Learning about Return and Risk from The Historical Record Chapter 5 1 Bodi Kane Marcus Ch 5.

Bodi Kane Marcus Ch 5

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Learning about Return and Risk from The Historical Record

Chapter 5


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Determinants of the level of Interests

•Real and Nominal rates of Interest•The equilibrium Real Rate of Interest•The equilibrium Nominal Rate of Interest•Taxes and Real Rate of Interest

more…

Real vs. Nominal Rates

Fisher effect: Approximationnominal rate = real rate + inflation premium

R = r + i or r = R - iExample r = 3%, i = 6%

R = 9% = 3% + 6% or 3% = 9% - 6%Fisher effect: Exact

r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)

Empirical Relationship:Inflation and interest rates move closely together

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Factors Influencing Rates

•Supply▫Households

•Demand▫Businesses

•Government’s Net Supply and/or Demand▫Federal Reserve Actions

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The equilibrium Real Rate of Interest

Q0 Q1

r0

r1

Funds

Inte

rest

R

ate

s

Supply

Demand

5Pergeseran kurva Demand ke kanan dapat terjadi karena pemerintah menerapkan budget deficit permintaan akan uang meningkat interest rate naik


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Risk and Risks Premium

•Holding Period Returns•Expected Return and Standard Deviation•Excess Returns and Risk Premiums

more……


Rates of Return: Single Period

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PDPPHPR

0

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HPR = Holding Period Return

P0 = Beginning price

P1 = Ending price

D1 = Dividend during period one

Holding Period Returns


Rates of Return: Single Period Example

Ending Price ($) = 48Beginning Price ($) =

40Dividend ($) = 2

HPR = (48 - 40 + 2 )/ (40) = 25%8


Characteristics of Probability Distributions

1) Mean: most likely value2) Variance or standard deviation3) Skewness

* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

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Characteristics of Probability Distributions

1) Mean:The simple mathematical average of a set of two or more numbers

2) Variance : A measure of the dispersion of a set of data points around their mean value. ▫Variance is a mathematical expectation of the

average squared deviations from the mean.

3) Skewness : an asymmetry in the distribution of

the data values


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The Coefficient of Variation (CV)•A statistical measure of the dispersion of data

points in a data series around the mean. It is calculated as follows:

The coefficient of variation represents the ratio of the standard deviation to the mean.

In the investing world, CV determine how much volatility (risk) in comparison to the amount of return you can expect from your investment.

Source: Investopedia


Symmetric distributionr

s.d. s.d.

Normal Distribution

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Subjective returns

p(s) = probability of a stater(s) = return if a state occurs1 to s states

E(r) = p(s) r(s)s

Measuring Mean: Scenario or Subjective Returns

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State Prob. of State r in State1 .1 -.052 .2 .053 .4 .154 .2 .255 .1 .35

E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)E(r) = .15

Numerical Example: Subjective or Scenario Distributions

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Standard deviation = [variance]1/2

Measuring Variance or Dispersion of Returns

Subjective or Scenario

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Variance = s

p(s) [rs - E(r)]2

Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]Var= .01199

S.D.= [ .01199] 1/2 = .1095

Using Our Example:


Annual Holding Period Returns

Geom. Arith. Stan.Series Mean% Mean% Dev.%

Large Stock10.5 12.5 20.4Small Stock12.6 19.0 40.4LongT Gov 5.0 5.3 8.0

T-Bills 3.7 3.8 3.3

Inflation 3.1 3.2 4.5 16


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Expected Return and Standard Deviation

•Spreadsheet 5.1

Purchase Price 100$ T-Bill Rate 6%

ProbabilityEnding Price($)

Dividends HPR p* HPR Dev2 p*Dev2

Excess Returns

Boom 0.3 129.5 4.5 0.34 0.10 0.04 0.012 0.28Normal growth 0.5 110 4 0.14 0.07 0 0 0.08Recession 0.2 80.5 3.5 -0.16 -0.03 0.09 0.018 -0.22Expected value (mean) 0.14Standard Deviations 0.1732


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Teknis Penghitungan

• Expected Value▫ Jumlahkan hasil perkalian probabilitas dengan HPR

• Standard Deviation of HPR▫ Deviasi= HPR dikurangi mean▫ Kuadratkan Deviasi▫ Kalikan probabilitas dg Dev2

▫ Jumlahkan (p* Dev2 ), kemudian pangkat-kan (0.5) atau diakar• Excess Return

▫ HPR dikurangi RFR• Squared Deviations (Dev2 ) Excess Return

▫ Excess Return dikurangi risk premium, kemudian dikuadratkan

• Standard Deviation of Excess Return▫ Kalikan probabilitas dg Dev2 Excess Return ▫ Jumlahkan (p* Dev2 ), kemudian pangkat-kan (0.5) atau diakar


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Expected Return and Standard Deviation

•Problem 5-7 page 151

ProbabilityEnding Price($)

HPR (incl Dividends) p* HPR Dev2

Boom 0.35 140 44.50% 0.16 0.093025Normal growth 0.3 110 14% 0.04 0Recession 0.35 80 -16.50% -0.06 0.093025Expected value (mean) 0.14


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Excess Returns and Risk Premiums

• Excess Returns : Returns in excess of the risk-free rate or in excess of a market measure, such as an index fund. ▫ When you have excess returns you are making more

money than if you put your money into an index fund like the Dow Jones Industrial Average (DJIA).

• Risk Premiums : The return in excess of the risk-free rate of return that an investment is expected to yield. ▫An asset's risk premium is a form of compensation for

investors who tolerate the extra risk - compared to that of a risk-free asset - in a given investment.



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Time Series Analysis of Past Rates of Return

•Time Series versus Scenario Analysis•Expected Returns and the arithmetic

Average•The Geometric (Time Weighted) Average

Return•The Reward to Volatility (Sharpe) Ratio


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Time Series Analysis of Past Rates of Return• Time Series versus Scenario Analysis• Time Series Analysis

▫useful to see how a given asset, security or economic variable changes over time or how it changes compared to other variables over the same time period

• Scenario Analysis: The process of estimating the expected value of a portfolio after a given period of time, assuming specific changes in the values of the portfolio's securities or key factors that would affect security values, such as changes in the interest rate.



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• The arithmetic Average: A mathematical representation of the typical value of a series of numbers, computed as the sum of all the numbers in the series divided by the count of all numbers in the series.



Day 1 2 3 4 5 Sum

Closing Price

$14.50 $14.80 $15.20 $14.00 $15.50 $74.00

the arithmetic mean of a stock's closing price = $ 74.00 / 5 = $14.80.

The arithmetic mean of a stock's closing price = $ 74.00 / 5 = $14.80.


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• The Geometric (Time Weighted) Average Return: The average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio.

• The Geometric = {(1+ r1)*(1+r2)*…*(1+rn)} 1/n -1

Average Return

more……

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Illustration Geometric Mean

Year Return

1 0.15

2 0.20

3 -0.20

•Geometric Mean•[(1.15) x(1.20) x (0.80)]1/3 – 1 •= (1.104) 1/3 -1 =0.03353 =

3.353%


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• The Reward to Volatility (Sharpe) Ratio: A ratio developed by Nobel laureate William F. Sharpe to measure risk-adjusted performance. ▫ The Sharpe ratio is calculated by subtracting the

risk-free rate from the rate of return for a portfolio and dividing the result by the standard deviation of the portfolio returns

(5.18) Return Excess of SD

PremiumRisk Ratio Sharpe



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The Normal Distribution

State ofEconomy

Probability ofEconomic State

Stock Performance

Probability of Stock Performance in Given Economic

Joint Probability

Good 0.3 Good 0.6 0.18Neutral 0.3 0.09Poor 0.1 0.03

Neutral 0.5 Good 0.4 0.20Neutral 0.3 0.15Poor 0.3 0.15

Poor 0.2 Good 0.2 0.04Neutral 0.3 0.06Poor 0.5 0.10

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Jawaban: Probabilitas perekonomian dalam keadaan neutral dan saham pada kondisi kinerja poor=

0.15

Problem 5.6/ CFA Problem/p.153


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Deviations from Normality

(5.19) E(r)] -E[r(s)

Skew3

3

Positively skewed

Negatively skewed


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The Historical Record of Returns on Equities and Long Term Bonds•Average Returns and Standard Deviations•Other statistics of Risky Portfolios•Sharpe Ratios•Serial Correlation•Skewness and Kurtosis•Estimates of Historical Risk Premiums•A Global View of the Historical Record

more……


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Average Returns and Standard Deviation

Answer CFA Problem 3 dan 4/ p.153

Probabilitystock X Exp Return Dev Dev2 Dev2*p

Bull market 0.3 50% 15.00%30.00% 0.090 0.0270

Normal Market

0.5 18% 9.00%-2.00% 0.000 0.0002

Bear Market 0.2 -20% -4.00% -40.00% 0.160 0.0320Exp Return 20.00%

Variance 0.0592St Dev 0.2433


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Measurement of Risk with non Normal Distributions

• Value at Risk (VaR) : A technique used to estimate the probability of portfolio losses based on the statistical analysis of historical price trends and volatilities.

• Conditional Tail Expectation (CTE) : an important actuarial risk measure and a useful tool in financial risk assessment.

• Lower Partial Standard Deviation (LPSD) : Compute expected lower partial moments for normal asset returns


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Lower Partial Standard Deviation (LPSD)•Measure of risk non normal distributions•The LPSD for the large and small stock

portfolios are not very different from value from the normal distributions because the skews are similar to those from the normal (see Table 5.5)Large US Stocks Small US Stocks

Lower Partial Standard Deviation (%)

History Normal History Normal

LPSD for 25 year HPR 4.34 4.23 7.09 7.14

LPSD for 1 year HPR 21.71 21.16 35.45 35.72

Average 1 –year HPR 12.13 12.15 17.97 17.95


Annual Holding Period Risk Premiums and Real Returns

Risk RealSeriesPremiums% Returns%Lg Stk 8.7 9.3Sm Stk 15.2 15.8LT Gov 1.5 2.1T-Bills 0 0.6Inflation 0.6

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Learning about Return and Risk from The Historical Record Chapter 5 1 Bodi Kane Marcus Ch 5.

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