Finance (FINA1027)

Introduction to Risk, Return and the Opportunity Cost of Capital

Rates of ReturnIn January 2008, I paid £10 for 1 share in ABC Ltd. In January 2009, the share had risen in price to £12. ABC paid a dividend of 50p at the end of 2009.Inflation was 2.5% in 2008

Percentage = capital gain + dividend Return Initial share price

= 2 + 0.50 10.00

= 25%

Dividend = dividend Yield Initial share price

= 0.50 10.00 = 5%

Percentage = capital gain Capital gain Initial share price

= 210.00

= 20%

Real Rates of Return

1 + real rate of return = 1 + nominal rate of return 1 + inflation rate

= 1.25 1.025 = 1.2195

The real rate of return is 21.95%

• Nominal rates of return measure how much more money you have at the end of the year

• In the interim, prices of goods will have increased in line with the rate of inflation

•Real rates of return indicate how much more you will be able to buy with your money at the end of the year

How can I estimate the future performance of ABC stock?

• Investing in shares is a risky business

• It is impossible to consistently and accurately predict future share prices

• Historical data gives some insight into the patterns of returns that can be expected and an indication of the bounds of returns

Market Indices• A market index measures the investment performance of

the overall market• Historical returns of market indices can indicate the

typical performance of different types of investment• Not all indices are the same• DJIA – The Dow – tracks a portfolio which has 1 share in

each of 30 large firms• Not representative of average performance• S&P 500 is capitalisation weighted, holds shares in

proportion to the number of shares issued to investors• S&P 500 shows the average performance of investors in

the top 500 firms

Why is the S&P 500 more representative?

Shares Outstanding (bn)

Share price move

GE 10 15%

Du Pont 1 10%

Dow Return 12.50%

S&P Return 14.55%

Average Investor Return 14.55%

• GE and Du Pont appear in both the Dow and the S&P 500

•GE has 10bn shares in public ownership, Du Pont has 1 bn shares

•The average investor therefore owns 10 GE shares for every Du Pont share

•Assume that GE shares increase by 15% and Du Pont by 10%, all other shares in both indices remain static

•The Dow would increase by 15% + 10% = 12.5% 2

•S&P 500 would increase by (15% x 10 + 10% x1) = 14.55%11

•The average investor would earn 14.55% so this number is of most interest

Different Investment Classes perform Differently

• The Chart illustrates the performance of three portfolios since 1962

• US T-bills have averaged 5.5% p.a.

•US 10 year bonds have averaged 7.2% p.a.

•S&P 500 Index has averaged 9.4% p.a.

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US 10 Year Bonds S&P 500 Index US T-bills

Data Source: Yahoo Finance

Dividend yields averaged over 10 year periods

Monthly compounding assuming income on 10 year bond equal to start yield and 3 month and 2 month US rates equal

Does that mean that equities are best?

• Portfolios are not equally risky

• T-bills are very risk averse– Issued by the US government, no default risk– Short term maturity means capital losses are limited– Guaranteed return after 3 months

• T-bonds are more risky– No default risk– Guaranteed Income through coupon payments– Price will fluctuate as interest rates change– Have to hold for 10 years to guarantee return at inception

• S&P 500 Index is most risky– No guarantee on return of capital– No guaranteed income

The Risk Premium

PortfolioAverage Annual Return

Average Premium

T-bills 5.5%

T-bonds 7.2% 1.7%

Equities 9.4% 3.9%

• T-bonds give higher returns that T-bills – the Maturity Premium - +1.7%

• This excess return compensates the investor for the additional risk

• Equity returns exceed T-bills by an average of 3.9% p.a. – the Equity Risk Premium

•History shows that investors receive a risk premium for investing in risky assets

•The risk premium varies according to the period of history considered

•To smooth market fluctuations, long period of data must be used e.g. 1987 crash

Data source – Yahoo finance, 1962 - 2009

Asset Class Risk

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S&P 500 Index US 10 Year Bonds US T-bills

•Plotting monthly returns illustrates the riskiness of different asset classes

•T-bills have delivered a safe, steady return

•T-bonds have fluctuated by more

•Equities have delivered the most extreme fluctuations in price – positive and negative

October 87 – equities

down 21.3%

August 98 – equities

down 14.3%

October 08 – equities

down 16%

October 74 – equities

up 16.6%

Cost of Capital

• Investment decisions require an estimation of the opportunity cost of capital

• This is the return which could be achieved elsewhere at an equivalent level of risk

• Suppose a project arises which has the same level of risk as the equity market

• To deduce the cost of capital, you must estimate the expected return on this market portfolio

Estimating Expected Return

• Historical Average?– Is average return a good estimate? It has been 9.4%

over the last 48 years – In August 1981, T-bills offered an annual return of

15.5%– We know equities are more risky, so why would any

sane and rational person invest for an expected return below the riskless alternative?

• Estimate must be relative to other asset classes

Expected Market Return

Expected Current Interest Long term Market = rate on + risk Return

T-bills premium

2009 Expected Market Return = 0.04% + 3.9% = 3.94%

1981 Expected Market Return = 15.5% + 3.9% = 19.4%

The expected return on an investment compensates investors for waiting (time value of money) and also for worrying (asset class risk)

Issues

• If the time period used to calculate the risk premium is extended, then the premium changes

• Which is the correct premium to use?• Would it be more sensible to look at the

premium over longer term bonds?• Risk premium varies from country to country e.g.

France ~10%, Spain ~7%, is the French equity market more risky than Spain?

Return distributions – T bills

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The chart illustrates the monthly returns of T-bills since 1960. The bars indicate how many times the monthly return has fallen within a specific range.

The returns are bucketed into 2% performance brackets with a minimum of -22% and a maximum of 20%

550 out of 575 of historical monthly return observations lie in the range 0% to 1%

Return distributions – T bonds

The chart illustrates the monthly returns of T-bonds since 1960. The bars indicate how many times the monthly return has fallen within a specific range.

The returns are bucketed into 2% performance brackets with a minimum of -22% and a maximum of 20%

The historical monthly returns show a much wider range of dispersion with a positive skew. The most common observation of return is still the 0-1% bucket, with 107 observations but there are many returns below and above this.

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Return distributions – Equities

The chart illustrates the monthly returns of equities since 1960. The bars indicate how many times the monthly return has fallen within a specific range.

The returns are bucketed into 2% performance brackets with a minimum of -22% and a maximum of 20%

The historical monthly returns show an even wider wider range of dispersion with a more positive skew but also more extreme observations. The most common observation of return is now in the 1-2% bucket, with 70 observations but the incidence of large negative and positive returns is much higher.

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How can we use this to calculate risk?

• Given a set of representative observations, it is possible to calculate the probability that a given return will be achieved

• Example– For T-bills, 550 out of 575 observations lie in

the 0-1% bucket of performance– Probability of next months performance lying

in this bucket = 550/575 = 95.6%

Variance and Standard Deviation

• Variance – average value of squared deviations from the mean

• Standard deviation – square root of variance

Measuring Risk

Example– I am stuck in a traffic jam, there is a 20% chance that I will reach

work in 10 minutes, 30% chance of 20 minutes, 40% chance of 15 minutes and 10% chance that I will be delayed for half an hour

What is my expected delay?

Expected delay = 0.2x10 + 0.3x20 + 0.4x15 + 0.1x30 = 17 minutes

However, there are deviations around this average. I could arrive 7 minutes earlier, or even 13 minutes later.

Variance & Standard Deviation

Variance = Sum of probability weighted squared dispersions from the mean

= (17-10)^2 x 20%

+(17-20)^2 x 30%

+ (17 -15)^2 x 40%

+(17-30)^2 x 10%

= 39.4

Standard Deviation = Square root of variance

= 6.28

Standard deviation provides an indication of the degree of dispersion around the 17 minutes expected delay to my journey.

Measuring Variation in Stock returns

Most analysts assume that the spread of past returns gives a good indication of future returns 0%

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US 10 Year Bonds S&P 500 Index US T-bills

The chart illustrates the annualised standard deviations of monthly returns for rolling 12 month periods since 1962. Equities are much more volatile than the other asset classes, but recent swings are by no means unprecedented adding weight to the validity of the argument that the past return distribution is representative of future returns

Risk and Diversification

• Diversification reduces risk• Ice cream seller reduces risk by

purchasing a hot dog cooker rather than another Mr Whippy machine

• The same principle applies across and within asset classes

• The aim is to find stocks which offer attractive returns but have less than perfect positive correlation

A note an correlation

Perfect positive correlationThe direction of returns is always the same – if the price of A increases, then the price of B will increase

Perfect negative correlationThe direction of returns is always the opposite – if the price of A increases, then the price of B will decrease

Most stocks lie somewhere between the two

Portfolio diversification works because prices of different stocks do not always move in the same direction

Real world example

• On October 10th 2008, the FTSE 100 fell by 9%. • On that day the banking shares themselves fell

by around 20%, indeed HBOS was down 23%.• By way of contrast, stocks which would be

considered to be defensive, that is stocks which are not subject to the vagaries of the market and the economy as a whole, were much less affected. For example – Cadbury – (don’t we all eat chocolate in a crisis!) was only down 2%

• A diversified investor would have enjoyed better returns than one invested 100% in banking stocks

Hypothetical Example

• Assume that Mr Finance has £1000 invested in Gas U Like - a new competitor to British Gas

• The expected return over the next year is 10% and the historic standard deviation of returns is 12%

• To diversify his portfolio, increase expected return and reduce risk, Mr Finance is considering reducing his holding in Gas U Like and investing £500 into an exciting new stock called Bagomania which sells designer luggage through the internet

• Bagomania is expected to return 15% over the next year with a standard deviation of 20%

• If Mr Finance wants to reduce risk, is he insane?

Not necessarily..

• Designer luggage retailers perform well in boom periods

• Utilities-such as gas providers – perform well during steady growth periods or in a downturn

• Gas U Like and Bagomania are, infact, negatively correlated, the correlation coefficient is -0.25, that is, their returns partly offset each other

Risk and Return – Before and After

InvestmentExpected Return

Standard deviation

Gas U Like £1,000 10% 12%Total £1,000 10% 12%

InvestmentExpected Return

Standard deviation

Bagomania £500 15% 20%Gas U Like £500 10% 12%Total £1,000 13% 10%

So Mr Finance is actually quite astute!

Unique Risk

• The risk that can be eliminated by diversification is called unique risk, specific to a company.

• Unique risk can result in good outcomes or bad outcomes with equal probability, therefore the expected return is zero

• Market risk stems from economy wide factors and cannot be eliminated by diversification