mbf2263 portfolio management lecture 8: risk and return in ... · eq.10.2 by compounding the...

MBF2263 Portfolio Management

Lecture 8: Risk and Return in Capital Markets

1. A First Look at Risk and Return

• We begin our look at risk and return by illustrating how the risk premium affects investor decisions and returns:

• Suppose you were given $10,000 in a raffle in December 1988 and decided to invest it all in a portfolio of Australian shares, with dividends being reinvested.

• By December 2008, 20 years later, your share portfolio would be worth $55,695 and a comparable portfolio of cash $41,134 as shown in Figure 10.1.– The impact of the stock market decline of 2007 and the global

slowdown that occurred from 2008 is evident in the sharp decline of the graph.

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Figure 1: Value of $10,00 invested in cash and Australian shares over 20 years from December 1988

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Table 10.1 Range of returns on Australian investments over 20 years from December 1988

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Table above shows returns of four investment classes with different risk profiles over 20 years.

General principle that investors do not like risk and demand a premium to bear it.

2. Historical Risks and Returns of Securities

• In this section, we explain how to calculate average returns and a measure of risk, or volatility, using historical stock market data.

• The distribution of past returns can be useful in estimating the possible future returns for investors.

• We start by first explaining how to calculate historical returns.

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• Individual Investment Realised Return

• The realised return is the total return that occurs over a particular time period.

• The realised return from your investment in the share from tto t+1 is:

(Eq. 10.1)

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FORMULA!


Problem:

• Metropolis Limited paid a one-time special dividend of $3.08 on 15 November 2010.

• Suppose you bought Metropolis share for $28.08 on 1 November 2010 and sold it immediately after the dividend was paid for $27.39.

• What was your realised return from holding the share?

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Example 1 - Realised return

Solution:

Plan:

• We can use Eq.10.1 to calculate the realised return.

• We need the purchase price ($28.08), the selling price ($27.39), and the dividend ($3.08) and we are ready to proceed.

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• Execute:

• Using Eq.10.1, the return from 1 Nov 1 2010 until 15 Nov 2010 is equal to:

• This 8.51% can be broken down into the dividend yield and the capital gain yield:

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Evaluate:• These returns include both the capital gain (or in this case a

capital loss) and the return generated from receiving dividends.

• Both dividends and capital gains contribute to the total realised return—ignoring either one would give a very misleading impression of Metropolis’ performance.

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• Individual Investment Realised Return

• For quarterly returns (or any four compounding periods that make up an entire year) the annual realised return, which can be observed over years, Rannual, is found by compounding:

1+ Rannual =(1+R1) (1+R2) (1+R3) (1+R4)

(Eq. 10.2)

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FORMULA!

Problem:

• Suppose you purchased Metropolis’ shares on 1 November 2010 and held them for one year, selling on 31 October 2011.

• What was your realised return?

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Example 2 - Compounding realised returns

Solution:

Plan:

• We need to analyse the cash flows from holding Metropolis’ shares for each quarter.

• In order to get the cash flows, we must look up Metropolis share price data at the start and end of both years, as well as at any dividend dates.

• From the data we can construct the following table to fill out our cash flow timeline:

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Plan (cont’d):

• Next, calculate the return between each set of dates using Eq.10.1.

• Then determine each annual return similarly to Eq.10.2 by compounding the returns for all of the periods in that year.

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Execute:

• In Example 10.1, we already calculated the realised return for 1 Nov to 15 Nov 2010 as 8.51%.

• We continue this for each period until we have a series of realised returns.

• For example, 15 from Nov 2010 to 11 Feb 2011, the realised return is:

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Execute (cont’d):

• We then determine the one-year return by compounding:

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Execute (cont’d):

• The table below includes the realised return at each period:

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Evaluate:• By repeating these steps, we have successfully calculated

the realised annual returns for an investor holding Metropolis shares over this one-year period.

• From this exercise we can see that returns are risky.

• Metropolis fluctuated up and down over the year and ended-up only slightly (2.75%) at the end.

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• Average Annual Returns• The average annual return of an investment during some

historical period is simply the average of the realised returns for each year.

• That is, if Rt is the realised return of a security in each year t, then the average annual return for years 1 through T is:

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(Eq. 10.3)

FORMULA!

• The average return provides a estimate of the return we should expect in any given year.

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Table 2 - Annual Returns on the Australian All Ordinaries Index 2004-08

• The Variance and Volatility of Returns

• To determine the variability, we calculate the standarddeviation of the distribution of realised returns, which is the square root of the variance of the distribution of realised returns.

• Variance measures the variability in returns by taking the differences of the returns from the average return and squaring those differences.

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FORMULA! (Eq. 10.4)

(Eq. 10.5)

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Variance estimate using realised returns

We have to square the difference of each return from

the average, because the unsquared differences from

an average must be zero.

Because we square the returns, the variance is in

units of ‘%2’ or percent-squared, which is not useful.

So we take the square root, to get the standard

deviation in units of ‘%’.


FORMULA!

• The standard deviation (represented by the Greek letter sigma, σ)

shows how much variation or dispersion from the average exists. A low

standard deviation indicates that the data points tend to be very close to

the mean (also called expected value); a high standard deviation

indicates that the data points are spread out over a large range of

values.

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Standard deviation- Definition

http://en.wikipedia.org/wiki/Sigma

http://en.wikipedia.org/wiki/Statistical_dispersion

http://en.wikipedia.org/wiki/Mean

Problem:

• Using the data from Table 10.2, what is the standard deviation of the return on the All Ordinaries Index for the years 2004-2008?

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Example 3 - Calculating historical volatility

Solution:

Plan:

• First, calculate the average return using Eq.10.3 because it is an input to the variance equation.

• Next, calculate the variance using Eq.10.4 and then take its square root to determine the standard deviation.

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Execute:

• In the previous section we already calculated the average annual return of the All Ordinaries during this period as 9.74%, so we have all of the necessary inputs for the variance calculation from Eq.10.4, we have:

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Execute (cont'd):

• Alternatively, we can break the calculation of this equation out as follows:

• Summing the squared differences in the last row, we get 0.3531.

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Execute (cont'd):

• Finally, dividing by (5 - 1= 4) gives us 0.3531/4 =0.0883.

• The standard deviation is therefore:

Evaluate:

• Our best estimate of the expected return for the All Ordinaries Index is its average return, 9.74%, but it is risky, with a standard deviation of 29.71%.

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Example 10.3 Calculating historical volatility

• The Normal Distribution:

• Standard deviations are useful for more than just ranking the investments from riskiest to least risky.

• It also describes a normal distribution, shown in Figure 10.2:

– About two-thirds of all possible outcomes fall within one standard deviation above or below the average;

– About 95% of all possible outcomes fall within two standard deviations above and below the average;

– Figure 10.2 shows these outcomes for the shares of a hypothetical company.

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Figure 2- Normal Distribution

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Because we are about 95% confident that next year’s

returns will be within two standard deviations of the

average:

(Eq. 10.6)

Table 3 - Summary of Tools for Working with Historical Returns

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• The returns of large portfolios

• Figure 10.3 plots the average returns vs. the volatility of US large company shares, US small shares, US corporate bonds, US Treasury bills and a world portfolio.

• Note that investments with higher volatility, measured by standard deviation, have rewarded investors with higher average returns.

• This is consistent with the view that investors are risk averse; risky investments must offer higher average returns to compensate for the risk.

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3. The Historical Trade-off Between Risk and Return

Figure 3 -The historical tradeoff between risk and return in large portfolios, 1926–2006

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• Returns of Individual Securities

• Although it will take more work to establish the relation between risk and return for individual shares, the following is true:

1. There is a relationship between size and risk—larger shares have lower volatility than smaller ones.

2. Even the largest shares are typically more volatile than a portfolio of large shares, such as the S&P 500.

3. All individual shares have lower returns and/or higher risk than the portfolios in Figure 10.3 — the individual shares all lie below the line in the figure.

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• Individual Securities

• While volatility (standard deviation) seems to be a reasonable measure of risk when evaluating a large portfolio, the volatility of an individual security doesn’t

explain the size of its average return.

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4. Common Versus Independent Risk

• Example: Theft vs. earthquake insurance

• Consider two types of home insurance: theft insurance and earthquake insurance.

• Assume that the risk of each of these two hazards is similar for a given home in KL – each year there is about a 1% chance the home will be robbed, and also a 1% chance the home will be damaged by an earthquake.

• Suppose an insurance company writes 100,000 policies of each type of insurance for homeowners in KL; are the risks of the two portfolios of policies similar?

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4. Common Versus Independent Risk

• Example: Theft vs. earthquake insurance

• Why are the portfolios of insurance policies so different when the individual policies themselves are quite similar?– Intuitively, the key difference between them is that an

earthquake affects all houses simultaneously, so the risk is linked across homes – common risk.

– The risk of theft is not linked across home, some homeowners are unlucky, others lucky – independent risk.

• Diversification: The averaging out of independent risk in a large portfolio.

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Table 4 - Summary of Types of Risk

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Problem:

• You are playing a very simple gambling game with your friend: a $1 bet based on a coin flip.

• That is, you each bet $1 and flip a coin: heads you win your friend’s $1, tails you lose and your friend takes your dollar.

• How is your risk different if you play this game 100 times in a row versus just betting $100 (instead of $1) on a single coin flip?

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Example 5 - Diversification

Solution:

Plan:

• The risk of losing one coin flip is independent of the risk of losing the next one: each time you have a 50% chance of losing, and one coin flip does not affect any other coin flip.

• We can calculate the expected outcome of any flip as a weighted average by weighting your possible winnings (+$1) by 50% and your possible losses (-$1) by 50%.

• We can then calculate the probability of losing all $100 under either scenario.

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Execute:

• If you play the game 100 times, you should lose about 50% of the time and win 50% of the time, so your expected outcome is: 50 (+$1) + 50 (-$1) = $0

• You should break-even.

• Even if you don’t win exactly half of the time, the probability that you would lose all 100 coin flips (and thus lose $100) is exceedingly small (in fact, it is 0.50100, which is far less than even 0.0001%).

• If it happens, you should take a very careful look at the coin!

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Execute (cont’d):• If instead, you make a single $100 bet on the outcome of

one coin flip, you have a 50% chance of winning $100 and a 50% chance of losing $100, so your expected outcome will be the same: break-even.

• However, there is a 50% chance you will lose $100, so your risk is far greater than it would be for 100 one dollar bets.

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Evaluate:

• In each case, you put $100 at risk, but by spreading-out that risk across 100 different bets, you have diversified much of your risk away compared to placing a single $100 bet.

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5. Diversification in Share Portfolios

• As the insurance example indicates, the risk of a portfolio depends upon whether the individual risks within it are common or independent.

• Independent risks are diversified in a large portfolio, whereas common risks are not.

• Our goal is to understand the relation between risk and return in the capital markets, so let’s consider the implication of this distinction for the risk of stock

portfolios.

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• Unsystematic vs. systematic risk• Share prices and dividends fluctuate due to two types of

news:

– Company or Industry-specific news: Good or bad news about a company (or industry) itself; e.g. a firm might announce that it has been successful in gaining market share within its industry.

– Market-wide news: This is news that affects the economy as a whole and therefore affects all shares; e.g. the Reserve Bank might announce that it will lower interest rates to boost the economy.

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• Unsystematic vs. systematic risk

• Fluctuations of a shares’ return that are due to company or industry-specific news are independent risks.

• Like theft across homes, these risks are unrelated across shares and are also referred to as unsystematic risk.

• On the other hand, fluctuations of a shares’ return that are due to market-wide news represent common risk, which affect all shares simultaneously.

• This type of risk is also called systematic risk.

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Figure 4 - Volatility of portfolios of type S and U shares

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• Unsystematic vs. systematic risk

• When firms carry both types of risk, only the unsystematic risk will be diversified away when we combine many firms into a portfolio.

• The volatility will therefore decline until only the systematic risk, which affects all firms, remains.

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Figure 5 - The effect of diversification on portfolio volatility

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• Diversifiable risk and the risk premium

• Competition among investors ensures that no additional return can be earned for diversifiable risk:

– The risk premium of a share is not affected by its diversifiable, unsystematic risk.

– The risk premium for diversifiable risk is zero.

– Thus, investors are not compensated for holding unsystematic risk.

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Table 5: The expected of type S and type U firms, assuming the risk-free rate is 5%

The risk premium of a security is determined by its systematic risk and does not depend on its diversifiable risk.

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Table 6: Systematic risk vs. unsystematic risk

Thus, there is no relationship between volatility and average returns for individual securities.

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mbf2263 portfolio management lecture 8: risk and return in ... · eq.10.2 by compounding the...

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