Risk, return, and portfolio theory

RRISKISK & R & RETURNETURN

The concept and measurement of Return:

Realized and Expected return. Ex-ante and ex-post returns

The concept of Risk: Sources and types of risk. Measurement of risk :

Range, Std Deviation and Co-Efficient of Variation.

Risk-return trade-offRisk-return trade-off

04

/11

/23

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Risk, Return &Portfolio Risk, Return &Portfolio TheoryTheory

Tu

esd

ay, A

pril 1

1, 2

02

3

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

INTRODUCTION TO RISK AND RETURN

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

INTRODUCTION TO RISK AND RETURNRisk and return are the two most

important attributes of an investment.

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

Risk isn’t just the potential loss of return, it is the potential loss of the entire investment itself (loss of both principal and interest).

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Return %

RF

Risk

Risk Premium

Real Return

Expected Inflation Rate

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:

Risk, Return and Portfolio Theory04/11/23

Risk, Return and Portfolio Theory

04/11/23

MEASURING RETURNSRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

MANY DIFFERENT MATHEMATICAL DEFINITIONS OF "RETURNS"...

04

/11

/23

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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.

MANY DIFFERENT MATHEMATICAL DEFINITIONS OF "RETURNS"...

TYPE 1: PERIOD-BY-PERIOD RETURNS . . .

"PERIODIC" RETURNSSIMPLE "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.

TYPE 2: MULTIPERIOD RETURN 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.

TYPE 2: MULTIPERIOD RETURN 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.

ADVANTAGES OF PERIOD-BY-PERIOD (TIME-WEIGHTED) RETURNS (CONT’D)

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

ADVANTAGES OF PERIOD-BY-PERIOD (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).

ADVANTAGES OF MULTI-PERIOD 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).

ADVANTAGES OF MULTI-PERIOD 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 YEARS BETWEEN TIME "t-1" AND "t". )Yr(* V = CF+V Y)V( - )CF + V( = r t1-ttt1-tttt EXPLNLN

EXAMPLE:

01/01/98 V = 100003/31/99 V = 1100 & CF = 50 PER ANNUMr = (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 = $0.95 in "1991$“

If rt = Nominal Total Return, year tit = Inflation, year tRt = 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 = $100,0001992 NET RENT = $10,0001992 PROPERTY VALUE = $101,0001992 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 . . .

0%

25%

50%

75%

100%

-10% -5% 0% 5% 10% 15% 20% 25% 30%

Returns

Pro

bab

ility

A

B

C

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% PROBABILITYOR:-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!

SO THE CAPITAL MARKETS COMPENSATE THEM BY PROVIDING HIGHER RETURNS (EX ANTE) ON MORE RISKY ASSETS . . .

Risk

ExpectedReturn

rf

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 = $100,000POSSIBLE 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 = $100,000POSSIBLE 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 BuildingKnown as of end

1998 Value = $100,000 Expected value end

99 = $100,000 Expected net rent

99 = $11,000 Ex ante risk

premium = 11% - 7% = 4%

B: Bowling AlleyKnown as of end

1998 Value = $100,000 Expected value end

99 = $100,000 Expected net rent

99 = $15,000 Ex ante risk

premium = 15% - 7% = 8%

EXAMPLE (CONT’D) – SUPPOSE THE FOLLOWING OCCURRED IN 1999

A: Office BuildingNot known until end

1999 End 99 Value =

$110,000 99 net rent =

$11,000 99 Ex post risk

premium = 21% - 7% = 14%

(“The Dice Rolled Favorably”)

B: Bowling AlleyNot known until end

1999 End 99 Value =

$80,000 99 net rent =

$15,000 99 Ex post risk

premium = -5% - 7% = -12%

(“The Dice Rolled Unfavorably”)

SUMMARY:

THREE USEFUL WAYS TO BREAK TOTAL RETURN INTO TWO COMPONENTS...1) TOTAL RETURN = CURRENT YIELD

+ GROWTHr = 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.


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.


ii

i

CFwBegVal

CFBegValEndValr

where:= sum of all net cash flows occurring in period t,

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.)

iCF

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)

%00.111)0087387.1(/

%87387.0//1

100

/1

0

/1

101000

12

12

11

2

yrIRR

moIRRmoIRRmoIRRmoIRR j

j

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 FORMULAFORMULA INCLUDES A TIME-WEIGHTED 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...

[Note: (1/3)NOI = (2/3)(1/3)NOI+(1/3)(1/3)NOI+(0)(1/3)NOI ]

NOICIPSBegVal

NOICIPSBegValEndValrNPI 3121

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.

ARITHMETIC VS. GEOMETRIC 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.

ARITHMETIC VS. GEOMETRIC MEAN (CONT’D)

The two are more similar:- The less volatility in returns

across time- The more frequent the

return intervalNote: "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

Year:

End of year asset value:

HPR:

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%


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%


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

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").

)IRR + (1

CF + . . . +

)IRR + (1

CF + IRR) + (1

CF + CF = 0N

N2

210

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)

where PV = -CF0, the initial cash "deposit" 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 more capital is in the investment will be weighted more heavily in determining the IRR.

NNNN CFIRRCFIRRCFIRRPV

111 11

1

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 INVESTMENT WAS MADE).

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 INVESTMENT (ON SAME SIDE OF "=" SIGN).

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).


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 CAPITAL VALUE.

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 PERIODIC RETURNS): IRRt,t+N=rt,t+N=yt,t+N+gt,t+N.


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 . . .

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

1996 1997 1998 1999

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

PERIODIC RETURN +10.00% -10.00% +10.00%

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

IRR = IRR(-2000,1100,0,1089) = 4.68%

END OF YEAR: 1996 1997 1998 1999

UNITS BOUGHT 2

UNITS SOLD 1 1

CASH FLOW -$2000 +$1100 0 $1089

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

IRR = IRR(-1000,-1100,990,1089) = -0.50%

END OF YEAR: 1996 1997 1998 1999

UNITS BOUGHT 1 1

UNITS SOLD 1 1

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 RETURNSINTRODUCTION

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

MEASURING RETURNSINTRODUCTION

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

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.

Yield loss)(or Gain Capital Yield Dividend / Income

0

1

g

P

Dkc


MEASURING RETURNSINCOME 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.

Where CF1 = the expected cash flow to be received

P0 = the purchase price

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

yield Income 0

1

P

CF[8-1]

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)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

EX POST VERSUS EX ANTE RETURNSMARKET INCOME YIELDS

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

8-1 FIGURE

Insert Figure 8 - 1

MEASURING RETURNSCOMMON SHARE AND LONG CANADA BOND YIELD GAP

Average Yield Gap (%)

1950s 0.821960s 2.351970s 4.541980s 8.141990s 5.512000s 3.55Overall 4.58

Table 8-1 Average Yield Gap

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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.

MEASURING RETURNSDOLLAR 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!)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

MEASURING RETURNSCONVERTING 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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

%808.$25

$25-$27 return (loss)gain Capital

0

01

P

PP[8-2]

MEASURING RETURNSTOTAL PERCENTAGE RETURN

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

%1212.008.004.025$

25$27$

25$

00.1$

yield loss)(or gain Capital yield Income return Total

0

01

0

1

0

011

P

PP

P

CF

P

PPCF

[8-3]

MEASURING RETURNSTOTAL PERCENTAGE RETURN – GENERAL FORMULA

The general formula for holding period return is:

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

yield loss)(or gain Capital yield Income return Total

0

01

0

1

0

011

P

PP

P

CF

P

PPCF

[8-3]

MEASURING AVERAGE RETURNSEX 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: Arithmetic average Geometric mean

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

MEASURING AVERAGE RETURNSARITHMETIC AVERAGE

Where:ri = the individual returnsn = the total number of observations

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

observations

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

(AM) Average Arithmetic 1

n

rn

ii

[8-4]

MEASURING AVERAGE RETURNSGEOMETRIC MEAN

Measures the average or compound growth rate over multiple periods.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

11111(GM)Mean Geometric 1

321 -)]r)...(r)(r)(r [( nn[8-5]

MEASURING AVERAGE RETURNSGEOMETRIC 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)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

MEASURING AVERAGE RETURNSAVERAGE INVESTMENT RETURNS AND STANDARD DEVIATIONS

Annual Arithmetic

Average (%)

Annual Geometric Mean (%)

Standard Deviation of Annual Returns

(%)

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

Source: Data are from the Canadian Institute of Actuaries

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

ESTIMATING EXPECTED RETURNSESTIMATING EX ANTE (FORECAST) RETURNS

The general formula

Where:ER = the expected return on an investmentRi = the estimated return in scenario i

Probi = the probability of state i occurring

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

)Prob((ER)Return Expected 1

i

n

iir[8-6]

ESTIMATING EXPECTED RETURNSESTIMATING EX ANTE (FORECAST) RETURNS

Example:

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

State of the EconomyProbability of Occurrence

Possible Returns on

Stock A in that State

Economic Expansion 25.0% 30%Normal Economy 50.0% 12%Recession 25.0% -25%


ESTIMATING EXPECTED RETURNSESTIMATING 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)

State of the EconomyProbability of Occurrence

Possible Returns on

Stock A in that State

Weighted Possible

Returns on 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 EXPECTED RETURNSESTIMATING 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.

7.25%

)25.0(-25%0.5)(12% .25)0(30%

)Prob(r)Prob(r )Prob(r

)Prob((ER)Return Expected

332211

1i

n

iir


MEASURING RISKRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

RISK

Probability of incurring harm For investors, risk is the probability of earning an

inadequate return. If investors require a 10% rate of return on a given

investment, then any return less than 10% is considered harmful.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

RISKILLUSTRATED

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm

The range of total possible returns on the stock A runs from -30% to more than +40%. If the required return on the stock is 10%, then those outcomes less than 10% represent risk to the investor.

A

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

DIFFERENCES IN LEVELS OF RISKILLUSTRATED

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

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

A is a much riskier investment than BB

A

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)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

MEASURING RISKANNUAL RETURNS BY ASSET CLASS, 1938 - 2005

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

FIGURE 8-2

REFINING THE MEASUREMENT OF RISKSTANDARD 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)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

MEASURING RISKEX POST STANDARD DEVIATION

nsobservatio ofnumber the

year in return the

return average the

deviation standard the

:

_

n

ir

r

Where

i


1

)(post Ex 1

2_

n

rrn

ii

[8-7]

MEASURING RISKEXAMPLE USING THE EX POST STANDARD DEVIATION

ProblemEstimate 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

Step 2 – Calculate the Standard Deviation

%0.85

40

5

10812-2410 (AM) Average Arithmetic 1

n

rn

ii


%88.121664

664

4

404002564

4

2020162

15

)814()88()812()824(8)-(10

1

)(post Ex

22222

222221

2_

n

rrn

ii

EX POST RISKSTABILITY 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 UNCERTAINTYEQUITIES VERSUS BONDS

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

FIGURE 8-3

MEASURING RISKEX ANTE STANDARD DEVIATION

A Scenario-Based Estimate of Risk

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

)()(Prob anteEx 2

1i ii

n

i

ERr

[8-8]

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

State of the Economy Probability

Possible Returns on Security A

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


GIVEN INFORMATION INCLUDES:

- Possible returns on the investment for different discrete states

- Associated probabilities for those possible returns

SCENARIO-BASED ESTIMATE OF RISKEX ANTE STANDARD DEVIATION – SPREADSHEET APPROACH

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

SCENARIO-BASED ESTIMATE OF RISKFIRST STEP – CALCULATE THE EXPECTED RETURN



Weighted Possible 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%


Determined by multiplying the probability times the

possible return.

Expected return equals the sum of the weighted possible returns.

SCENARIO-BASED ESTIMATE OF RISKSECOND STEP – MEASURE THE WEIGHTED AND SQUARED DEVIATIONS




Deviation of Possible

Return from Expected

Squared Deviations

Weighted and

Squared Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531

Expected Return = 10.3% Variance = 0.0420

Standard Deviation = 20.50%




Deviation of Possible

Return from Expected

Squared Deviations

Weighted and

Squared Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070Economic 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 from the mean.The sum of the weighted and square deviations

is the variance in percent squared terms.The standard deviation is the square root

of the variance (in percent terms).

First calculate the deviation of possible returns from the expected.

Now multiply the square deviations by their probability of occurrence.

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

%5.20205.

0420.

)06126(.25.)00141(.5.)10401(.25.

)8.24(25.)8.3(5.)3.32(25.

)3.1035(25.)3.1014(5.)3.1022(25.

)()()(

)()(Prob anteEx

222

222

2331

2222

2111

2

1i

ERrPERrPERrP

ERr ii

n

i




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%

MODERN PORTFOLIO THEORYRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

DIVERSIFICATION

Diversification has two faces:

1. Diversification results in an overall reduction in 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.)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

EXPECTED RETURN OF A PORTFOLIOMODERN 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:

The portfolio weight of a particular security is the percentage of the portfolio’s total value that is invested in that security.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

)( n

1i

iip ERwER[8-9]

EXPECTED RETURN OF A PORTFOLIOEXAMPLE

%288.8%284.4%004.4

) %6(.714)%14(.286)( n

1i

iip ERwER

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Portfolio value = $2,000 + $5,000 = $7,000rA = 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%

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry


Example 1:

Assume ERA = 8% and ERB = 10%

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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


Ex

pe

cte

d R

etu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

8 - 4 FIGURE

ERA=8%

ERB= 10%



8 - 4 FIGURE

Ex

pe

cte

d R

etu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

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)



Ex

pe

cte

d R

etu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

8 - 4 FIGURE

ERA=8%

ERB= 10%

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



Ex

pe

cte

d R

etu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

8 - 4 FIGURE

ERA=8%

ERB= 10%

The expected return on the portfolio if 100% is invested in Asset A is 8%.

%8%)10)(0(%)8)(0.1( BBAAp ERwERwER



8 - 4 FIGURE

Ex

pe

cte

d R

etu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

The expected return on the portfolio if 100% is invested in Asset B is 10%.

%10%)10)(0.1(%)8)(0( BBAAp ERwERwER



8 - 4 FIGURE

Ex

pe

cte

d R

etu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

The expected return on the portfolio if 50% is invested in Asset A and 50% in B is 9%.

%9%5%4

%)10)(5.0(%)8)(5.0(

BBAAp ERwERwER


Example 1:

Assume ERA = 14% and ERB = 6%

(See the following 2 slides )

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

A graph of this relationship is found on the following slide.

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

Weight of Asset A

Weight of Asset B

Expected Return on the

Portfolio0.0% 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%50.0% 50.0% 10.0%60.0% 40.0% 10.8%70.0% 30.0% 11.6%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%

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Range of Portfolio Returns

0.00%2.00%4.00%6.00%8.00%

10.00%12.00%14.00%16.00%

Weight Invested in Asset A

Ex

pe

cte

d R

etu

rn o

n T

wo

A

ss

et

Po

rtfo

lio

EXPECTED PORTFOLIO RETURNSEXAMPLE OF A THREE ASSET PORTFOLIO

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

K. Hartviksen

Relative Weight

Expected Return

Weighted Return

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

RISK IN PORTFOLIOSRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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

))()((2)()()()( ,2222

BABABBAAp COVwwww

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

[8-11]

Risk of Asset A adjusted for weight

in the portfolio

Risk of Asset B adjusted for weight

in the portfolio

Factor to take into account comovement of returns. This factor

can be negative.

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

))()()()((2)()()()( ,2222

BABABABBAAp wwww

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

[8-15]

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:

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

BABABABBAAp wwww ,2222 2

RISK OF A THREE-ASSET PORTFOLIO

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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

B C

ρa,b

ρb,c

ρa,c

CACACACBCBCBBABABACCBBAAp wwwwwwwww ,,,222222 222

RISK OF A FOUR-ASSET PORTFOLIO

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

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

C

B D

ρa,b ρa,d

ρb,c ρc,d

ρa,c

ρb,d

COVARIANCE

A statistical measure of the correlation of the fluctuations of the annual rates of return of different investments.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

)-)((Prob _

,1

_

,i BiB

n

iiiAAB kkkkCOV

[8-12]

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

BA

ABAB

COV

[8-13]

COVARIANCE AND CORRELATION COEFFICIENT

Solving for covariance given the correlation coefficient and standard deviation of the two assets:

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

BAABABCOV [8-14]

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

AFFECT OF PERFECTLY NEGATIVELY CORRELATED RETURNSELIMINATION OF PORTFOLIO RISK

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Time 0 1 2

If returns of A and B are perfectly negatively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would be no variabilityof the portfolios returns over time.

Returns on Stock A

Returns on Stock B

Returns on Portfolio

Returns%

10%

5%

15%

20%

EXAMPLE OF PERFECTLY POSITIVELY CORRELATED RETURNSNO DIVERSIFICATION OF PORTFOLIO RISK

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Time 0 1 2

If returns of A and B are perfectly positively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be risky. There would be no diversification (reduction of portfolio risk).

Returns%

10%

5%

15%

20%

Returns on Stock A

Returns on Stock B

Returns on Portfolio

AFFECT OF PERFECTLY NEGATIVELY CORRELATED RETURNSELIMINATION OF PORTFOLIO RISK

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Time 0 1 2

If returns of A and B are perfectly negatively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would be no variabilityof the portfolios returns over time.

Returns%

10%

Returns on Portfolio5%

15%

20%

Returns on Stock B

Returns on Stock A

AFFECT OF PERFECTLY NEGATIVELY CORRELATED RETURNSNUMERICAL EXAMPLE

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

%10%5.2%5.7

) %5(.5)%15(.5)( n

1i

iip ERwER

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

YearReturn on

Asset AReturn on

Asset B

Expected Return on the

Portfolioxx07 5.0% 15.0% 10.0%xx08 10.0% 10.0% 10.0%xx09 15.0% 5.0% 10.0%

Perfectly Negatively Correlated Returns over time

%10%5.7%5.2

) %15(.5)%5(.5)( n

1i

iip ERwER

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

EXAMPLE OF PORTFOLIO COMBINATIONS AND CORRELATION

AssetExpected

ReturnStandard Deviation

Correlation Coefficient

A 5.0% 15.0% 1B 14.0% 40.0%

Weight of A Weight of BExpected


100.00% 0.00% 5.00% 15.0%90.00% 10.00% 5.90% 17.5%80.00% 20.00% 6.80% 20.0%70.00% 30.00% 7.70% 22.5%60.00% 40.00% 8.60% 25.0%50.00% 50.00% 9.50% 27.5%40.00% 60.00% 10.40% 30.0%30.00% 70.00% 11.30% 32.5%20.00% 80.00% 12.20% 35.0%10.00% 90.00% 13.10% 37.5%0.00% 100.00% 14.00% 40.0%

Portfolio Components Portfolio Characteristics

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Perfect Positive

Correlation – no

diversification

Both portfolio returns and risk are bounded by the range set by the constituent assets when ρ=+1


AssetExpected



A 5.0% 15.0% 0.5B 14.0% 40.0%



100.00% 0.00% 5.00% 15.0%90.00% 10.00% 5.90% 15.9%80.00% 20.00% 6.80% 17.4%70.00% 30.00% 7.70% 19.5%60.00% 40.00% 8.60% 21.9%50.00% 50.00% 9.50% 24.6%40.00% 60.00% 10.40% 27.5%30.00% 70.00% 11.30% 30.5%20.00% 80.00% 12.20% 33.6%10.00% 90.00% 13.10% 36.8%0.00% 100.00% 14.00% 40.0%


Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Positive Correlation –

weak diversification

potential

When ρ=+0.5 these portfolio combinations have lower risk – expected portfolio return is unaffected.


AssetExpected



A 5.0% 15.0% 0B 14.0% 40.0%



100.00% 0.00% 5.00% 15.0%90.00% 10.00% 5.90% 14.1%80.00% 20.00% 6.80% 14.4%70.00% 30.00% 7.70% 15.9%60.00% 40.00% 8.60% 18.4%50.00% 50.00% 9.50% 21.4%40.00% 60.00% 10.40% 24.7%30.00% 70.00% 11.30% 28.4%20.00% 80.00% 12.20% 32.1%10.00% 90.00% 13.10% 36.0%0.00% 100.00% 14.00% 40.0%


Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

No Correlation –

some diversification

potential

Portfolio risk is lower than the risk of either asset A or B.


AssetExpected



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



100.00% 0.00% 5.00% 15.0%90.00% 10.00% 5.90% 12.0%80.00% 20.00% 6.80% 10.6%70.00% 30.00% 7.70% 11.3%60.00% 40.00% 8.60% 13.9%50.00% 50.00% 9.50% 17.5%40.00% 60.00% 10.40% 21.6%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%


Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Negative Correlation –

greater diversification

potential

Portfolio risk for more combinations is lower than the risk of either asset


AssetExpected



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



100.00% 0.00% 5.00% 15.0%90.00% 10.00% 5.90% 9.5%80.00% 20.00% 6.80% 4.0%70.00% 30.00% 7.70% 1.5%60.00% 40.00% 8.60% 7.0%50.00% 50.00% 9.50% 12.5%40.00% 60.00% 10.40% 18.0%30.00% 70.00% 11.30% 23.5%20.00% 80.00% 12.20% 29.0%10.00% 90.00% 13.10% 34.5%0.00% 100.00% 14.00% 40.0%


Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Perfect Negative

Correlation – greatest

diversification potential

Risk of the portfolio is almost eliminated at 70% invested in asset A

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ryDiversification of a Two Asset Portfolio

Demonstrated Graphically

The Effect of Correlation on Portfolio Risk:The Two-Asset Case

Expected Return

Standard Deviation

0%

0% 10%

4%

8%

20% 30% 40%

12%

B

AB= +1

A

AB = 0

AB = -0.5

AB = -1

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

EXPECTED PORTFOLIO RETURNIMPACT OF THE CORRELATION COEFFICIENT


8 - 7 FIGURE

15

10

5

0

Sta

nd

ard

De

via

tio

n (

%)

of

Po

rtfo

lio

Re

turn

s

Correlation Coefficient (ρ)

-1 -0.5 0 0.5 1

ZERO RISK PORTFOLIO

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

Becomes:

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

BAp ww )1( [8-16]

))()()()((2)()()()( ,2222

BABABABBAAp wwww [8-15]

AN EXERCISE TO PRODUCE THE EFFICIENT FRONTIER USING THREE ASSETSRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

AN EXERCISE USING T-BILLS, STOCKS AND BONDS

Base Data: Stocks T-bills BondsExpected Return(%) 12.73383 6.151702 7.0078723

Standard Deviation (%) 0.168 0.042 0.102

Correlation Coefficient Matrix:Stocks 1 -0.216 0.048T-bills -0.216 1 0.380Bonds 0.048 0.380 1

Portfolio Combinations:

Combination Stocks T-bills BondsExpected Return Variance

Standard Deviation

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%3 80.0% 20.0% 0.0% 11.4 0.0177 13.3%4 70.0% 30.0% 0.0% 10.8 0.0134 11.6%5 60.0% 40.0% 0.0% 10.1 0.0097 9.9%6 50.0% 50.0% 0.0% 9.4 0.0067 8.2%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%

Weights Portfolio

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

Historical averages for

returns and risk for three asset

classes

Historical correlation coefficients

between the asset classes

Portfolio characteristics for each combination of securities

Each achievable portfolio combination is plotted on expected return, risk (σ) space, found on the following slide.

ACHIEVABLE PORTFOLIOSRESULTS USING ONLY THREE ASSET CLASSES

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 5.0 10.0 15.0 20.0

Standard Deviation of the Portfolio (%)

Po

rtfo

lio E

xpec

ted

Ret

urn

(%

) Efficient Set

Minimum Variance

Portfolio

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

The plotted points are attainable portfolio

combinations.

The efficient set is that set of achievable portfolio

combinations that offer the highest rate of return for a

given level of risk. The solid blue line indicates the efficient

set.

ACHIEVABLE TWO-SECURITY PORTFOLIOSMODERN PORTFOLIO THEORY

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

8 - 9 FIGURE

Ex

pe

cte

d R

etu

rn %

Standard Deviation (%)

13

12

11

10

9

8

7

6

0 10 20 30 40 50 60

This line represents the set of portfolio combinations that are achievable by varying relative weights and using two non-correlated securities.

DOMINANCE

It is assumed that investors are rational, wealth-maximizing and risk averse.

If so, then some investment choices dominate others.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

INVESTMENT CHOICESTHE CONCEPT OF DOMINANCE ILLUSTRATED

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

A B

C

Return%

Risk

10%

5%

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

A dominates B because it offers the same return but for less risk.

A dominates C because it offers a higher return but for the same risk.

20%5%

EFFICIENT FRONTIERTHE TWO-ASSET PORTFOLIO COMBINATIONS

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

A is not attainable

B,E lie on the efficient frontier and are attainable

E is the minimum variance portfolio (lowest risk combination)

C, D are attainable but are dominated by superior portfolios that line on the line

above E

8 - 10 FIGURE

Ex

pe

cte

d R

etu

rn %


A

E

B

C

D

EFFICIENT FRONTIERTHE TWO-ASSET PORTFOLIO COMBINATIONS

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

8 - 10 FIGURE

Ex

pe

cte

d R

etu

rn %


A

E

B

C

D

Rational, risk averse investors will only want to hold portfolios such as B.

The actual choice will depend on her/his risk preferences.

DIVERSIFICATIONRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

DIVERSIFICATIONDOMESTIC DIVERSIFICATION

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

8 - 11 FIGURE

14

12

10

8

6

4

2

0

Sta

nd

ard

Dev

iati

on

(%

)

Number of Stocks in Portfolio

0 50 100 150 200 250 300

Average Portfolio RiskJanuary 1985 to December 1997

DIVERSIFICATIONDOMESTIC DIVERSIFICATION


Average Monthly Portfolio

Return (%)

Standard Deviation of Average

Monthly Portfolio Return (%)

Ratio of Portfolio Standard Deviation to

Standard Deviation of a Single Stock

Percentage of Total Achievable Risk Reduction

1 1.51 13.47 1.00 0.002 1.51 10.99 0.82 27.503 1.52 9.91 0.74 39.564 1.53 9.30 0.69 46.375 1.52 8.67 0.64 53.316 1.52 8.30 0.62 57.507 1.51 7.95 0.59 61.358 1.52 7.71 0.57 64.029 1.52 7.52 0.56 66.17

10 1.51 7.33 0.54 68.30

14 1.51 6.80 0.50 74.1940 1.52 5.62 0.42 87.2450 1.52 5.41 0.40 89.64100 1.51 4.86 0.36 95.70200 1.51 4.51 0.34 99.58222 1.51 4.48 0.33 100.00

Source: Cleary, S. and Copp D. "Diversification with Canadian Stocks: How Much is Enough?" Canadian Investment Review (Fall 1999), Table 1.

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

TOTAL RISK OF AN INDIVIDUAL ASSETEQUALS THE SUM OF MARKET AND UNIQUE RISK

This graph illustrates that total risk of a stock is made up of market risk (that cannot be diversified away because it is a function of the economic ‘system’) and unique, company-specific risk that is eliminated from the portfolio through diversification.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

[8-19]

Sta

nd

ard

Dev

iati

on

(%

)


Average Portfolio Risk

Diversifiable (unique) risk

Nondiversifiable (systematic) risk

risk )systematic-(non Uniquerisk c)(systematiMarket risk Total [8-19]

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.)

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

DIVERSIFICATIONINTERNATIONAL DIVERSIFICATION

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

8 - 12 FIGURE

100

80

60

40

20

0

Pe

rce

nt

ris

k

Number of Stocks

0 10 20 30 40 50 60

International stocks

U.S. stocks

11.7

SUMMARY AND CONCLUSIONS

In this chapter you have learned:How to measure different types of returnsHow to calculate the standard deviation

and interpret its meaningHow 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.

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

CONCEPT REVIEW QUESTIONSRisk, Return and Portfolio Theory

Tu

esd

ay, A

pril 1

1, 2

02

3R

isk, Re

turn

an

d P

ortfo

lio T

he

ory

CONCEPT REVIEW QUESTION 1EX ANTE AND EX POST RETURNS

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

Risk, R

etu

rn a

nd

Po

rtfolio

Th

eo

ry

COPYRIGHTCopyright © 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.


Risk, return, and portfolio theory

Economy & Finance

risk return trade

measurement of return

concept of risk

greater risk

measurement of risk

concept of investment

risk return101212

risk andreturnrisk