Summary Investments & Risk management
Chapter 7 Introduction to risk and return
7.1 Over a century of capital market history in one easy lesson
Dimson, Marsh, Staunton (2002) measure 3 historical portfolios of US securities
1. A portfolio of Treasury bills (=maturity <1 year)
2. A portfolio of US government bonds
3. A portfolio of US common stocks
Treasury bills are a safe investment:
- No risk of default
- Short maturity, so stable prices
However, there is still uncertainty about inflation, so no real rate of return.
Long-term government bonds have fluctuating interest rates. Bond price when interest rate and bond price when
interest rate .
Treasury bills provide lowers average return. Common stocks provide most average return and they offer additional
risk premium.
Arithmetic averages and compound annual returns
Arithmetic average return = sum of averages / number of averages
Compound annual return = rate of return taking into consideration the compounding effect of the investment for each
year
If a common stock is €100, there is an equal chance that at the end of the year the stock will be worth €90, €110 or €130. Thus, the
return could be -10%, +10% or +30% (no dividend). The expected return is 1/3 (-10 + 10 + 30) = +10%.
If we discount the expected cash flow by the expected rate of return, we get a value of:
110
PV = =€ 100
1.10
Thus, the expected return of 10% is the correct rate of discount.
If we observe returns over a large number of years and assume that 1/3 will be -10%, +10% and then +30%, we can calculate the
arithmetic average as follows:
−10+10+30
=+10 %
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Meaning that the arithmetic average of returns is correctly measuring the opportunity cost of capital. The average annual compound
return would be:
(.9 x 1.1 x 1.3)1/ 3−1=.088 ,∨8,8 %
This is less than the opportunity cost of capital. Investors would not be willing to invest in a project that offers 8.8% return if they could
get an expected return of 10% in capital markets. The net present value of such a project would be
108.8
NPV =100+ =−1.1
1.1
MORAL: if the cost of capital is estimated from historical returns or risk premiums, use arithmetic averages, not compound annual rates
of return.
Using historical evidence to evaluate today’s cost of capital
If an investment project has the same degree of risk as the market portfolio, what rate should you use to discount this
project’s forecasted cash flows? The currently expected rate of return on the market portfolio (r m ).
r m =r f + risk premium
Even with more than 100 years of data, we can’t estimate the market risk premium exactly, nor can we be sure that
investors today are demanding the same risk reward as 50/100 years ago. There are two reasons that history may
exaggerate the risk premium that investors demand today.
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,Summary Investments & Risk management
1. The US has been among the world’s most prosperous countries. By focusing on equity return in the US, we may
obtain a biased view of what investors expect.
2. US stock prices have outpaced the growth in company dividends or earnings for some years now.
Steady growth (g):
¿1
PV =
(r −g)
¿1
Dividend yield= =r−g
PV
The dividend yield measures the difference between the discount rate and the expected growth rate. If dividend yields
decline, it can be because:
- Investors have increased their forecast of future growth
- Investors are content with a lower expected return
Now, have investors raised their forecast or future dividend growth?
- They may anticipate a forthcoming golden age of prosperity and surging profits
- Companies have increasingly preferred to distribute cash by stock repurchase
The effect of any decline in the expected market risk premium is to increase the realized rate of return. Assume that
the market portfolio dividend is 400 billion and it is expected to grow indefinitely at 6% per year. If the yield on these
stocks is 2%, the expected total rate of return is r (6+2).
¿1 =400 billion
g=.06
r =2 %+6 %=0.08
If you plug these numbers into the constant-growth dividend-discount model the value of the market portfolio is:
400
PV = =20.000 billion
(0,08−0,06)
Suppose that investors now see the stock market as a safer investment than before, so they revise their required risk premium downward
from 7% to 6.5% and the required return from 8% to 7.5%. The market portfolio then becomes:
400
PV = =26.667 billion
(0.075−0.06)
The dividend yield falls to:
¿1 400
= =0.015∨1.5 %
PV 26.667
In other words, a fall of 0.5% in the risk premium demanded would cause a 33% rise in the market value, from 20.000 to 26.667 billion.
The total return to investors including the 2% dividend yield is then 2 + 33 = 35%. With a 1% interest rate, the risk premium earned is 35 –
1 = 34%, much greater than investors expected.
If the above example enters our sample of past risk premiums we may be led to a double mistake:
1. We will overestimate the risk premium that investors required in the past
2. We will fail to recognize that investors require a lower expected risk premium when looking at the future
Diversification and portfolio risk
To estimate discount rates for assets that do not fit simple cases, we need to learn two things:
- How to measure risk
- The relationship between risks borne and risk premiums demanded
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Variance and standard deviation
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Variance ( Rm ) =the expected valueof (Rm−r m)
Where Rm is the actual return and r m is the expected return.
Standard deviation ( σ ) of R m=√ variance(Rm )
Measuring variability
On the right we can see the standard deviations and variances observed
for our three portfolios over the period 1900-2017. Treasury bonds are
the most stable safest. Variability is never stable.
How diversification reduces risk
The market portfolio is made up of individual stocks, so why doesn’t its
variability reflect the average variability of its components? Because
diversification reduces variability. Diversification works because prices of
different stocks do not move exactly together.
Specific risk = The risk that can be eliminated by diversification. Perils that
surround individual companies and immediate competitors.
Market risk = Unavoidable risk. Economywide perils that threaten all
businesses.
If you have only one stock, specific risk is very important; once you have a portfolio of +20 stocks, diversification has
done the bulk of its work.
Calculating portfolio risk
Calculating the expected portfolio return is easy. The hard part is to work out the risk of your portfolio. The exact
procedure for calculating the risk of a two-stock portfolio can be seen below
Stock 1 Stock 2
Stock 1 2
x 1 ❑1
2
x 1 x 2 ❑12=x 1 x2 ❑12 ❑1 ❑2
Stock 2 x 1 x 2 ❑12=x 1 x2 ❑12 ❑1 ❑2 x 22 ❑22
x1, x2 = proportions invested in stocks 1 and 2
2 2
❑1 ,❑2 = variance of stock returns
❑12 = covariance of returns (❑12❑1 ❑1)
❑12 = correlation between returns on stock 1 and 2
The entries in the diagonal boxes depend on the variances of stocks 1 and 2; the entries in the other two boxes depend
on their covariance. For the most part stocks tend to move together, which leads to positive covariance. If they are
unrelated, both the correlation coefficient and the covariance would be zero. If the stocks were to move in opposite
directions, the correlation coefficient and covariance would be negative.
Once all four boxes are complete, we add the entries to obtain the portfolio variance:
Portfolio variace=x12 ❑12+ x 22 ❑22+2( x1 x2 ❑12 ❑1 ❑2)
When there is perfect negative correlation, there is always a portfolio strategy that will completely eliminate risk.
Unfortunately, perfect negative correlation doesn’t really occur between common stocks.
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