is the square root
from deviation
is the variance in percent (in percent terms).
of the variance squared terms.

SCENARIO-BASED ESTIMATE OF
RISK
EXAMPLE USING THE EX ANTE STANDARD
DEVIATION FORMULA

Possible Weighted
State of the Returns on Possible
Economy Probability Security A Returns

Recession 25.0% -22.0% -5.5%
Normal 50.0% 14.0% 7.0%
Economic Boom 25.0% 35.0% 8.8%
Expected Return = 10.3%
n
Ex ante σ = ∑ (Prob ) × (r − ER )
i =1
i i i
2

= P (r1 − ER1 ) 2 + P2 (r2 − ER2 ) 2 + P (r3 − ER3 ) 2
1 1

= .25(−22 − 10.3) 2 + .5(14 − 10.3) 2 + .25(35 − 10.3) 2
= .25(−32.3) 2 + .5(3.8) 2 + .25(24.8) 2
= .25(.10401) + .5(.00141) + .25(.06126)
= .0420
= .205 = 20.5%

MODERN PORTFOLIO

THEORY

PORTFOLIOS

 A portfolio is a collection of different securities such as stocks and
bonds, that are combined and considered a single asset

 The risk-return characteristics of the portfolio is demonstrably
different than the characteristics of the assets that make up that
portfolio, especially with regard to risk.

 Combining different securities into portfolios is done to achieve
diversification.

DIVERSIFICATION
Diversification has two faces:

Diversification results in an overall reduction in

1.
portfolio risk (return volatility over time) with little
sacrifice in returns, and
2. Diversification helps to immunize the portfolio from
potentially catastrophic events such as the outright
failure of one of the constituent investments.

(If only one investment is held, and the issuing firm
goes bankrupt, the entire portfolio value and returns
are lost. If a portfolio is made up of many different
investments, the outright failure of one is more than
likely to be offset by gains on others, helping to make
the portfolio immune to such events.)

EXPECTED RETURN OF A
PORTFOLIO
MODERN PORTFOLIO THEORY

The Expected Return on a Portfolio is simply the
weighted average of the returns of the individual assets
that make up the portfolio:

n
[8-9] ER p = ∑ ( wi × ERi )
i =1

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%

n
ER p = ∑ ( wi × ERi ) = (.286 ×14%) + (.714 × 6% )
i =1

= 4.004% + 4.284% = 8.288%

RANGE OF RETURNS IN A TWO
ASSET PORTFOLIO

In a two asset portfolio, simply by changing the weight of the

constituent assets, different portfolio returns can be achieved.

Because the expected return on the portfolio is a simple
weighted average of the individual returns of the assets, you
can achieve portfolio returns bounded by the highest and the
lowest individual asset returns.

ASSET PORTFOLIO

Example 1:

Assume ERA = 8% and ERB = 10%

(See the following 6 slides based on Figure 8-4)

EXPECTED PORTFOLIO RETURN
AFFECT ON PORTFOLIO RETURN OF CHANGING RELATIVE
WEIGHTS IN A AND B

8 - 4 FIGURE

10.50

10.00 ERB= 10%

9.50

9.00

8.50

8.00 ERA=8%
%n u e R de ce px E

7.50
t

7.00
0 0.2 0.4 0.6 0.8 1.0 1.2
r t

Portfolio Weight


WEIGHTS IN A AND B

8 - 4 FIGURE

A portfolio manager can select the relative weights of the two
assets in the portfolio to get a desired return between 8%
(100% invested in A) and 10% (100% invested in B)
10.50

10.00 ERB= 10%

9.50

9.00

8.50

8.00 ERA=8%
%n u e R de ce px E

7.50
t

7.00
0 0.2 0.4 0.6 0.8 1.0 1.2
r t

Portfolio Weight


WEIGHTS IN A AND B

8 - 4 FIGURE

10.50

ERB= 10%
10.00

9.50 The potential returns of
the portfolio are
bounded by the highest
9.00 and lowest returns of
the individual assets
8.50 that make up the
portfolio.

8.00
%n u e R de ce px E

ERA=8%
7.50
t

7.00
0 0.2 0.4 0.6 0.8 1.0 1.2
r t

Portfolio Weight


WEIGHTS IN A AND B

8 - 4 FIGURE

10.50

ERB= 10%
10.00

9.50

9.00
The expected return on
the portfolio if 100% is
8.50 invested in Asset A is
8%.
8.00 ER p = wA ER A + wB ERB = (1.0)(8%) + (0)(10%) = 8%
%n u e R de ce px E

ERA=8%
7.50
t

7.00
0 0.2 0.4 0.6 0.8 1.0 1.2
r t

Portfolio Weight


WEIGHTS IN A AND B

8 - 4 FIGURE

10.50 The expected return on
invested in Asset B is ERB= 10%
10.00 10%.

9.50

9.00

8.50
ER p = wA ER A + wB ERB = (0)(8%) + (1.0)(10%) = 10%
8.00
%n u e R de ce px E

ERA=8%
7.50
t

7.00
0 0.2 0.4 0.6 0.8 1.0 1.2
r t

Portfolio Weight


WEIGHTS IN A AND B

8 - 4 FIGURE

10.50 The expected return on
invested in Asset A and ERB= 10%
10.00 50% in B is 9%.

9.50
ER p = wA ERA + wB ERB
9.00
= (0.5)(8%) + (0.5)(10%)
8.50 = 4% + 5% = 9%
8.00
%n u e R de ce px E

ERA=8%
7.50
t

7.00
0 0.2 0.4 0.6 0.8 1.0 1.2
r t

Portfolio Weight


ASSET PORTFOLIO

Example 1:

Assume ERA = 14% and ERB = 6%

(See the following 2 slides )

RANGE OF RETURNS IN A TWO ASSET
PORTFOLIO
E(R)A= 14%, E(R)B= 6%

Expected return on Asset A = 14.0%
Expected return on Asset B = 6.0%

Expected
Weight of Weight of Return on the
Asset A Asset B Portfolio
0.0% 100.0% 6.0%
10.0% 90.0% 6.8%
20.0% 80.0% 7.6%
30.0% 70.0% 8.4%
40.0% 60.0% 9.2% A graph of this
50.0% 50.0% 10.0% relationship is
60.0% 40.0% 10.8%
found on the
70.0% 30.0% 11.6%
following slide.
80.0% 20.0% 12.4%
90.0% 10.0% 13.2%
100.0% 0.0% 14.0%

RANGE OF RETURNS IN A TWO ASSET
PORTFOLIO
E(R)A= 14%, E(R)B= 6%

Range of Portfolio Returns

Expected Return on Two

16.00%
14.00%
Asset Portfolio

12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%

%
%

%

%

%

%

%

%

%

%
0%

0
.0

.0

.0

.0

.0

.0

.0

.0

.0

0.
0.

10

20

30

40

50

60

70

80

90

10
Weight Invested in Asset A

EXPECTED PORTFOLIO RETURNS
EXAMPLE OF A THREE ASSET PORTFOLIO

Relative Expected Weighted
Weight Return Return
Stock X 0.400 8.0% 0.03
Stock Y 0.350 15.0% 0.05
Stock Z 0.250 25.0% 0.06
Expected Portfolio Return = 14.70%

K. Hartviksen

RISK IN PORTFOLIOS

MODERN PORTFOLIO THEORY - MPT

 Prior to the establishment of Modern Portfolio Theory (MPT),

most people only focused upon investment returns…they
ignored risk.

 With MPT, investors had a tool that they could use to
dramatically reduce the risk of the portfolio without a
significant reduction in the expected return of the portfolio.

EXPECTED RETURN AND RISK FOR
PORTFOLIOS
STANDARD DEVIATION OF A TWO-ASSET PORTFOLIO USING
COVARIANCE

[8-11] σ p = ( wA ) 2 (σ A ) 2 + ( wB ) 2 (σ B ) 2 + 2( wA )( wB )(COVA, B )

Risk of Asset A Risk of Asset B Factor to take into
adjusted for weight adjusted for weight account comovement
in the portfolio in the portfolio of returns. This factor
can be negative.

EXPECTED RETURN AND RISK FOR
PORTFOLIOS
STANDARD DEVIATION OF A TWO-ASSET PORTFOLIO USING
CORRELATION COEFFICIENT

[8-15] σ p = ( wA ) 2 (σ A ) 2 + ( wB ) 2 (σ B ) 2 + 2( wA )( wB )( ρ A, B )(σ A )(σ B )

Factor that takes into
account the degree of
comovement of returns.
It can have a negative
value if correlation is
negative.

GROUPING INDIVIDUAL ASSETS
INTO PORTFOLIOS
 The riskiness of a portfolio that is made of different risky
assets is a function of three different factors:
 the riskiness of the individual assets that make up the

portfolio
 the relative weights of the assets in the portfolio
 the degree of comovement of returns of the assets making
up the portfolio
 The standard deviation of a two-asset portfolio may be
measured using the Markowitz model:

σ p = σ w + σ w + 2 wA wB ρ A, Bσ Aσ B
2
A
2
A
2
B
2
B

RISK OF A THREE-ASSET
PORTFOLIO

The data requirements for a three-asset portfolio grows
dramatically if we are using Markowitz Portfolio selection formulae.

We need 3 (three) correlation coefficients between A and B; A and
C; and B and C.
A
ρa,b ρa,c
B C
ρb,c

σ p = σ A wA + σ B wB + σ C wC + 2wA wB ρ A, Bσ Aσ B + 2 wB wC ρ B ,Cσ Bσ C + 2 wA wC ρ A,Cσ Aσ C
2 2 2 2 2 2

RISK OF A FOUR-ASSET PORTFOLIO

The data requirements for a four-asset portfolio grows dramatically

if we are using Markowitz Portfolio selection formulae.

We need 6 correlation coefficients between A and B; A and C; A
and D; B and C; C and D; and B and D.

A
ρa,b ρa,d
ρa,c
B D
ρb,d
ρb,c ρc,d
C

COVARIANCE
 A statistical measure of the correlation of the
fluctuations of the annual rates of return of

different investments.

n _ _
[8-12] COV AB = ∑ Prob i (k A,i − ki )(k B ,i - k B )
i =1

CORRELATION
 The degree to which the returns of two stocks co-
move is measured by the correlation coefficient (ρ).

 The correlation coefficient (ρ) between the returns
on two securities will lie in the range of +1 through
- 1.
+1 is perfect positive correlation
-1 is perfect negative correlation

COV AB
[8-13] ρ AB =
σ Aσ B

COVARIANCE AND CORRELATION
COEFFICIENT
 Solving for covariance given the correlation
coefficient and standard deviation of the two
assets:

[8-14] COV AB = ρ ABσ Aσ B

IMPORTANCE OF CORRELATION

 Correlation is important because it affects the

degree to which diversification can be achieved
using various assets.
 Theoretically, if two assets returns are perfectly
positively correlated, it is possible to build a
riskless portfolio with a return that is greater than
the risk-free rate.

AFFECT OF PERFECTLY NEGATIVELY
CORRELATED RETURNS
ELIMINATION OF PORTFOLIO RISK
Returns
If returns of A and B are
%
20% perfectly negatively correlated,
a two-asset portfolio made up of

equal parts of Stock A and B
would be riskless. There would
15% be no variability
of the portfolios returns over
time.

10%

Returns on Stock A
Returns on Stock B
5%
Returns on Portfolio

Time 0 1 2

EXAMPLE OF PERFECTLY POSITIVELY
CORRELATED RETURNS
NO DIVERSIFICATION OF PORTFOLIO RISK
Returns
If returns of A and B are
%
20% perfectly positively correlated,
a two-asset portfolio made up of

equal parts of Stock A and B
would be risky. There would be
15% no diversification (reduction of
portfolio risk).

10%
Returns on Stock A
Returns on Stock B
5%
Returns on Portfolio

Time 0 1 2

AFFECT OF PERFECTLY NEGATIVELY
CORRELATED RETURNS
NUMERICAL EXAMPLE

Weight of Asset A = 50.0%
Weight of Asset B = 50.0%

n
Expected ER p = ∑ ( wi × ERi ) = (.5 × 5%) + (.5 ×15% )
i =1
Return on Return on Return on the = 2.5% + 7.5% = 10%
Year Asset A Asset B Portfolio
n
xx07 = ∑ (wi5.0% = (.5 ×15%) 15.0% )
ER p × ERi ) + (.5 × 5% 10.0%
i =1
xx08 = 7.5% + 2.5% = 10%
10.0% 10.0% 10.0%
xx09 15.0% 5.0% 10.0%

Perfectly Negatively
Correlated Returns
over time

DIVERSIFICATION POTENTIAL

 The potential of an asset to diversify a portfolio is dependent

upon the degree of co-movement of returns of the asset with
those other assets that make up the portfolio.
 In a simple, two-asset case, if the returns of the two assets are
perfectly negatively correlated it is possible (depending on the
relative weighting) to eliminate all portfolio risk.
 This is demonstrated through the following series of
spreadsheets, and then summarized in graph format.

EXAMPLE OF PORTFOLIO
COMBINATIONS AND CORRELATION
Perfect
Positive
Correlation –
no

Expected Standard Correlation
Asset Return Deviation Coefficient diversification
A 5.0% 15.0% 1
B 14.0% 40.0%

Portfolio Components Portfolio Characteristics Both
Expected Standard
Weight of A Weight of B Return Deviation portfolio
100.00% 0.00% 5.00% 15.0% returns and
90.00% 10.00% 5.90% 17.5%
80.00% 20.00% 6.80% 20.0%
risk are
70.00% 30.00% 7.70% 22.5% bounded by
60.00% 40.00% 8.60% 25.0%
50.00% 50.00% 9.50% 27.5%
the range set
40.00% 60.00% 10.40% 30.0% by the
30.00% 70.00% 11.30% 32.5%
20.00% 80.00% 12.20% 35.0%
constituent
10.00% 90.00% 13.10% 37.5% assets when
0.00% 100.00% 14.00% 40.0%
ρ=+1

Positive
Correlation –
weak
diversification

Asset Return Deviation Coefficient potential
A 5.0% 15.0% 0.5
B 14.0% 40.0%

Portfolio Components Portfolio Characteristics
Expected Standard When ρ=+0.5
Weight of A Weight of B Return Deviation
100.00% 0.00% 5.00% 15.0% these portfolio
90.00% 10.00% 5.90% 15.9% combinations
80.00% 20.00% 6.80% 17.4%
70.00% 30.00% 7.70% 19.5%
have lower
60.00% 40.00% 8.60% 21.9% risk –
50.00% 50.00% 9.50% 24.6% expected
40.00% 60.00% 10.40% 27.5%
30.00% 70.00% 11.30% 30.5% portfolio return
20.00% 80.00% 12.20% 33.6% is unaffected.
10.00% 90.00% 13.10% 36.8%
0.00% 100.00% 14.00% 40.0%

No
Correlation –
some
diversification

A 5.0% 15.0% 0
B 14.0% 40.0%

Expected Standard
Portfolio
100.00% 0.00% 5.00% 15.0%
90.00% 10.00% 5.90% 14.1% risk is
80.00% 20.00% 6.80% 14.4% lower than
70.00% 30.00% 7.70% 15.9%
60.00% 40.00% 8.60% 18.4%
the risk of
50.00% 50.00% 9.50% 21.4% either
40.00% 60.00% 10.40% 24.7%
asset A or
30.00% 70.00% 11.30% 28.4%
20.00% 80.00% 12.20% 32.1% B.
10.00% 90.00% 13.10% 36.0%
0.00% 100.00% 14.00% 40.0%

Negative
Correlation –
greater
diversification

A 5.0% 15.0% -0.5
B 14.0% 40.0%

Expected Standard
Portfolio risk
100.00% 0.00% 5.00% 15.0% for more
90.00% 10.00% 5.90% 12.0% combinations
80.00% 20.00% 6.80% 10.6%
70.00% 30.00% 7.70% 11.3% is lower than
60.00% 40.00% 8.60% 13.9% the risk of
50.00% 50.00% 9.50% 17.5%
40.00% 60.00% 10.40% 21.6%
either asset
30.00% 70.00% 11.30% 26.0%
20.00% 80.00% 12.20% 30.6%
10.00% 90.00% 13.10% 35.3%
0.00% 100.00% 14.00% 40.0%

Perfect
Negative
Correlation –
greatest
diversification

A 5.0% 15.0% -1
B 14.0% 40.0%

Expected Standard
100.00% 0.00% 5.00% 15.0%
90.00% 10.00% 5.90% 9.5%
80.00% 20.00% 6.80% 4.0%
Risk of the
70.00% 30.00% 7.70% 1.5% portfolio is
60.00% 40.00% 8.60% 7.0% almost
50.00% 50.00% 9.50% 12.5% eliminated at
40.00% 60.00% 10.40% 18.0%
30.00% 70.00% 11.30% 23.5% 70% invested in
20.00% 80.00% 12.20% 29.0% asset A
10.00% 90.00% 13.10% 34.5%
0.00% 100.00% 14.00% 40.0%

Diversification of a Two Asset Portfolio
Demonstrated Graphically
The Effect of Correlation on Portfolio Risk:
The Two-Asset Case

Expected Return B

ρAB = -0.5
12%
ρAB = -1

8%
ρAB = 0

ρAB= +1

A
4%

0%

0% 10% 20% 30% 40%

Standard Deviation

IMPACT OF THE CORRELATION
COEFFICIENT

 Figure 8-7 (see the next slide) illustrates the
relationship between portfolio risk (σ) and the

correlation coefficient
 The slope is not linear a significant amount of
diversification is possible with assets with no correlation (it
is not necessary, nor is it possible to find, perfectly
negatively correlated securities in the real world)
 With perfect negative correlation, the variability of portfolio
returns is reduced to nearly zero.

EXPECTED PORTFOLIO
RETURN
IMPACT OF THE CORRELATION COEFFICIENT
8 - 7 FIGURE

15

10

5
) % not a ve D d a dna S
s n u e R ol o t r o Pf o
t
i f
r

0
-1 -0.5 0 0.5 1
( i i
r t

Correlation Coefficient (ρ)


ZERO RISK PORTFOLIO
 We can calculate the portfolio that removes all risk.
 When ρ = -1, then

[8-15] σ p = ( wA ) 2 (σ A ) 2 + ( wB ) 2 (σ B ) 2 + 2( wA )( wB )( ρ A, B )(σ A )(σ B )

 Becomes:

[8-16] σ p = wσ A − (1 − w)σ B

Friday, October 12, 2012
AN EXERCISE TO PRODUCE
THE EFFICIENT FRONTIER
USING THREE ASSETS

AN EXERCISE USING T-BILLS,
STOCKS AND BONDS
Base Data: Stocks T-bills Bonds Historical
Expected Return(%) 12.73383 6.151702 7.0078723 averages for
Standard Deviation (%) 0.168 0.042 0.102
returns and risk for
three asset

Correlation Coefficient Matrix:
Stocks 1 -0.216 0.048
Each achievable
classes
T-bills -0.216 1 0.380 portfolio
Bonds 0.048 0.380 1
combination is
Historical
Portfolio Combinations: plotted correlation
on
Weights Portfolio
expected return,
coefficients
Expected Standard risk between the asset
(σ) space,
classes
Combination Stocks T-bills Bonds Return Variance Deviation found on the
1 100.0% 0.0% 0.0% 12.7 0.0283 16.8%
2 90.0% 10.0% 0.0% 12.1 0.0226 15.0% following slide.
3 80.0% 20.0% 0.0% 11.4 0.0177 13.3% Portfolio
4 70.0% 30.0% 0.0% 10.8 0.0134 11.6% characteristics for
5 60.0% 40.0% 0.0% 10.1 0.0097 9.9% each combination
6 50.0% 50.0% 0.0% 9.4 0.0067 8.2%
of securities
7 40.0% 60.0% 0.0% 8.8 0.0044 6.6%
8 30.0% 70.0% 0.0% 8.1 0.0028 5.3%
9 20.0% 80.0% 0.0% 7.5 0.0018 4.2%
10 10.0% 90.0% 0.0% 6.8 0.0014 3.8%

ACHIEVABLE PORTFOLIOS
RESULTS USING ONLY THREE ASSET CLASSES

Attainable Portfolio Combinations The efficient set is that set of
and Efficient Set of Portfolio Combinations achievable portfolio
combinations that offer the

highest rate of return for a
14.0
Efficient Set given level of risk. The solid
Portfolio Expected Return (%)

12.0 blue line indicates the efficient
Minimum Variance
Portfolio
set.
10.0
8.0
The plotted points are
6.0
attainable portfolio
4.0 combinations.

2.0
0.0
0.0 5.0 10.0 15.0 20.0
Standard Deviation of the Portfolio (%)

ACHIEVABLE TWO-SECURITY
PORTFOLIOS
MODERN PORTFOLIO THEORY
8 - 9 FIGURE

This line
represents

13 the set of
12
portfolio
combinations
11
that are
10 achievable by
9
varying
relative
8 weights and
%n u e R de ce px E

7 using two
non-
t

6
0 10 20 30 40 50 60
correlated
r t

Standard Deviation (%) securities.

DOMINANCE

 It is assumed that investors are rational, wealth-

maximizing and risk averse.
 If so, then some investment choices dominate others.

INVESTMENT CHOICES
THE CONCEPT OF DOMINANCE ILLUSTRATED

Return A dominates B
% because it offers

A B the same return
10% but for less risk.
A dominates C
C because it offers a
5% higher return but
for the same risk.

5% 20% Risk
To the risk-averse wealth maximizer, the choices are clear, A dominates B,
A dominates C.

EFFICIENT FRONTIER
THE TWO-ASSET PORTFOLIO COMBINATIONS
8 - 10 FIGURE

A is not attainable

B,E lie on the
efficient frontier and
are attainable
A B E is the minimum
variance portfolio
C (lowest risk
combination)

C, D are
E attainable but are
%n u e R de ce px E

D dominated by
superior portfolios
t

that line on the line
above E
r t

Standard Deviation (%)

EFFICIENT FRONTIER
THE TWO-ASSET PORTFOLIO COMBINATIONS
8 - 10 FIGURE

Rational, risk
averse

investors will
only want to
A B hold portfolios
such as B.
C

The actual
E choice will
%n u e R de ce px E

D depend on
her/his risk
t

preferences.
r t

Standard Deviation (%)

DIVERSIFICATION

DIVERSIFICATION
 We have demonstrated that risk of a portfolio can be
reduced by spreading the value of the portfolio across,
two, three, four or more assets.

 The key to efficient diversification is to choose assets
whose returns are less than perfectly positively
correlated.
 Even with random or naïve diversification, risk of the
portfolio can be reduced.
 This is illustrated in Figure 8 -11 and Table 8 -3 found
on the following slides.
 As the portfolio is divided across more and more securities,
the risk of the portfolio falls rapidly at first, until a point is
reached where, further division of the portfolio does not
result in a reduction in risk.
 Going beyond this point is known as superfluous

diversification.

DIVERSIFICATION
DOMESTIC DIVERSIFICATION
8 - 11 FIGURE

Average Portfolio Risk
January 1985 to December 1997

14

12

10

8

6

4

2

0
t

0 50 100 150 200 250 300
r

Number of Stocks in Portfolio
( i i

DIVERSIFICATION
DOMESTIC DIVERSIFICATION

Table 8-3 Monthly Canadian Stock Portfolio Returns, January 1985 to December 1997

Number of Average Standard Deviation Ratio of Portfolio Percentage of
Stocks in Monthly of Average Standard Deviation to Total Achievable
Portfolio Portfolio Monthly Portfolio Standard Deviation of a Risk Reduction
Return (%) Return (%) Single Stock
1 1.51 13.47 1.00 0.00
2 1.51 10.99 0.82 27.50
3 1.52 9.91 0.74 39.56
4 1.53 9.30 0.69 46.37
5 1.52 8.67 0.64 53.31
6 1.52 8.30 0.62 57.50
7 1.51 7.95 0.59 61.35
8 1.52 7.71 0.57 64.02
9 1.52 7.52 0.56 66.17
10 1.51 7.33 0.54 68.30
14 1.51 6.80 0.50 74.19
40 1.52 5.62 0.42 87.24
50 1.52 5.41 0.40 89.64
100 1.51 4.86 0.36 95.70
200 1.51 4.51 0.34 99.58
222 1.51 4.48 0.33 100.00
So urce: Cleary, S. and Co pp D. "Diversificatio n with Canadian Sto cks: Ho w M uch is Eno ugh?" Canadian Investment Review (Fall 1999), Table 1.

TOTAL RISK OF AN INDIVIDUAL
ASSET
EQUALS THE SUM OF MARKET AND UNIQUE RISK

Average Portfolio Risk
 This graph illustrates
that total risk of a stock
is made up of market
risk (that cannot be

diversified away
Diversifiable because it is a function
(unique) risk
of the economic
[8-19] ‘system’) and unique,
company-specific risk
Nondiversifiable that is eliminated from
(systematic) risk the portfolio through
t

diversification.
Number of Stocks in Portfolio
( i i r

[8-19] Total risk = Market (systematic) risk + Unique (non - systematic) risk

INTERNATIONAL DIVERSIFICATION
 Clearly, diversification adds value to a portfolio
by reducing risk while not reducing the return on
the portfolio significantly.

 Most of the benefits of diversification can be
achieved by investing in 40 – 50 different
‘positions’ (investments)
 However, if the investment universe is expanded
to include investments beyond the domestic
capital markets, additional risk reduction is
possible.
(See Figure 8 -12 found on the following slide.)

DIVERSIFICATION
INTERNATIONAL DIVERSIFICATION
8 - 12 FIGURE

100

80

60

40
ks r t necr e P

U.S. stocks
20
International stocks
11.7
0
i

0 10 20 30 40 50 60
Number of Stocks

SUMMARY AND CONCLUSIONS

In this chapter you have learned:
 How to measure different types of returns

 How to calculate the standard deviation and
interpret its meaning
 How to measure returns and risk of portfolios
and the importance of correlation in the
diversification process.
 How the efficient frontier is that set of
achievable portfolios that offer the highest rate
of return for a given level of risk.

CONCEPT REVIEW
QUESTIONS

CONCEPT REVIEW QUESTION 1
EX ANTE AND EX POST RETURNS

What is the difference between ex ante and ex
post returns?


COPYRIGHT
Copyright © 2007 John Wiley & Sons
Canada, Ltd. All rights reserved.
Reproduction or translation of this work
beyond that permitted by Access
Copyright (the Canadian copyright
licensing agency) is unlawful. Requests
for further information should be
addressed to the Permissions
Department, John Wiley & Sons
Canada, Ltd. The purchaser may make
back-up copies for his or her own use
only and not for distribution or resale.
The author and the publisher assume no
responsibility for errors, omissions, or
damages caused by the use of these files
or programs or from the use of the
information contained herein.


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