introduction to risk and return - financial markets and...

Motivations and ObjectivesRisky Asset Returns

Risk and HorizonHistorical Risk and Return

Introduction to Risk and ReturnFinancial Markets and Intermediaries

Paolo Vitale

LUISS University

[email protected]

Paolo Vitale Introduction to Risk and Return



Outline

Motivation and Objectives

Risky Asset Returns

Characterizing Asset Returns

Key Assumptions

Historical Risk and Return




Motivations and Objectives

In the next lectures we examine the following questions:

How do we measure risk?

How does the financial market determine the price of risk?




Motivations and Objectives (cont.ed)

The purpose of the following three Sections,

1 Introduction to Risk and Return,

2 Portfolio Theory

3 and Capital Asset Pricing Model

is to acquire the tools needed to deal with the risk-returntrade-off.

The final result will be a model that allows us to compute therequired return of any security and project given its risk chara-cteristics.

This model is the Capital Asset Pricing Model.




Review of Probability and StatisticsCharacterizing Asset Returns: An ExampleUsing Historical DataAsset Returns and Preferences

Risky Asset Returns

Asset returns over a given period are uncertain:

r =P1 − P0

P0where

P0 is the price of the asset at the beginning of the period,

P1 is the price at the end of the period - uncertain.

Here tildas (˜) refer to uncertain numbers.





Risky Asset Returns (cont.ed)

Return on an asset is a random variable, characterized by

? all the possible outcomes,

? and the probabilities attached to each outcome.

All these possible outcomes and attached probabilities canhowever be characterized by only a few statistical numbers.

The statistical numbers are: the mean, the variance, thestandard deviation, the covariance and the correlation.

Let us go once over their definitions and calculations.





Review of Probability and Statistics

Consider two random variables, x and y . Assume their jointdistribution is given in the following Table:

State 1 2 · · · nProbability p1 p2 · · · pn

Value of x x1 x2 · · · xn

Value of y y1 y2 · · · yn

wheren∑

i=1

pi = 1.





Moments

Definition (Mean)

The expected (or mean) value of a random variable:

E [x ] ≡ x =∑n

i=1 pi xi .

Definition (Variance)

The variance measures how much the realized outcome is likely todiffer from the expected value:

Var [x ] ≡ σ2x =

∑ni=1 pi

(xi − x

)2.

Definition (Standard Deviation)

The standard deviation is the square root of the variance:

StDev [x ] ≡ σx =√

Var [x ].





Moments (cont.ed)

Definition (Covariance)

The covariance measures the degree to which two random variablesvary together:

Cov [x , y ] ≡ σx ,y =∑n

i=1 pi

(xi − x

) (yi − y

).

Definition (Correlation)

The correlation is a standardized measure of covariation:

Corr [x , y ] ≡ ρx ,y =σx ,y

σx σy.





Moments (cont.ed)

Notice that

1 ρx ,y must lie between -1 and +1.

2 The two random variables are

Perfectly positively correlated if ρx,y = +1.

Perfectly negatively correlated if ρx,y = −1.

Uncorrelated if ρx,y = 0.





Characterizing Asset Returns: An Example

Let us go once over the calculations of the mean, the variance, thestandard deviation, the covariance and the correlation in a specificscenario.

Example: Let us assume that the monthly returns on IBM andExxon stock have the following characteristics

Outcome rj 1 2 3Probability pj 0.20 0.60 0.20

Return on IBM -9% 2.5% 8%Return on Exxon -7% 1.4% 7%





Characterizing Asset Returns: Mean

The expected values of the returns on IBM and Exxon are

E [rIBM] ≡ rIBM = p1 × r1,IBM + p2 × r2,IBM + p3 × r3,IBM

= 0.20× (−0.09) + 0.60× (0.025) + 0.20× (0.08)

= 0.013,

E [rExxon] ≡ rExxon = 0.20× (−0.07) + 0.60× (0.014) + 0.20× (0.07)

= 0.0084.





Characterizing Asset Returns: Variance and StandardDeviation

The variances of the returns on IBM and Exxon are:

Var [rIBM] ≡ σ2IBM = p1 × (r1,IBM − rIBM)2 +

p2 × (r2,IBM − rIBM)2 + p3 × (r3,IBM − rIBM)2

= 0.20× (−0.08− 0.013)2 + 0.60× (0.025− 0.013)2 +

0.20× (0.08− 0.013)2 = 0.0031,

Var [rExxon] ≡ σ2Exxon = 0.20× (−0.07− 0.0084)2 + 0.60× (0.014− 0.0084)2 +

0.20× (0.07− 0.0084)2 = 0.0020.

The standard deviations for IBM and Exxon are:

StD [rIBM] ≡ σIBM =√

0.0031 = 0.056,

StD [rExxon] ≡ σExxon =√

0020 = 0.045.





Characterizing Asset Returns: Covariance and Correlation

The covariance for IBM and Exxon is:

Cov [rIBM, rExxon] ≡ σIBM,Exxon =

p1 × (r1,IBM − rIBM)× (r1,Exxon − rExxon) +

p2 × (r2,IBM − rIBM)× (r2,Exxon − rExxon) +

p3 × (r3,IBM − rIBM)× (r3,Exxon − rExxon)

= 0.20× (−0.09− 0.013)× (−0.07− 0.0084) +

0.60× (0.025− 0.013)× (0.014− 0.0084) +

0.20× (0.08− 0.013)× (0.07− 0.0084) = 0.0025.

The correlation is hence equal to:

Corr [rIBM, rExxon] ≡ ρIBM,Exxon =σIBM,Exxon

σIBM × σExxon= 0.9921.





Characterizing Asset Returns: Historical Data

Clearly, to estimate the expected rate of return on a risky as-set we must

? consider all the possible outcomes

? and attach a probability to each of these possible outcomes.

This is a difficult task.

In practice, the best way to overcome this difficulty is to lookat historical prices over a certain sample period.

Idea: What happened historically is the best indication avail-able of what should happen in the future.





Characterizing Asset Returns: IBM

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Characterizing Asset Returns: Exxon

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NEWS & INFOHeadlinesFinancial BlogsCompany EventsMessage BoardsMarket Pulse NEW!

COMPANYProfileKey StatisticsSEC FilingsCompetitorsIndustryComponents

ANALYST COVERAGEAnalyst OpinionAnalyst EstimatesResearch ReportsStar Analysts

OWNERSHIPMajor HoldersInsider TransactionsInsider Roster

FINANCIALSIncome StatementBalance SheetCash Flow

Enter name(s) or symbol(s) GET CHART COMPARE EVENTS TECHNICAL INDICATORS CHART SETTINGS RESET

Last Trade: 79.82

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52wk Range: 55.94 - 88.23

Volume: 0

Avg Vol (3m): 17,550,600

Market Cap: 393.20B

P/E (ttm): 11.37

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Characterizing Asset Returns: Data for IBM and Exxon

Take the monthly stock returns on IBM and Exxon from January2006 to June 2011 (see spreadsheet IBM-Exxon-2006-2011.xls):

Date Open High Low Adj Close Return Open High Low Adj Close ReturnJune‐11 168.9 169.58 166.5 166.56 ‐0.01402948 83.55 83.65 78.33 79.82 ‐0.043728286May‐11 172.11 173.54 165.9 168.93 ‐0.005240843 88.1 88.13 79.42 83.47 ‐0.045839049April‐11 163.7 173 162.19 169.82 0.046076137 84.72 88 82.38 87.48 0.045661009March‐11 163.15 167.72 151.71 162.34 0.007321916 86.41 86.5 78.8 83.66 ‐0.016343327February‐11 162.11 166.25 159.03 161.16 0.003237052 81.14 88.23 81.04 85.05 0.065789474January‐11 147.21 164.35 146.64 160.64 0.10382739 73.72 80.82 73.64 79.8 0.103276649December‐10 143.61 147.5 143.51 145.53 0.037499109 70.38 73.69 70.38 72.33 0.051155355November‐10 143.64 147.53 141.18 140.27 ‐0.010580518 66.72 71.9 66.63 68.81 0.052784578October‐10 135.51 144 134.39 141.77 0.070527826 62.32 66.81 61.8 65.36 0.076061903September‐10 125.31 136.11 124.52 132.43 0.089420862 60.04 62.44 59.72 60.74 0.045438898August‐10 129.25 132.49 122.28 121.56 ‐0.036308863 60.64 62.99 58.05 58.1 ‐0.002575107July‐10 123.55 131.6 120.61 126.14 0.039901072 56.98 61.88 55.94 58.25 0.045780969June‐10 124.69 131.94 122.82 121.3 ‐0.014221861 60.38 64.5 56.92 55.7 ‐0.056092188May‐10 129.39 133.1 116 123.05 ‐0.02403236 68.11 68.22 58.46 59.01 ‐0.101826484April‐10 128.95 132.28 127.12 126.08 0.005823694 67.27 70 66.85 65.7 0.011858925March‐10 127.5 130.73 125.2 125.35 0.008609591 65.36 67.89 65.08 64.93 0.030471354February‐10 123.23 128.27 121.61 124.28 0.043580485 65.77 67.23 63.56 63.01 0.015471394January‐10 131.18 134.25 121.9 119.09 ‐0.065007459 68.72 70.6 64.02 62.05 ‐0.055124105

IBM EXXON

July‐06 77.54 78.53 72.73 71.05 0.007658488 61.8 67.94 61.63 60.96 0.104147799June‐06 79.89 80.87 76.06 70.51 ‐0.038587401 60.4 62.65 56.64 55.21 0.007297938May‐06 82.59 83.69 79.06 73.34 ‐0.026158545 63.4 64.77 59.15 54.81 ‐0.029567989April‐06 82.72 84.45 80.63 75.31 ‐0.001590879 61.36 65 60.43 56.48 0.036330275March‐06 80.2 84.99 79.51 75.43 0.027796703 59.59 61.92 58.44 54.5 0.025206922February‐06 80.9 82.24 78.93 73.39 ‐0.010516381 62.77 63.08 58.6 53.16 ‐0.048845947January‐06 82.45 85.03 80.21 74.17 ‐0.010934791 56.42 63.96 56.42 55.89 0.117129722

From the realized returns of IBM and Exxon stocks we can deriveestimates of their mean, variance and covariance.





Basic Statistics: Sample Mean and Variance

In fact:

Let ri ,t denote the realized return on asset i , in period t.

Suppose we observe the realized return ri ,t over the periods1, 2, · · · , t, · · · ,T .

Then, we can define the sample mean and sample variance.

Definition (Sample Mean and Variance)

The sample mean and sample variance of the return ri are

ˆri =1

T

T∑t=1

ri ,t and σ2i =

1

T − 1

T∑t=1

(ri ,t − ˆri

)2.





Basic Statistics: Sample Covariance

In addition:

Let rj ,t denote the realized return on a second asset j , in pe-riod t.

Suppose we also observe the realized return rj ,t over the pe-riods 1, 2, · · · , t, · · · ,T .

Then, we can define the sample covariance.

Definition (Sample Covariance)

The sample covariance of the returns ri and rj is

σi ,j =1

T − 1

T∑t=1

(ri ,t − ˆri

)(rj ,t − ˆrj

).





Basic Statistics for IBM and Exxon

Given these definitions, we can calculate the sample means,variances and covariance for the returns on IBM and Exxonstocks, using the information contained in the spreadsheetIBM-Exxon-2006-2011.xls.

These statistics are reported in the following Tables:

Mean ReturnStock

IBM 0.01375Exxon 0.00845

Variance/CovarianceStock IBM Exxon

IBM 0.0031364 0.0006717Exxon 0.0006717 0.0027495





Characterizing Asset Returns: The Use of Historical Data

We have seen how by looking at historical prices over a cer-tain sample period we can estimate the expected rate of re-turn on a risky asset.

It is however important to notice that doing so implicitly assu-mes that

? the realized returns in the sample constitute all the possibleoutcomes and that

? they each have the same probability of occurring in the future.





Key Assumptions on Investors’ Preferences

1 Higher mean in return is preferred:

r ≡ E [r ].

2 Higher standard deviation in return is disliked:

σ ≡√

E [r − r ].

3 Investors care only about mean and standard deviation (orvariance).





Key Assumptions on Investors’ Preferences (cont.ed)

Under the above Assumptions 1-3, standard deviation (StD)gives a measure of risk. Sometimes, risk is also measured us-ing variance (StD2). Clearly, this is equivalent.

The standard deviation as a risk measure is a reasonable sim-plification. However, other elements, such as asymmetries inthe distributions of returns or in the investors’ preferences,may also matter.




The IID AssumptionThe Implications of the IID Assumption

Risk and Horizon

So far, we considered return and risk over a fixed horizon.However, in many cases, we need to know:

? How do risk and return vary with horizon?

? How do risk and return change over time?

To answer these questions, we need to know how successiveasset returns are related. The following IID assumption is agreat simplification.

Definition (IID Returns)

Asset returns are IID when successive returns are independent andidentically distributed.





The IID Assumption: An Example

1 Prices can go “up” by 5% or “down” by 2.5% at each node.

2 Probabilities of “up” or “down” are the same at each node.





The IID Assumption: An Example (cont.ed)

For the above binomial price process:

Successive returns are IID, independent and identically dist-ributed.

If “up” and “down” are equally likely, the expected returnover one period is

(5%− 2.5%)/2 = 1.25%.

The variance of the return on one-period is

(5%−1.25%)2×0.50 + (−2.5%−1.25%)2×0.50 = (0.0375)2.

The variance of the return over t periods is

(0.0375)2 × t.





The Implications of the IID Assumption

1 Returns are serially uncorrelated.

2 No predictable trends, cycles or patterns in returns exist

3 Risk (measured by variance) accumulates linearly over time:

? Var [r1 + r2 + · · · + rt ] = t × Var [r1].

? The annual variance is 12 times the monthly variance.

? The annual StD is√

12 times the monthly StD.





The Implications of the IID Assumption

Advantages of the IID assumption:

1 The return’s distribution on a particular horizon providessufficient information on returns for all horizons.

2 The return’s distribution is easy to estimate from past returns.

3 The IID assumption is consistent with informationally efficientfinancial markets.

Weaknesses of the IID assumption:

1 Returns may be serially correlated.

2 Risk may not accumulate linearly over time.




Historical Risk and Return

Three central facts appear from history of US financial markets:

1. Returns on riskier assets have been higher on averagethan returns on safer assets.

Average Annual Total Returns from 1926 to 1996 (Real)

Asset Mean StD

T-bills 0.7% 4.2%Long Term T-bonds 2.4% 10.5%Long Term Corp. Bonds 2.9% 10.0%Large Stock 9.4% 20.4%Small Stock 14.1% 33.5%




Historical Risk and Return (cont.ed)





2. Returns on risky assets can be highly correlated to eachother.

Cross Correlations of Annual Real Returns (1926 to 1996)

T-bills T-bonds C-bonds L stock S stock

T-bills 1.00 0.24 0.22 - 0.04 - 0.09Long Term T-bonds 1.00 0.94 0.18 0.03Long Term C-bonds 1.00 0.25 0.11Large stock 1.00 0.81Small stock 1.00





3. Returns on risky assets are serially uncorrelated.

Serial Correlations of Annual Real Returns (1926 to 1996)

Asset Serial Correlation

T-bills 0.66Long Term T-bonds 0.07Long Term Corp. bonds 0.21Large stock - 0.02Small stock 0.06


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