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P.V. VISWANATH
FOR A FIRST COURSE IN INVESTMENTS
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 How are interest rates determined?
 How can we compare rates of return for different
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holding periods?
What is the relation between inflation rates and
interest rates?
How do we analyze a time series of returns?
How do we characterize the distribution of a series of
returns?
How do we construct risk measures when the
probability distribution is not normal?
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 We are interested in determining if there is a
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relationship between risk and return in US capital
markets. If there is one, then it needs to be taken
into account in portfolio construction.
To do this, we need to define what we mean by
return and what we mean by risk.
Let’s take the notion of return.
This is easiest to do when discussing a risk-free
security where the investor puts up his money at one
point in time and then gets a payoff in the next
period.
This return is called an interest rate.
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 The real interest rate is the price of access to real
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resources today as opposed to a point later in time.
The nominal interest rate is the price of access to money
today as opposed to later.
Money, of course, is needed in a money economy to
obtain access to real resources.
However, the total amount of money can change over
time depending upon the actions of the Federal Reserve.
If the supply of money increases over time without a
change in the amount of goods, then there is a larger
amount of money chasing the same amount of goods.
As a result prices increase.
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 Investors are interested in return in terms of
purchasing power.
 However, risk-free securities are risk-free only in
nominal terms.
 To compute the return in nominal terms, we need to
know the change in the purchasing power of the unit
of currency, i.e. the rate of inflation, i.
 The real rate of return r, is approximately equal to R,
the nominal rate of return less the rate of inflation, i.
r=R-i
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 Because of this, nominal interest rates are not the same
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as real interest rates.
If we assume that people care about the consumption of
real goods, then the markets would be expected to fix the
real interest rate.
This is embodied in the Fisher Equation, which says that
the nominal interest rate is simply the real interest rate
plus the expected inflation rate.
This is based on the assumption that the real and
nominal sectors of the economy are independent and
unrelated.
Here’s how interest rates are determined according to
this approach.
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 The equilibrium real rate of interest is the point at
which the Demand Curve for Funds intersects the
Supply curve for loanable funds.
 The Supply curve consists of the supply of funds
from savers, primarily households
 The Demand for funds consists of the demand from
businesses who invest in plant, equipment and
inventories (real assets).
 The government is sometimes a net demander and
sometimes a net supplier of loanable funds.
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 Both suppliers and demanders of capital thus
determine their supply and demand for capital at
different real rates of interest.
 However, since loan contracts are nominal, they
have to contract for a nominal rate of return. To
ensure that the resulting real return is consistent
with what they are looking for, they forecast the
inflation rate and add that to the desired real rate of
return and then come up with the desired nominal
rate of return or nominal rate of interest.
 In this way, they generate their nominal supply and
demand curves.
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 Assuming that investors’ time preference for real resources do
not vary too much over time, changes in the nominal interest
rate will simply track changes in the inflation rate.
 However, this assumes that the inflation rate is easy to predict.
Changes in the money supply are the primary determinant of the
inflation rate and unfortunately, changes in the money supply
can be difficult to forecast.
 Furthermore, there is a certain amount of money illusion.
People think and contract, to some extent, in terms of nominal
prices and nominal quantities. As a result, changes in money
supply could also have an impact on real quantities.
 In addition, money is needed as a medium of exchange. Hence it
has a “real” role in the economy, as well. Hence the nominal side
of the economy could affect the real economy.
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 If there were complete inflation illusion and the nominal
interest rate were constant, the inflation rate would be
uncorrelated with the nominal interest rate.
 On the other hand, if the Fisher Hypothesis held, and if
expected real interest rates were relatively constant, then
nominal interest rates would move one-to-one with inflation
rates.
 Upto 1967, there seems to have been money illusion, since the
correlation of the inflation rate with the nominal T-bill rate
was close to zero (r = -0.17). However, since 1968, the
correlation has increased (r = 0.64). This suggests that
markets are less subject to money illusion.
 The next graph shows the much closer connection between
interest rates and inflation in the post-1967 period.
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 If the Fisher Hypothesis held, and the real interest rate were
constant, then the realized real T-bill rate (i.e. nominal minus
inflation) should be perfectly negative correlated with the inflation
rate, because any increase in inflation would reduce the realized real
rate by an equal amount.
R = r + E(i) = r + i + E(i) – i or
R-i = r + E(i) – i
If r and E(i) are constant, then Corr(R-i , i) should be -1.0
Corr(R-i, i) = Corr(r, i) + Corr(E(i), i) –Corr(i, i) = 0+0-1 = -1.0
 In practice, this correlation has been about -0.44 after 1967,
compared to -1.0 before 1967.
 This suggests that the nominal rates are not keeping pace with
expected inflation. This would happen if real rates and inflation
rates were positively correlated. That is, the real and nominal sectors
of the economy are connected – Corr(r, i) > 0.
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 http://www.crestmontresearch.com/docs/i-rate-
history7.xls
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 Arithmetic and Geometric Averages
 The arithmetic average is the estimated average
return over the next one year.
 The geometric average is the actual yearly return
over the next n years, where n is the number of years
over which the average is computed.
 So if we wanted the average annual return over the
next five years, how should we compute it?
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 http://www.youtube.com/watch?v=GPDH1e-rxlY
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Ibbotson and Matthew Rettick
Arithmetic and Geometric Averages
E(Geom Average) = E(Arith Average) – Variance/2
Watch this youtube video and see what’s wrong with the
arguments made.
 The next graphs will show that bond and equity returns
are volatile. As a result, it’s not sufficient for an investor
to simply look at average returns; it’s necessary to look at
average returns in comparison to volatility.
 We have seen that financial portfolio returns are volatile.
Investors need to consider the average return in
comparison with the risk/volatility of portfolios.
 The Sharpe Portfolio compares the average return on
portfolios over and above the risk-free rate of return as a
multiple of the volatility of this excess return.
 Sharpe Ratio for Portfolios:
[E(R)-Rf]/sd(R)
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Risk Premium

SD of Excess Returns
Note that this is not appropriate for individual assets
 Investment management is easier when returns are
normal.
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Standard deviation is a good measure of risk when returns are
symmetric.
If security returns are symmetric, portfolio returns will be, too.
Future scenarios can be estimated using only the mean and the
standard deviation.
 What if excess returns are not normally distributed?
 Standard deviation is no longer a complete measure of risk
 Sharpe ratio is not a complete measure of portfolio
performance
 Need to consider skew and kurtosis
Skew
Equation 5.19
_ 3


 R  R  
skew  average  ^  


3
 

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
Kurtosis
 Equation 5.20
_ 4


R

R
 

kurtosis  average  ^    3
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
4
 
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
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 If return distributions are not asymmetric, then:
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We should look at negative outcomes separately
Because an alternative to a risky portfolio is a risk-free investment
vehicle, we should look at deviations of returns from the risk-free
rate rather than from the sample average
Fat tails should be accounted for
 LPSD: similar to usual standard deviation, but uses only
negative deviations from rf
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The LPSD is the square root of the average squared deviation,
conditional on a negative excess return.
However, this measure ignores the frequency of negative excess
returns!
 Sortino Ratio replaces Sharpe Ratio
Sortino Ratio = Average excess return/LPSD
 A measure of loss most frequently associated
with extreme negative returns
 VaR is the quantile of a distribution below
which lies q % of the possible values of that
distribution
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The 5% VaR , commonly estimated in practice, is the
return at the 5th percentile when returns are sorted from
high to low.
 Also called conditional tail expectation (CTE)
 More conservative measure of downside risk
than VaR
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VaR takes the highest return from the worst cases
ES takes an average return of the worst cases
This is a version of the LPSD
 Returns appear normally distributed
 Returns are lower over the most recent half of
the period (1986-2009)
 SD for small stocks became smaller; SD for
long-term bonds got bigger
 Better diversified portfolios have higher Sharpe
Ratios
 Negative skew