An Introduction to Risk and Risk Premiums, and The Historical Record

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An Introduction to Risk and Risk Premiums, and
The Historical Record
B,K M Chapter 5 End-of-chapter problems 1-15
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Determinants of the Level of Interest Rates

• Interest rates, of course, are important inputs
  to many economic decisions
• Forecasting interest rates is difficult, but
  they are determined by the forces of supply and
  demand (as would be expected in any competitive
  market)
• Inflationary expectations are critical since
  lenders will demand compensation for anticipated
  losses in purchasing power

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Real and Nominal Interest Rates

• The real rate of interest is the nominal
  (reported) interest rate reduced by the loss of
  purchasing power due to inflation
• - r is (approximately) R- inf
• Where r is the real interest rate
• R is the nominal interest rate
• inf is the inflation rate
• The exact relationship (when reported rates are
  compounded annually) is given below
• although the approximation is good if the
  inflation rate is not too high

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The Equilibrium Real Rate of Interest

• Supply, demand, and government actions determine
  the real rate while the nominal rate is the real
  rate plus the expected rate of inflation
• The fundamental determinants of the real rate
  are the propensity of households to borrow and to
  save, the expected productivity (profitability)
  of physical capital, and the propensity of the
  government to borrow or save
• In Class Exercise
• - Use demand and supply analysis to predict the
  change in the real rate given an increase in each
  of its above mentioned determinants

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The Equilibrium Nominal Rate of Interest

• As noted, the nominal rate differs from the real
  rate because of inflation
• The Treasury and the Fed have the ability to
  influence short-term interest rates by
  controlling the flow of new funds into the
  banking system, however the influence on
  long-term rates is not always favorable because
  of the potential impact of expansionary monetary
  policy on expected inflation
• It is, of course, an over-simplification to
  speak of a single interest rate because in
  reality there are many rates which depend on term
  to maturity and default risk

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Fisher Effect

• The fisher effect is the relationship between
  real and nominal rates
• The basic intuition is that investors will
  require compensation for inflation in order to
  hold securities whose returns are in nominal
  terms. The expected real rate is thus the nominal
  rate minus expected inflation
• If real interest rates are relatively constant,
  then fluctuations in nominal rates will be due to
  changes in expected inflation

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Fisher Effect

• In fact, short-term realized real rates are quite
  variable (Graph below)

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Taxes, the Real Rate of Interest, and Realized
Returns

• In Class Exercise
• Are you indifferent between earning 10 when
  inflation is 8 and 2 when inflation is 0?
• It is critical to remember that the real
  after-tax rate is (approximately) the after-tax
  nominal rate minus the inflation rate

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Risk and Risk Premiums
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Holding Period Returns (HPR)

• Risk means uncertainty about what your realized
  holding period return will be
• We can quantify the uncertainty using probability
  distributions

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Holding Period Returns (HPR)

• Example Assume there is considerable
  uncertainty with respect to the end of year price
  of an index stock fund which currently sells for
  100, although we expect a dividend of 4
• If the normal growth prevails, then the HPR is
• (110 4 - 100) / (100) .14 or 14
• The expected return is the probability weighted
  average of all possible outcomes

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Holding Period Returns (HPR)

• In the above example, the expected return is
  calculated as follows
• The standard deviation (?) is a measure of risk.
  It is defined as the square root of the variance
• Which in this example is calculated as follows
• Therefore the standard deviation (?) is 21.21

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Holding Period Returns (HPR)
Would this investment be attractive to a risk
averse investor? This will generally depend on
the risk premium it affords, where the risk
premium is the excess of the expected return over
the risk-free rate. The risk-free rate is the
return on competitive risk-free assets such as
T-Bills.
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The Historical Record
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Risk in a Portfolio Context

• A fundamental principle of financial economics
  is that you cannot assess the riskiness of an
  investment by examining only its own standard
  deviation!
• Risk must always be considered in a portfolio
  context, that is, taking into account the
  standard deviation of your entire portfolio after
  adding the asset in question
• Example
• Your 100,000 home will burn down with a prob.
  .002. Your expected loss (due to your home
  burning down) is .002 x 100,000, or 200.
• An insurance policy (no deductible) costs 220

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Risk in a Portfolio Context

• A. What is the expected profit of the
  investment in the policy?
• The expected profit is -20, with an expected
  return of 20/220 or 9.09
• B. What is the standard deviation of profit of
  an investment in the policy?
• The standard deviation (?) is therefore 4467.8

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Risk in a Portfolio Context

• Who wants to buy an asset with a negative
  expected return and a high standard deviation?
• In fact, this may be a valuable addition to a
  portfolio because of its impact on portfolio risk
• In Class Exercise
• - What is the standard deviation of the value of
  the complete portfolio which includes the
  insurance policy?

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Risk, Risk Aversion, and Portfolio Risk and Return
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Risk and Risk Aversion

• Investors avoid risk and demand a reward for
  investing in risky investments
• The proper measure of the risk of an asset is
  the marginal impact of the asset on the riskiness
  of the entire portfolio in which it is held
• A Simple Example
• Assume you have initial wealth of 100,000
• You can invest it in a risky portfolio or in
  risk-free T-Bills

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Risk and Risk Aversion

• The risky portfolio has an expected return of
• .6 x 50 .4 x 25 20
• The risk-free portfolio has an expected return
  of 8
• The risk premium is therefore 20 - 8 12
• The investors choice will depend upon his/her
  attitude toward risk

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Overview

• In a world of certainty, rational choice entails
  choosing the bundle of consumption that maximizes
  utility subject to budget constraint
• In finance we focus on the utility of
  end-of-period wealth (or rate of return given the
  current level of wealth)
• The Utility Function characterizes the
  preferences of an individual investor over the
  distribution of the rate of return on the
  portfolio.
• The individual chooses the portfolio to maximize
  utility.
• An example of a simple utility function follows
• where U is the utility value, A is the investors
  degree of risk aversion, ER is the expected
  rate of return on the portfolio, and ?r2 is the
  variance of the rate of return on the portfolio

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Overview

• If an investor is risk averse
• - He/she prefers a certain outcome to an
  uncertain outcome with the same rate of return
• The utility function is concave (U(W) as a
  function of W)
• This representation of utility ignores higher
  moments of the return distribution such as
  skewness and kurtosis to simplify the math
• In fact, investors probably prefer skewed
  distributions with long positive tails

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Certainty Equivalent Rate

• A risky portfolio utility value is the rate that
  a risk-free portfolio would have to earn to be
  equally attractive to the risky portfolio.
• The risky portfolio is only desirable if its
  certainty-equivalent is equal to or higher than
  the risk-free rate
• A less risk-averse investor would assign a
  higher certainty-equivalent to the same risky
  portfolio
• A risk-neutral investor (A 0) cares only about
  the expected rate of return

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Certainty Equivalent Rate

• Without knowing more about an investor than
  he/she is risk averse, any portfolio to the
  northwest of another portfolio will be
  preferred because it has both higher expected
  return and lower risk. Any portfolio to the
  southeast is dominated and is preferred by the
  mean variance criterion

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Empirical Evidence on Us

• Very strong evidence that investors prefer more
  to less
• Very strong evidence that investors are risk
  averse (Agt0)
• Some view casino gambling that has negative
  expected return as consumption rather than
  investment

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Portfolio Returns
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Portfolio Returns

• To compute the return on a portfolio, use the
  same formula you use for the return on a single
  asset
• is the period t return on asset A
• The return on the portfolio is the weighted
  average of the individual security returns
• If we are concerned with the ex ante expected
  return of a portfolio, the above formula applies
  as well
• The historical average return is often used as a
  proxy for expected return

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The Variance and Standard Deviation of an
individual Security

• The variance and standard deviations are
  measures of the dispersion of returns from its
  expected value
• If we do not know the probability of each state
  of the world, we can estimate the variance of an
  asset using historical date (sample variance)
• where T is the number of time periods of data
• In class exercise
• - Why is the denominator in the above formula T-1
  rather than T?

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Portfolio Variance
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Portfolio Variance

• Variance of the portfolio is more complicated
  because the extent to which the randomness in the
  different securities tends to reduce overall risk
  must be accounted for
• Example Consider 2 stocks, A B. Both have
  identical variances and expected returns. If we
  hold a portfolio that consists of equal weights
  of both stocks, the variance of the portfolio
  will depend upon the extent to which the two
  stock returns move together. If A has below
  normal returns when B has above normal return,
  and vice verse, then it will be possible to form
  a portfolio with very low variance relative to
  the variance of either A or B
• If both tend to be above average or below
  average together, portfolio variance may not be
  appreciably less than the variance of either A or
  B

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Portfolio Variance

• The formula for the variance of a portfolio is
  derived by evaluating the following expectation,
  using the definitions we developed above for the
  return and expected return of a portfolio
• The measure of co-movement which emerges is the
  covariance, where the covariance is the expected
  value of the products of deviations from the
  sample means
• In practice we must estimate the covariance
• We will first see how to estimate the covariance
  between two assets in a portfolio, then we show
  how we use these covariances to estimate the
  portfolio variance

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Portfolio Variance

• The sample covariance of stock returns is the
  average of the products of the deviations from
  the sample means
• What is the relation between the sample
  covariance and the sample correlation coefficient
  (a statistic you may be more familiar with)?
• The sample correlation coefficient of two assets
  is simply the scaled covariance

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Portfolio Variance

• Note that
• The covariance and the correlation coefficient
  have the same sign
• ?A,B gt 0 when A and B tend to move together
• ?A,B lt 0 when A and B tend to move in opposite
  directions
• ?A,B 0 when A and B are independent
• The larger ?A,B , the more closely related are
  the returns of A and B
• The larger is ?A,B , the narrower is the
  cloud formed by plotting the returns of A on
  the horizontal axis, and B on the vertical axis.
  The cloud is a straight line when ?A,B 1
• Finally, note that covA,B ?A,B?A?B. We will
  use this later on

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Portfolio Variance

• Following is the formula for the variance of
  portfolio returns in a form you can use
• Where
• wi is the proportion of the portfolio invested
  in asset i at the beginning of the period
• ?i,j is the covariance of returns between i and
  j
• ?i,j ?i2 if and only if i j
• ?i is the standard deviation of asset i
• ?i,j is the correlation of returns between
  assets i and j

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Portfolio Variance

• Some find matrix notation to be easier. First
  form a matrix with the covariances between the
  assets returns and the portfolio weights
• Below is an example for a 3 asset portfolio

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Portfolio Variance

• Each element gives the covariance of returns for
  the intersection of the columns stock and the
  rows stock. (The diagonal from the NW to the SE
  is the covariance between each stock and itself.
  This is, by definition, the stocks variance.)
• Note as well that the matrix is symmetric for
  each weight and covariance above the diagonal
  there is an equal covariance and weight below the
  diagonal
• Calculating the variance of the portfolio is
  done by multiplying each of the covariances in
  the matrix by the weight at the top of its column
  and the weight at the left side of its row
• You then obtain the variance of the portfolio by
  summing these products

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Portfolio Variance

• Note that the formula for a two asset risky
  portfolio is therefore as follows
• In class exercise What is the variance of the
  three asset portfolio?
• Note that the number of elements is the square
  of the number of assets in the portfolio. How
  many covariance estimates are necessary to
  estimate the variance of a 100 asset portfolio?
• Note as well that as the number of assets in the
  portfolio increases, the relative importance of
  the variance terms diminishes (because the ratio
  of stocks on the diagonal to stocks off the
  diagonal diminishes).

 
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The Case of Independent Stock Returns An Example
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An Example

• Assume returns are independent (?i,j 0 if i is
  not equal to j), and all standard deviations are
  30
• What is the standard deviation of a portfolio of
  three stocks with equal weights?
• (Note that all covariance terms are zero by
  assumption)

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Example (cont.)

• As we increase the number of assets in a
  portfolio, the importance of the variance terms
  diminishes, but that of the covariance term does
  not. If all covariance terms are 0, then the
  standard deviation of the portfolio approaches
  zero as the number of assets becomes large, or in
  the above example
• N ?p
• 10 9.4868
• 100 3.0
• 1000 0.94868
• In fact, most covariances between pairs of
  stocks are positive, so we are not able to ignore
  the covariance terms
• This means that it is virtually impossible to
  reduce portfolio variance to zero (for portfolios
  of risky assets)