Lecture 7: Return & Risk C. L. Mattoli 1 (c) Red Hill Capital Corp. 2009

Lecture 7:Return & Risk

C. L. Mattoli

1(c) Red Hill Capital Corp. 2009

Intro We have discussed the inputs of returns,

including, maturity, inflation, and risks. Moreover, we have been using RRR’s, but

we have said little, so far, about how they are determined for a particular investment.

In order to explore this question, more fully, we will have to quantify risk.

Then, we can take a closer look at returns and their relationship to risk.


Intro We start out, following chapter 10, looking at

the historical record of Australian Capital markets.

From that study, we will be able to get a handle on the possibilities for returns, in the real world.

One conclusion will be that there is a reward for bearing risk, and the greater the risk, the more the return that will be expected from bearing that risk.

In the text book: read chapters 10 and 11.


Learning outcomes

On successful completion of this module you should be able to:

calculate the return on an investment discuss the historical returns on various

important types of investments calculate the risk of an investment discuss the historical risks on various

important types of investments


Learning outcomes

calculate the (arithmetic and geometric) average of a series of returns

discuss concept of market efficiency calculate expected returns and variance

for an individual security and a portfolio discuss impact of diversification explain security market line and the risk-

return trade-off.


Return

As noted earlier, in the course, return on investment is how much money you make on the investment.

That return is comprised of two components:

1. Some sort of cash income payments, like dividends on stock or interest from bonds, or other cash flows from other investments, and


Return2. A capital gain or loss from increased of

decreased value of your investment holding. $ Return = Income + CG (capital gain/loss) The holding period rate of return will be the

total return on investment divided by the initial investment. HPR = [INC + CG]/IIO = INC/IIO + CG/IIO = Income yield + capital gains yield.

That can be converted to an annual rate for comparison to other returns.


Example Returns Suppose ABC stock shares are currently

selling in the market at $25, and you purchase 100 shares.

Then, your initial investment out flow is (100 shares)x($25/share) = $2,500.

Assume, during the year, the stock pays $1.50/share in dividends (D). So, your income = $1.50/share x 100 shares = $150.


Example Returns Also, at the end of the year, the shares are

selling in the market for $26/share. Your investment is worth $26/share 100shares = $2600.

That means that the difference in value of your shares is – Pbegin = CG = $2600 - $2500 = $100.

If you sell your shares you will have a realized capital gain; if you keep them, you will have a unrealized capital gain, a paper gain (a gain, on paper, but not in cash flow).


Example Returns Your $return = INC + CG = $150 + $100 =

$250. If you sold the shares, then, you would have

total cash in hand of cash inflow = income + sale of stock = $150 + $2600 = $2750.

You can figure out your simple holding period rate of return (HPR) as follows.

HPR = (Cash Inflow – cash outflow)/cash outflow = (cash inflow – investment outflow)/investment outflow = $2750 – $2500)/$2500.


Example Returns

Rewriting, HPR = [(INC + Pend)– Pbegin ]/Pbegin = [INC +CG]/IIO = 10.0%.

Breaking that up between the two components of return, we have HPR = income yield + capital gain yield = $150/$2500 + $100/$2500 = 6% + 4%.

In the case of dividends, the first component is called the dividend yield, as we learned in an earlier module.


Return Origins: RRR


The Build up: to date

Person gives up current consumption and saves.

He requires a return for his trouble (RRR) Wants to be able to consume more with

the money when he gets it back, in the future.

Thus, RRR will include inflation, to maintain his buying power, plus a real return.


The Build up: to date Money comes in future CF (FCF), and the

future is uncertain. As a result, there must also be at least

two other components of RRR’s: term structure and risk.

First, there will be a term to maturity part, and, usually, the farther until payment, the higher the required return.

There are several logical reasons to assume that it should be that way.


The Build up: to date

The farther out the future payment, the higher the probability that you will not get paid.

Also, as it turns out, mathematically, the farther out the payment, the more sensitive the PV of that future CF will be to changes in interest rates (YTM’s)

Second, there is risk of not getting paid, any time you get a promise of future payment.


Term & Risk structure of Interest Rates

(c) Red Hill Capital Corp. 2009 16

Securities, RRR and Risk So far, we have covered risk, lightly, and it is

the major topic of this module, but lets take a look at everything we have learned, logically.

Ok, given that people invest their savings, some will invest in the securities of companies.

Remember, it’s not that mysterious: securities are pieces of paper that a company hands out to evidence borrowing = loans [bonds, notes, bills, debentures] or equity = ownership [stocks].


Securities, RRR and Risk Companies start out with nothing; then,

they get money to start their business by issuing securities to investors.

Then, the company invests the money in business projects.

Investors determine the price they pay for the securities, which, theoretically, is PV of FCF as PV = DFCF, using a discounting RRR to get the price, PV.


Securities, RRR and Risk Investors, in aggregate, in the markets,

determine what the proper RRR’s are for companies’ securities.

We know that, in general, RRR = inflation+real return+term structure+risk.

The term structure part will depend on, for example, when a bond matures and also if it makes intermediate coupon interest payments because then the risk of not getting paid is moderated by intermediate payments.

The bigger question is: how do they factor in risk in RRR?


Risk, RRR and Company X

Lets look at the securities of a mythical company X.

Suppose it issues one bond (means one series, not 1 single piece of paper) and common ordinary shares of stock.

First of all, debt is always less risky than stock for several reasons.

Debt has a maturity/stock does not;



Debt payments have to be paid or the company will be in default of the debt contract, and debt holders can take the company to court and even bankruptcy.

Stock has no payment obligations. If the company liquidates or is forced

into bankruptcy, debt holders get paid before equity stock holders.


Risk, RRR and Company X The bottom line is that the markets will

look at the company and its businesses and determine the perceived risk of the company.

That will lead the markets to pick and apply an appropriate RRR to the company, call it RRRX.

Then, suppose that the capital structure of the company is WD + WE =100%, WD

percent debt, and WE percent equity.22(c) Red Hill Capital Corp. 2009


The markets will then take RRRX and use it to figure out RRRD and RRRE, and RRRX = WDRRRD+WERRRE.

That will be the essence of cost of capital in an upcoming module.

In fact, we used that, implicitly, in the last module when we evaluated projects.

So, lets take a further look at that.


Pictures

StakeholdersGovernment

Company gets money by issuing securities

Reinvestment

We summarize the corporation, in terms of the cash flows that we discussed, in this chapter.

CF from assets

Payments to security holders

Firm investsmoney in assets

Financial markets

Invest in and value firm


CF from A=CF to debt+ CF to stock One final connection: CF from A. CF has to come from assets, including

plant & WC. The corp. does its business which includes

investment and divestment of assets. Then CF can go to: investment in PP&E &

WC. In the Op CF part of CF from A, we omit

interest payments because they are outside of operations.


CF from A=CF-to-debt+CF-to-stock They come after along with payments to other

people who gave the company $, including ordinary and pfd stockholders.

In fact, the CF from assets is FCF: everything is done except financing transactions, like paying interest to bonds & dividends to SH.

So, after the company does its business, including selling, paying off costs and investing in itself, it has to pay the investors in bonds and stock.

Before the end of it all, too, the company can issue other issues of bonds and more stock.


Table 2.5 from text book: CF identity


CF from A=CF-to-debt+CF-to-stock So, the final component of the other side of

the equation is taking in more money from new investors in new securities in the primary market.

In the end, on one side you get CF from assets; then, you use the FCF to pay off old investors (COC), and you might need more inflows from new investors.

We show a picture of the corporate machine, in the next slide.


The Corporate Machine

Investors in

securities

CashDebt Equity

SecuritiesIssued

CashFunding

Securities Markets

Cash &PhysicalAssets

Debt Equity

Businesses

BusinessProjectsMarkets

Reinvestment

InventoryPP&E Return on

Corporate Investment

Dividends

29

EPS

Interest

(c) Red Hill Capital Corp. 2009

RRR and the Company The RRR of the firm’s securities, bonds and

stock, is determined by the markets of people who buy them (primary markets).

Remember that on the other side of that transaction, investors giving $ to co. for securities, the company has to pay that RRR to investors.

For example, suppose a company issues $10 million in bonds that pay a 10% annual coupon for 10 years, and investors RRRD is higher than 10%, so they give the company less than $10 million.


RRR and the Company Suppose that the investor’s RRR was high

enough that they gave the company $9 million. Then, each year, the company will have to pay

$1 million in coupon interest to the investors, and in 10 years it will also have to pay them the $10 million principal.

Over the 10 years it will be paying out RRRD to bond investors.

Suppose that there is also $20 million in stock sold and investors had RRRE of 15%.


RRR and the Company Then, every year forever, technically, the

company would have to pay stockholders 15%x$20 million = $3 million.

Thus, the company will have to pay out over $4 million/year as RRR’s to its investors who gave it money to invest in its business projects.

If it has $29 million in assets that it bought with their money, it will need to earn a return of about $4.1 million/$29 million = 14.1% = RRRX on those assets to make enough to pay annual payments to bond and stock holders.


Three Categories of CF’s for project CF’s will fit into 3 basic slots for project

evaluation. In year 0, there will be investment in: PP&E

plus WC. In the middle, there will be the estimated CF

from operation of the project; there might also be investment or divestment of WC and there might be more CF injections required for the project.

In the last year, there might also be WC reversal and AT sale of equipment.


Year 0: Invest in Project

34

Installed cost

Purchase price -250 000

Insurance and transport -5 000

Installation -3 000

Sale of existing equipment 10 000

-248 000

– Change in net working capital

Add increase in accounts receivable

30 000

Less increase in accounts payable -20 000

Add increase in inventory 10 000

20 000

= Initial outlay -268 000


Installed cost is all it takes to get and install the equipment. Then, that whole number is depreciated.There may also be required investment in WC, e.g., buying inventory or making room for future A/R

WC Initial investment

You might need things like inventory and “room” to make sales on credit.

This is money you need upfront just as you do for equipment.

It can be subtle and then there might be inflows and outflow changes during intermediate years.


WC Initial investment

For example ass first year sales of $1 million. Your current WC is $50,000 and you want WC to be 10% of sales, then, before the first year you will need to increase WC by $50,000.

If future year sales increase, you will need to add WC investment in each year. If sales decline, you will get an CF inflow from WC.

Take a closer look at this factor in project investment and make sure that you understand it clearly.


In the middle: operating CF

Sales (50,000 units at $4.00/unit) $200,000

Variable Costs ($2.50/unit) 125,000

Gross profit $ 75,000

Fixed costs 12,000

Depreciation ($90,000 / 3) 30,000

EBIT $ 33,000

Taxes (30%) 9,900

Net Income $ 23,100

37

Final year: TV sale PP&E; WC reverse In the last year, you sell the equipment. The Terminal value CF of sale of equipment is

CFTV = Sale price – Tx[Sale – BV]. You also have to reverse out any WC that was

put in during the life of the project. The WC might have been put in at the beginning

and some more added or taken out over time. In the final year, you must reverse out whatever

is left because it all gets used up in the op CF’s, and the only thing you have left at the end is, possibly, cash.


WC recovery Thus, we have IIO = – $100,000; CF = $100,000

(sales) – $100,000 (cost) = 0. However, that would mean that we invested

$100,000 and ended up with $0, which is not correct.

We actually got our $100,000 back. It’s just that the accounting for CF gave us $0 OCF, so we have to reverse out the inventory investment to make things right.

After all we have no inventory because Inventory (0) = $100,000 and inventory(1) = $0.

39

WC with changing sales

Working capital will be 9% of revenues for each year. The working capital investment has to be made at the start of each period. All working capital will be recovered.


t=0 year 1 year 2 year 3 year 4

WC level (%Sales) 9,900 10,350 10,800 8,100 0

WC investment -9 900 -450 -450 2 700 8 100

Sales 110 000 115 000 120 000 90 000

RRRX and Projects

RRRX is basically the company’s cost of capital (COC).

That is the cost that it pays in percentage terms to the investors who give capital to the company to fund its operations.

Actually, COC will be slightly greater than RRRX because the company will have to pay costs to investment bankers, lawyers, regulatory agencies, accountants, brokers, etc. to help it issue the bonds and stock.


RRRX and Projects Thus, the company will take in slightly less than

$29 million, and since return is inversely related to price, COC> RRRX

The company, then, takes the capital that it raised from investors in the capital markets and invests to buy machinery and other things for its business.

It will use DFCF methods, like NPV, to value acquiring those business projects.

Since it has to pay COC to investors, it should use COC as the return it needs to earn on the projects it invests in.


Historical Returns


Historical Returns Chinese proverb: to dwell on the past robs

the present; to forget the past, robs the future.

There is a whole field of market analysis, called technical analysis, that tries to find patterns in the markets to predict the future of the markets.

Although that causes great debate among academics, the fact remains that we can certainly see some broad patterns and learn something from the past of financial markets.


Historical Returns We take a first look at returns as compiled in

the book “Triumph of the Optimists: 101 years of Global Investment Returns,” by Dimson, March and Staunton.

In the graph in the next slide (log-scale) are shown what $1 invested at the beginning of the 101 year period in various asset portfolios would have been worth at the end.

Included in the graph are equities, bonds, cash as an investment in 30-day bank bills, plus inflation is also shown.


101 Years of Investment Prices


Graph Analysis (Text: Figure 10.4) From the graph, you can see that the equity

investment was, by far, the most fruitful. $1 became $83,364.

The bond portfolio resulted in $1 becoming only $166 by the end.

Cash investment in ST BAB’s became $85. Moreover, inflation of prices over the period

can be interpreted as $56 was needed by the end of the period to purchase the same goods and services that cost only $1 at the beginning.


Detailed Returns

The authors of the textbook have compiled quarter-to-quarter returns for classes of Australian assets (over 20 years).

The portfolios for which they have calculated returns are:

1. ASX All Ordinaries Index, which includes shares of the 500 largest listed public Australian companies (in terms of market capitalization).


Detailed Returns

2. A portfolio of Australian Government bonds with 10 years to maturity

3. Cash as investment in a portfolio of 30-day BAB’s, and

4. The CPI, the consumer price index, which is a broad measure of inflation in a typical basket of consumer goods and services.

In the next few slides, we show the quarter-to-quarter returns of the various portfolios.


Q-to-Q Returns on ASX All Ord. ASX All Ordinaries Index


Q-to-Q Returns for 10 T-bonds 10-year T-bonds


Q-to-Q return on “Cash”Cash investment


Q-to-Q InflationAustralian CPI


Overview of the Return Data

According to the charts, the cash rate was pretty consistent. It was in the 4-5% range, in the late 1980’s, when world inflation was just coming under control. Then, it dropped to the 1.5-2% region from the early 1990’s.

At the time that cash was in the upper region, inflation was around2-3%; it dropped to the region of around 1%, in the 1990’s, except for a spike around the U.S World Trade Center incident.


Overview of the Return Data

Government bonds had a peak return of almost 12% in the March 1986 quarter, and a low of almost – 4%.

Equities had the largest positive and negative returns. The low return was around 40% during the global stock market crash of the fall of 1987. The high was almost 30%, in the quarter just preceding the crash.


Average Returns

As a first step in analyzing the historical record is to take a simple average of returns over the 20 year period, included in the graphs.

Then, the average return on cash was 8.4%. The average return for 10-year T-bonds was

10.6%. The average for equities over the period was

14.4%. In addition, inflation averaged 3.9%.


Real Returns

The averages, except for inflation, are nominal returns (inflation+real).

Since we have also computed the inflation rate, we can find real returns.

Thus, the real average return on cash was 4.5% (= 8.4% – 3.9%).

The average real return on 10-year T-bonds was 6.7%%.

And the real average rate of return on equities was 10.5%.


Risk Premiums The last thing that we should look at is risk

premiums. BAB’s are bank-guaranteed debt of large

corporations with good credit, and we have limited maturity to 30 days.

Moreover, the trading market in BAB’s is highly liquid.

Thus, the cash rate, discussed above, is virtually default-risk free, and we shall designate it as the riskless rate of return.


Risk Premiums Then, when we compare the riskless rate to

the rate of return on equities, we find that equities had an excess return of 5%.

Since that is the price for bearing the risk of owning equities, we interpret it as the risk premium, the reward for bearing the risk, for holding equities.

Thus, we have found a risk component of returns, in addition to the real rate of return for investing.


Summary of ReturnsAverage Returns 1982 -2002

Category Average Return

Real

Return

Risk Premium

All Ordinaries

14.4% 10.5% 6.0%

10 Year T-Bonds

10.6% 6.7% 2.2%

Cash 8.4% 4.5% 0.0%

Inflation 3.9% -- --


Variability of Returns:

Risk


Variability & Risk In the last section, we talked of risk

premiums of return. We concluded that the risk premium for T-

bonds was about 2% and for equities, 6%. So, if the risk premium is higher, then, so

must be the risk, somehow. Risk is, basically, the chance that the

future will turn out differently than expected.


Variability & Risk To quantify that concept for returns, we look

at the variability of return. In that regard, for example, we can see that,

while the average return on equities was 14.4%, the range of returns was – 40% to + 30%, a range of about 70%.

On the other hand, the average for T-bonds was 10.6% and the range was only – 4% to 12%, or a range of 16%.


Variability & Risk

In fact a first measure of variability of returns is those ranges of returns.

Thus, there seems to be a correspondence between our initial conception of risk and the market’s assignment of a bigger premium, in retrospect looking at past returns, for bearing the risk.


Histogram Frequency Distribution Just looking at the range of returns, in one

dimension, does not render a lot of information.

The next step, then, is a 2-dimensional look. Frequency distributions, or histograms,

organize data into a vertical-axis frequency-of-occurrence of a number versus the number on the horizontal axis.

In that way, we find out how many times a certain return occurred in the data set.


Histogram Frequency Distribution For example, assume that over the past

20 years, you observed that returns were in the ranges: – 30% to 0, 2 years; 0% to 5%, 4 years; 5% to 15%, 10 years; 15% to 25%, 3 years; and 25% to 40%, 1 year.

Then, we plot those ranges against the number of years that they show up in the data as:


Histogram Frequency Distribution

67

-30 0 5 15 25 40

1

5

10

Example Histogram Frequency Plot

Times

Return


Histogram Frequency Distribution We pick ranges of returns, instead of just one

number because it makes a better picture. There is, actually, no reason to pick a certain

range except, perhaps, because it fits the data best.

If we picked different ranges, the general shape will stay relatively the same, although it might change, if you make the ranges too small.

In the end, we have a more useful picture of returns than with just the whole range.


Histogram Frequency Distribution In the next slide, we show a histogram

frequency plot of ranges of return for return data for the All Ord.’s index, the ASX stock market index, monthly between 1985 to 1995.

The range of returns, over the period, was about – 22% to about +30%

We choose ranges, for simplicity, and also because it makes a nice picture that shows what we wanted to demonstrate.


Histogram Frequency Distribution Thus, to construct the histogram plot, we

count how many times returns fell into one of the sub-ranges, and plot that frequency number against the ranges.

In the plot, you can see that the range 3.59%-12.11% had the highest frequency, followed by the – 4.93-3.58% range.

The other ranges have much less frequency, and the histogram spreads out, while going down.


Histogram of Returns: fig. 10.9


Standard Deviation: Spread Another name for a histogram plot is a

distribution, which takes its name from probability and statistical analysis.

In fact, we borrow many concepts from that field to analyze risk and return.

We looked at the plain range of returns as a first approximation of variation.

However, the histogram shows us that there is also concentration of the returns, in the middle, and the frequency dies out as we move away from the middle.


Analysis of the Spread The statistical/probability concept that

captures the spirit of what we want is the variance standard deviation of returns.

The variance measures the spread of a distribution, while also accounting for the height, the frequency or probability, of the distribution, in the calculation.

First, we recalibrate our measurement by measuring the spread from the central, highest frequency return.


Analysis of the Spread The central return is calculated, statistically, as a

simple average of past returns. The central value is called the mean return = sum[i=1 to N]Ri /N.

Another way that we could calculate the mean is by taking each return, finding the frequency of that return, the number of times that it occurred in the data, then multiply frequency of each return by the return, summing up, and dividing by the total number of observations.

We consider a simple example, in the next slide.


Example: Mean and Frequency Consider the following set of observed

returns.

We calculate mean return = [10% + 4% + 10% + 6% + 3% + 1%]/6 = [2x10% + 4% + 6% + 3% + 1%]/6 = 5.67%.

So, we can see how it works.

75

Period 1 2 3 4 5 6

Return 10% 4% 10% 6% 3% 1%


The Meaning of Mean Because the mean is, basically, a

frequency-weighted average of the variable in the distribution, what it gives us is the balancing point of a distribution.

It is the point in the frequency distribution at which, if the distribution was, for example, cut out of cardboard, you could perfectly balance the distribution by hanging it by a string at that point or by balancing it on a fulcrum.

We show this, diagrammatically, in the next 2 slides.


The Mean as the Balancing Point of a Distribution

77

The Mean as a Balancing Point

Hung by the Mean

Fulcrumed at the Mean


Mean of Distribution From the picture, it appears that

the weight of the distribution on each side of the balance point is the same.

Return % Frequency

12 20

8 60

2 15

-5 5

Total=100


Variance Defined We must be careful how we define the

measure of spread from the mean. Consider, for example, a distribution that was

perfectly symmetrical about the mean, and consider a mean of exactly zero, as below.

79

Gaussian Distribution:

P(x) =(2-1/2exp[(x - m)2/2

-4 -3 -2 -1 0 1 2 3 4


Variance Defined If we just took a simple average spread of

returns from the center of that distribution, we would get zero for the answer since the negative side would exactly cancel out the positive side.

Thus, in our formal definition of spread from the mean, we take the spread, square it, and then construct a sum of the squares of the plain spread.

That number is called the variance of the distribution, and it is given by 2 formulas, depending on whether we are analyzing past data (called “ex post”) or using probabilities and future projected data (ex ante).


Ex post variance The ex post variance is the weighted

average of the deviations of observed returns from the mean return

Where T is the number of past observations and we use Mean(O) = Ō

1

)(

)()(

1

2

2

T

RR

rVarRVarianceT

tt


Example – mean ex post variance Given the following returns, calculate the

variance. The mean is 1.8%

Month Return %

July -4.96

August 6.14

September -3.12

October 9.15


Example – ex post variance calc What is the variance of returns given in the

example?

00457.0

3

005402.0002421.0001884.000457.014

)018.00915.0()018.00312.0()018.00614.0()018.00496.0(

1

)(

2

2

22222

1

2

2

T

RRT

tt


Standard deviation The standard deviation is the square root of

the variance. The standard deviation, then, is the

average spread from the mean. For ex post data, then:

For the example σ = 6.9%.

1

)(1

2_

T

RRSD

T

tt


SD as a Proxy for Risk Returning to the question of risk, we use

SD or variance or some other measure of variation as a proxy for the riskiness of things.

If you look at distributions, the more spread out a distribution, the more possible outcomes that can occur.

You can observe that, graphically, in looking at the distributions in the next slide.


Riskiness and the Spread of Outcomes

Gaussian Curves for Three SD's

1086420-2-4-6-8-10

Least Risky

Most Risky


The Normal Distribution A distribution that is useful in many fields of

study, including physics and finance, is the normal distribution, also called the bell-shaped curve.

The distribution is symmetric about the center, and its peak is at the mean.

Interpreting that is quite simple: it is equally likely that an observation of the variable is below or above the mean.

In that regard, it describes a fair game, one in which you are equally likely to win or to lose.

Often, people like to think that financial markets, like the stock and bond markets, are fair games.


Further Properties of the Bell Curve In general, distributions can correspond to

complicated mathematical formulas. One of the most simplifying properties of

the normal distribution is that it can be completely described by its mean and SD.

Another useful property is that the likelihood that an observation is within 1 SD of the mean is that 68%, and over 99% of the possible outcomes is within 3 standard deviations from the center.

SEE GRAPH, BELOW:


Possibilities for a Normal Distribution

SD’s from central expected value peak.

68.26%

95.46%99.74%

Gaussian Distribution: P(x) = (2πσ )-½ exp(-x2/2σ)


Textbook Figure 10.10


Historical Risk and Return A summary of the author’s look at the

markets between 1985 and 2005 are given in the preceding slide.

Thus, if they were truly described by normal distributions, we could expect, for example, that quarterly returns for the All Ord.’s would be between – 4.93% and 12.11% (mean ± SD = 3.6% ± 8.5%) 68% of the time.

There would be only a 5% chance that returns would fall outside the range – 13.45% and 20.63% (3.6% ± 2x8.5%)


A Note on Annualizing Risk/Return To annualize historical data for return and

risk, the following rules apply. It’s simple to annualize returns. For example,

if you used return monthly data, annualize returns by multiplying by 12. For quarterly data, multiply quarterly returns by 4.

The corresponding multiplication factors for SD, however, will be the square root of the factor that we use to annualize returns, e.g., monthly factor = 12; quarterly 4 = 2.

It is a statistical thing.


Geometric vs. Average Returns There is a basic problem that arises by

using average returns. Suppose, for example, that you start out

by investing $100. The first year, you lose $50, but by the end

of the second year, you are back to exactly $100.

If you consider that, you expect that your average return was 0%.


Geometric vs. Average Returns However, the actual average return is (– 50%

+ 100$%)/2 = 25% = RA. Your intuitive previous answer was actually

the geometric average return found from (1+RG)2 = (1+R1)(1+R2). It is equal to 0 for the example.

It is found the same way that we find EAR and nominal from real return and inflation.

The general formula for N years would be RG = [(1+R1)(1+R2) … (1+RN)]1/N – 1.


Geometric vs. Average Returns The geometric average will always be

smaller than the simple arithmetic average. An approximate formula is RG = RA – ½ SD. We summarize the historic arithmetic and

geometric returns, in the table, below:

95

Category Average Return

Geometric Return

Standard Deviation

All Ord 14.4% 12.8% 17.04%

10 Yr T-Bnds 10.6% 10.5% 5.64%

Cash 8.4% 8.4% 2.11%

Inflation 3.9% 3.9% 1.66


Which One?

If you have an estimate of an average return for a year, then, it is the best guess for next year’s return.

In general the average return will be too high an estimate for long periods of time

The geometric return will underestimate the return for short periods of time.

In some cases, the logic of the situation will dictate which you should use.


SD for the NYSE over 35 Years


Normal or Not Normal

Like many things in economics and finance, the normal distribution is an idealized perfect world supposition of the markets should act.

There are problems, as can be seen from the next slide, which shows returns for a stock versus what they would have been if returns followed a normal distribution.

For one thing, the most you can lose is 100% of investment, so there is a natural bias from the beginning.


Normal or Not Normal

Moreover, more recent studies of returns have found that most are actually skewed to the downside.

In that regard, the tail on the left is longer and more spread out.

The upside is usually scrunched into a smaller range above the hump in the distribution.


Histogram of actual returns vs. Normally distributed Returns


Market Efficiency


Intro A topic of great discussion in finance

is market efficiency. Market prices change all the time. The basic reason is the arrival and

dissemination of information to market participants.

The most important thing in investing is information.


Intro As new information arrives in the market,

people reassess their outlooks. The question of market efficiency

becomes how fast and accurately do markets adjust.

In an efficient market, prices should fully reflect all of the available information.

In other words, based on the information, available, there is no reason to believe that prices are either too high or too low.


Possible Reactions & Adjustments Suppose that a company announces a new

project that management has figured will greatly increase the PV of its shares.

In a completely efficient market, the price will adjust quickly to the news.

Other possibilities are delayed reaction, taking several days to assimilate the information, or overreaction and subsequent adjustment.

We show the 3 possibilities in the next slide from the textbook.


Figure 10.12


The Efficient Market Hypothesis The gist of the EMH is that well-developed

financial markets, like the ASX or the NYSE, are effectively efficient.

The proponents of the EMH might further state that, while there are anomalies, the anomalies are small and rare.

In an efficient market, all investments are zero NPV investments because market price equals value.


The Efficient Market Hypothesis The leftover, inefficient stocks will be just

enough to keep those people in business whose specialty is finding, analyzing and investing in undervalued investments.

In practice, competition among investors in information gathering and processing will move more and more prices to their proper levels.

As investors analyze the information, they may conclude that a price is too high or too low. Their action, their vote, in the market, buying or selling shares will help to move the price towards its true level.


Common EMH Misconceptions You can argue either side of market

efficiency, but in the end, lets just say that some markets are more efficient than others.

Prices fluctuate because information and analysis changes.

It does not mean that any investment will be as good as any other.

It does mean that patterns that you may find will either be ephemeral or not real.


Forms of the EMH

Over the years he EMH has been presented in 3 forms: strong, semi-strong, and weak.

In the strong form, the hypothesis states that all information is reflected in price. There is no such thing as inside information.


Forms of the EMH

However, inside information, necessarily exists, and profits, although illegal and unethical, could be made with such information.

In the semi-strong for, the statement is that all publically-available information is reflected in prices.


Forms of the EMH Arguments against this form are that, in that

case, security analysis has no value. There are no undervalued gems to mine.

The final form, weak efficiency, states that present price reflects the history of prices.

Then, studying the past, as is done in so-called technical stock analysis, which seeks to discover trading patterns, will not find any patterns to project future price.

Although the EMH is a centerpiece of modern financial theory, many studies over the past decade or so have turned up anomalies, like momentum trends, over and under reaction, and others.


Learning activity

Attempt all of the critical thinking and concepts review questions on pages 316 to 317.

Attempt questions 1 to 10 and 20 to 28 on pages 317 to 319 of your text.


END


Documents

Lecture 7: Return & Risk C. L. Mattoli 1 (c) Red Hill Capital Corp. 2009