Interest Rates, Risk Premiums, Risk Aversion, and Portfolio Arithmetic

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```					 Interest Rates, Risk
Premiums, Risk
Aversion, and Portfolio
Arithmetic
B,K & M Chapter 5, 6
Group Project 3
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
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- infl
-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:
1 r
r         1
1  inf
although the approximation is good if the inflation rate is not too high
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
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 some 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
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
Fisher Effect
In fact, short-term realized real rates are quite variable (Graph below)
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
Risk and Risk
Premiums
Holding Period Returns (HPR)

When is an investment 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.
The Historical
Record
TBills, CPI, and Real Rates
Historical Equity Returns Around the
World (1900-2000)
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
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?
-  2  .002(100,000  220  20) 2  .998(220  20) 2  19,961600
- The standard deviation () is therefore 4467.8
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?
Risk, Risk Aversion,
and Portfolio Risk
and Return
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
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 investor’s choice will depend upon his/her attitude toward risk
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:

U  E[r ]  .5 A r2
where U is the utility value, A is the investors degree of risk aversion,
E[R] is the expected rate of return on the portfolio, and r2 is the
variance of the rate of return on the portfolio
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)
- U/A < 0
- This representation of utility “ignores” higher moments of
the return distribution such as skewness and kurtosis to
simplify the math
- Investors probably prefer skewed distributions with long
positive tails
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
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
Empirical Evidence on U’s

 Very strong evidence that investors prefer more to less

 Very strong evidence that investors are risk averse (A>0)

 Some view casino gambling that has negative expected
return as consumption rather than investment
Portfolio Returns
Portfolio Returns
 To compute the return on a portfolio, use the same formula you use for the return
on a single asset:      P D P
rAt 
At      At           At 1

PAt 1
is the period t return on asset A
 The return on the portfolio is the weighted average of the individual security
returns:
rp  wA rA  wB rB  wC rC    …

 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:
T
E[rp ]   wi E[ri ]
i 1
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):              T
(ri  r )
2
 i2  
t 1   T 1
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?
Portfolio Variance
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
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:
 r2  E[rp  E[rp ]]2
p

 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
Portfolio Variance
 The sample covariance of stock returns is the average of the
products of the deviations from the sample means:
T      (rAt  rA )( rBt  rB )
 A, B  
ˆ
t 1           T 1

 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:    A, B where  1    1
ˆ
ˆ
 A B
A, B
A, B
ˆ ˆ
Portfolio Variance
   Note that:
1) The covariance and the correlation coefficient have the same
sign
2) A,B > 0 when A and B tend to move together
3) A,B < 0 when A and B tend to move in opposite directions
4) A,B = 0 when A and B are independent
5) The larger |A,B |, the more closely related are the returns of A
and B
6) 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
7) Finally, note that covA,B = A,BAB. We will use this later on
Portfolio Variance
 Following is the formula for the variance of portfolio returns in a
form you can use:
N    N                  N   N
 r2   w j wi ij
p
 r2   w j wi i j ij
p
i 1 j 1              i 1 j 1
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
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

weight         wa                 wb                 wc
weight STOCK           A                  B                  C

wa       A      Cov(ra,ra) = a2      cov(rb,ra)         cov(rc,ra)

wb       B         cov(ra,rb)      cov(rb,rb) = b2      cov(rc,rb)

wc       C         cov(ra,rc)         cov(rb,rc)      cov(rc,rc) = c2
Portfolio Variance
 Each element gives the covariance of returns for the intersection of
the column’s stock and the row’s stock. (The diagonal from the NW
to the SE is the covariance between each stock and itself. This is, by
definition, the stock’s 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
Portfolio Variance
 Note that the formula for a two asset risky portfolio is therefore as
follows:
 p  w12 12  w2 2  2w1w2 cov1, 2
2              2 2

 p  w12 12  w2 2  2w1w2 1 2 1, 2
2              2 2

 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).
The Case of
Independent Stock
Returns: An Example
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?
1 2 1 2 1 2 1
 p  ( .3  .3  .3 ) 2 = .17
9    9    9

(Note that all covariance terms are zero by assumption)
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:
.32 1
 p  ( )2
N
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)

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