INVESTMENTS AND PORTFOLIO ANALYSIS This lecture: Real vs Nominal Interest Rate Risk & Return, and...
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INVESTMENTS AND PORTFOLIO ANALYSIS This lecture: Real vs Nominal Interest Rate Risk & Return, and Portfolio Mechanics BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 1
INVESTMENTS AND PORTFOLIO ANALYSIS This lecture: Real vs Nominal Interest Rate Risk & Return, and Portfolio Mechanics BAHATTIN BUYUKSAHIN, JHU INVESTMENT
INVESTMENTS AND PORTFOLIO ANALYSIS This lecture: Real vs
Nominal Interest Rate Risk & Return, and Portfolio Mechanics
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 1
Slide 2
REAL VS. NOMINAL RATES AND RISK Intuitively real rate = nominal
rate - expected inflation Formally Rate guarantees nominal or real?
expectations vs. realizations Taxes BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 2
Slide 3
REAL VS. NOMINAL RATES Intuitively real rate (r) = nominal rate
(R) - expected inflation (i) r R - E[i] example: negative real
rates vs. nominal rates? Formally (1+R) = (1+r) (1+ E[i]) Rate
guarantees nominal or real? expectations vs. realizations BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 3
Slide 4
REAL VS. NOMINAL RISK Risk volatility vs. downside Risk-free
rate Risk premium for asset i E[R i ] - R f Excess return R i - R f
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 4
Slide 5
REAL VS. NOMINAL RATE DETERMINANTS Determinants of the real
rate supply of funds by savers demand of funds by businesses
governments net supply/demand of funds Determinants of the nominal
rate nominal rates as predictors of inflation real rate volatility
historical record BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 5
Slide 6
EQUILIBRIUM REAL RATE OF INTEREST Determined by: Supply Demand
Government actions Expected rate of inflation BAHATTIN BUYUKSAHIN,
JHU INVESTMENT AND PORTFOLIO ANALYSIS 6
Slide 7
FIGURE 5.1 DETERMINATION OF THE EQUILIBRIUM REAL RATE OF
INTEREST BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS
7
Slide 8
EQUILIBRIUM NOMINAL RATE OF INTEREST As the inflation rate
increases, investors will demand higher nominal rates of return If
E(i) denotes current expectations of inflation, then we get the
Fisher Equation: BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 8
Slide 9
TAXES Problem no inflation adjustment for taxes Intuitively tax
code hurts after-tax real rate of return Formally R(1-t) - i =
r(1-t) - i.t Historical record BAHATTIN BUYUKSAHIN, JHU INVESTMENT
AND PORTFOLIO ANALYSIS 9
Slide 10
ASSET RISK AND RETURN HPR = Holding Period Return r = capital
gain yield + dividend yield HPR = Holding Period Return P0 =
Beginning price P1 = Ending price D1 = Dividend during period one
assumptions dividend paid at end of period no reinvestment of
intermediate cash-flows BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 10
Slide 11
Ending Price =48 Beginning Price = 40 Dividend = 2 HPR = (48 -
40 + 2 )/ (40) = 25% RATES OF RETURN: SINGLE PERIOD EXAMPLE
Slide 12
TYPES OF RATES Treasury rates (the rates an investor earns on
Treasury bills or bonds) LIBOR (London Interbank Offered Rate)
rates: rate of interest at which the bank or other financial
institutions is prepared to make a large wholesale deposits with
other banks. LIBID (London Interbank Bid Rate) the rate at which
the bank will accept deposits from other banks. Repo (Repurchasing
Agreement) rates: The price at which securities are sold and the
price at which they are repurchased is referred to as repo rate.
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 12
Slide 13
MEASURING INTEREST RATES The compounding frequency used for an
interest rate is the unit of measurement The difference between
quarterly and annual compounding is analogous to the difference
between miles and kilometers BAHATTIN BUYUKSAHIN, JHU INVESTMENT
AND PORTFOLIO ANALYSIS 13
Slide 14
CONTINUOUS COMPOUNDING (PAGE 77) In the limit as we compound
more and more frequently we obtain continuously compounded interest
rates $100 grows to $ 100e RT when invested at a continuously
compounded rate R for time T $100 received at time T discounts to $
100e -RT at time zero when the continuously compounded discount
rate is R BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 14
Slide 15
MEASURING INTEREST RATE Effect of the compounding frequency on
the value of $1000 at the end of 10 year when the interest rate is
5% per year BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 15 Compounding frequencyValue of $1000 at the end of 10
year Annually (m=1)1628.895 Semi-annual (m=2)1643.616 Quarterly
(m=4)1643.619 Monthly (m=12)1647.009 Weekly (m=52)1648.325 Daily
(m=365)1648.665 Continuous1648.721
Slide 16
EFFECT OF COMPOUNDING FREQUENCY Effect of compounding
frequency: How much you should invest in order to get $1000 at the
end of 10 year when the interest rate is 5% per year BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 16
Slide 17
FUTURE VALUE OF MONEY BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 17
Slide 18
FUTURE VALUE AND INTEREST EARNED Future Value and Interest
Earned BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS
18
Slide 19
FREQUENCY OF COMPOUNDING Interest rates are usually stated in
the form of an annual percentage rate with a certain frequency of
compounding. Since the frequency of compounding can differ, it is
important to have a way of making interest rates comparable. This
is done by computing effective annual rate (EFF), defined as the
equivalent interest rate, if compounding were only once per year.
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 19
Slide 20
CONVERSION FORMULAS What if we want to find the equivalent
interest rate, if compounding is done continuously? Define R c :
continuously compounded rate R m : equivalent rate with compounding
m times per year BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 20
Slide 21
EXPECTED RETURN Expected return formulas expected return on
individual asset 1 period considered with a number of states
denoted s expected return based on time series from t=1 to T
expected return on portfolio of N assets BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 21
Slide 22
EXPECTED RETURN 2 Computing expected returns in practice
calculate by hand or use Excel built-in functions example 1:
expected value of a gamble state:sbadgood wealth:W(s)$80$150
probability: p(s)0.40.6 BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 22
Slide 23
EXPECTED RETURN 3 Risk premium vs. Excess return Excess return
= realized HPR - risk free rate Risk premium = expected HPR - risk
free rate= expected excess return risk-free asset inflation holding
period vs. investor horizon sources of risk business risk
(operations) vs. financial risk (leverage) BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 23
Slide 24
ASSET RISK Risk formulas variance of return on individual asset
1 period considered with a number of states denoted s expected
return based on time series from t=1 to T risk of portfolio of N
assets each with weight w i BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 24
Slide 25
ASSET RISK 2 Computing risk in practice calculate by hand or
use Excel built-in functions example 1: risk of a gamble average
across expected SOWs (states of the world) state:sbadgood
wealth:W(s)$80$150 probability: p(s)0.40.6 BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 25
Slide 26
ASSET RISK 2 Computing risk in practice example 2: stdev. of
several managers portf. Returns average across observations from a
sample BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS
26
Slide 27
ASSET RISK & RETURN: HISTORICAL DATA 1926-2005 (BKM7 Table
5.3) security:small stockslarge stocksLT bonds mean 17.95% 12.15%
5.68% stdev. 38.71% 20.26% 8.09% Interpreting return distributions
(Figs.5.4 & 5.5) 1 out of 6 years, return could be less than
-7.91% BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS
27
Slide 28
ASSET RETURNS Efficient Market Hypothesis: Current prices
convey all relevant information about the asset Any change in the
asset price is due to new news which are impossible to predict This
implies that changes in asset prices are unpredictable Random Walk
s t = ln[S t ] s t = s t-1 + t t ~ ( , 2 ) s t = R t = t If the
distribution of t is constant over time t (and R t ) are
independently and identically distributed (i.i.d.) BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 28
Slide 29
ASSET RETURNS When returns are uncorrelated (autocorrelation is
zero for all lags), the volatility increases as the horizon
increases, following the square root of time Autocorrelation
function: if (R t, R t-i ) > 0 movements in one direction one
day are followed by movements in the same direction trend if (R t,
R t-i ) < 0 movements in one direction one day are followed by
movements in the opposite direction mean reversion BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 29
Slide 30
STYLIZED FACTS OF ASSET RETURNS: MEAN AND STANDARD DEVIATION
The standard deviation of returns dominates the mean of returns at
short horizons such as daily If we test the null hypothesis that
the mean daily return is equal to zero, we fail to reject it!
Standard deviation of daily return is much higher than the mean
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 30
Slide 31
STYLIZED FACTS OF ASSET RETURNS: AUTOCORRELATION Daily returns
have very little autocorrelation (R t, R t-i ) 0 for i = 1,2,3, T
Returns are impossible to predict from their own past Market
efficiency!!! BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 31
Slide 32
STYLIZED FACTS OF ASSET RETURNS: SKEWNESS Stock market exhibits
occasional very large drops but not equally large up-moves the
distribution of asset returns is not symmetric Skewness: scaled
third moment FX market tends to show less evidence of skewness
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 32
Slide 33
STYLIZED FACTS OF ASSET RETURNS: SKEWNESS BAHATTIN BUYUKSAHIN,
JHU INVESTMENT AND PORTFOLIO ANALYSIS 33
Slide 34
STYLIZED FACTS OF ASSET RETURNS: KURTOSIS The unconditional
distribution of daily returns has fatter tails than the normal
distribution higher probability of large losses Kurtosis: scaled
fourth moment BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 34
Slide 35
STYLIZED FACTS OF ASSET RETURNS: KURTOSIS BAHATTIN BUYUKSAHIN,
JHU INVESTMENT AND PORTFOLIO ANALYSIS 35
Slide 36
DESCRIPTIVE STATISTICS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 36
Slide 37
STANDARD DEVIATION The standard deviation of returns dominates
the mean of returns at short horizons. It is not possible to reject
zero mean in short horizon. Standard deviations seem to be more
volatile over time. It reaches the peak of 11% around the collapse
of Lehman Brothers in September 2008. BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 37
Slide 38
STANDARD DEVIATIONS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 38
Slide 39
STYLIZED FACTS OF ASSET RETURNS: SQUARED RETURNS Squared
returns variance 2 = E(x 2 ) [E(x)] 2 s t = R t = t t ~ (0, 2 ) E(R
t ) = 0 2 = E(x 2 ) Squared returns exhibit positive
autocorrelation The autocorrelations of squared returns tend to be
positive for short lags and decay exponentially to zero as the
number of lags increases. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 39 (R 2 t, R 2 t-i ) > 0 for i = 1,2,3,
T
Slide 40
AUTOCORRELATION FUNCTIONS: GE BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 40
Slide 41
ACF: MSFT BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 41
Slide 42
ACF: IBM BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 42
STYLIZED FACTS OF ASSET RETURNS: LEVERAGE EFFECT Equity and
equity indices display negative correlation between variance and
returns Leverage effect A drop in the stock price will increase the
leverage of the firm and therefore the risk (variance) BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 44
Slide 45
STYLIZED FACTS OF ASSET RETURNS: CORRELATION BETWEEN ASSETS
Correlation between assets is not constant over time i.e. it
changes Empirical evidence shows that assets are more correlated
during crashes!!! Covariance: E(xy) = E[(x E(x)) (y E(y))] if E(x)
= 0 and E(y) = 0 E(xy) = E(x y) Cov( R i,t, R j,t ) = E ( R i,t, R
j,t ) Covariance between asset returns may be estimated by the
product of the returns BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 45
Slide 46
STYLIZED FACTS OF ASSET RETURNS: RETURN HORIZON As the return
horizon increases, the unconditional return distribution changes
and looks increasingly like a normal distribution BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 46
Slide 47
MONTHLY RETURNS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 47
Slide 48
UNCONDITIONAL DISTRIBUTION DAILY RETURNS S&P500 BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 48
Slide 49
RISK AND UNCERTAINTY Risk and uncertainty have a rather short
history in economics The formal incorporation of risk and
uncertainty into economic theory was only accomplished in 1944,
when John Von Neumann and Oskar Morgenstern published their Theory
of Games and Economic Behavior The very idea that risk and
uncertainty might be relevant for economic analysis was only really
suggested in 1921, by Frank H. Knight in his formidable treatise,
Risk, Uncertainty and Profit. BAHATTIN BUYUKSAHIN, JHU INVESTMENT
AND PORTFOLIO ANALYSIS 49
Slide 50
RISK AND UNCERTAINTY Indeed, he linked profits,
entrepreneurship and the very existence of the free enterprise
system to risk and uncertainty. Much has been made of Frank H.
Knight's (1921: p.20, Ch.7) famous distinction between "risk" and
"uncertainty". In Knight's interpretation, "risk" refers to
situations where the decision-maker can assign mathematical
probabilities to the randomness which he is faced with. In
contrast, Knight's "uncertainty" refers to situations when this
randomness "cannot" be expressed in terms of specific mathematical
probabilities. As John Maynard Keynes was later to express
it:Knight "By `uncertain' knowledge, let me explain, I do not mean
merely to distinguish what is known for certain from what is only
probable. The game of roulette is not subject, in this sense, to
uncertainty...The sense in which I am using the term is that in
which the prospect of a European war is uncertain, or the price of
copper and the rate of interest twenty years hence...About these
matters there is no scientific basis on which to form any
calculable probability whatever. We simply do not know." (J.M.
Keynes, 1937)Keynes BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 50
Slide 51
RISK AND UNCERTAINTY Nonetheless, many economists dispute this
distinction, arguing that Knightian risk and uncertainty are one
and the same thing. For instance, they argue that in Knightian
uncertainty, the problem is that the agent does not assign
probabilities, and not that she actually cannot, i.e. that
uncertainty is really an epistemological and not an ontological
problem, a problem of "knowledge" of the relevant probabilities,
not of their "existence". Going in the other direction, some
economists argue that there are actually no probabilities out there
to be "known" because probabilities are really only "beliefs". In
other words, probabilities are merely subjectively-assigned
expressions of beliefs and have no necessary connection to the true
randomness of the world (if it is random at all!). BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 51
Slide 52
RISK AND UNCERTAINTY Nonetheless, some economists, particularly
Post Keynesians such as G.L.S. Shackle (1949, 1961, 1979) and Paul
Davidson (1982, 1991) have argued that Knight's distinction is
crucial. In particular, they argue that Knightian "uncertainty" may
be the only relevant form of randomness for economics - especially
when that is tied up with the issue of time and information. In
contrast, situations of Knightian "risk" are only possible in some
very contrived and controlled scenarios when the alternatives are
clear and experiments can conceivably be repeated -- such as in
established gambling halls. Knightian risk, they argue, has no
connection to the murkier randomness of the "real world" that
economic decision-makers usually face: where the situation is
usually a unique and unprecedented one and the alternatives are not
really all known or understood. In these situations, mathematical
probability assignments usually cannot be made. Thus, decision
rules in the face of uncertainty ought to be considered different
from conventional expected utility. BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 52
Slide 53
RISK AND UNCERTAINTY The "risk versus uncertainty" debate is
long-running and far from resolved at present. As a result, we
shall attempt to avoid considering it with any degree of depth
here. What we shall refer throughout as "uncertainty" does not
correspond to its Knightian definition. Instead, we will use the
term risk and uncertainty interchangeably. BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 53
Slide 54
RISK AND UNCERTAINTY After Knight, economists finally began to
take it into account: John Hicks (1931), John Maynard Keynes (1936,
1937), Michal Kalecki (1937), Helen Makower and Jacob Marschak
(1938), George J. Stigler (1939), Gerhard Tintner (1941), A.G. Hart
(1942) and Oskar Lange (1944), appealed to risk or uncertainty to
explain things like profits, investment decisions, demand for
liquid assets, the financing, size and structure of firms,
production flexibility, inventory holdings, etc. The great barrier
in a lot of this early work was in making precise what it means for
"uncertainty" or "risk" to affect economic decisions. How do agents
evaluate ventures whose payoffs are random? How exactly does
increasing or decreasing uncertainty consequently lead to changes
in behavior? These questions were crucial, but with several
fundamental concepts left formally undefined, appeals risk and
uncertainty were largely of a heuristic and unsystematic nature.
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 54
Slide 55
RISK AND UNCERTAINTY The great missing ingredient was the
formalization of the notion of "choice" in risky or uncertain
situations. Already Hicks (1931), Marschak (1938) and Tintner
(1941) had a sense that people should form preferences over
distributions, but how does one separate the element of attitudes
towards risk or uncertainty from pure preferences over outcomes?
Alternative hypotheses included ordering random ventures via their
means, variances, etc., but no precise or satisfactory means were
offered up. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 55
Slide 56
RISK AND UNCERTAINTY Surprisingly, Daniel Bernoulli's (1738)
notion of expected utility which decomposed the valuation of a
risky venture as the sum of utilities from outcomes weighted by the
probabilities of outcomes, was generally not appealed to by these
early economists. Part of the problem was that it did not seem
sensible for rational agents to maximize expected utility and not
something else. Specifically, Bernoulli's assumption of diminishing
marginal utility seemed to imply that, in a gamble, a gain would
increase utility less than a decline would reduce it. Consequently,
many concluded, the willingness to take on risk must be
"irrational", and thus the issue of choice under risk or
uncertainty was viewed suspiciously, or at least considered to be
outside the realm of an economic theory which assumed rational
actors. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS
56
Slide 57
RISK AND UNCERTAINTY The great task of John von Neumann and
Oskar Morgenstern (1944) was to lay a rational foundation for
decision-making under risk according to expected utility rules. The
novelty of using the axiomatic method - combining sparse
explanation with often obtuse axioms - ensured that most economists
of the time would find their contribution inaccessible and
bewildering. Indeed, there was substantial confusion regarding the
structure and meaning of the von Neumann- Morgenstern expected
utility itself. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 57
Slide 58
RISK AND UNCERTAINTY In the von Neumann-Morgenstern hypothesis,
probabilities are assumed to be "objective" or exogenously given by
"Nature" and thus cannot be influenced by the agent. However, the
problem of an agent under uncertainty is to choose among lotteries,
and thus find the "best" lottery in (X), where (X) is the set of
simple lotteries on X (outcomes). One of von Neumann and
Morgenstern's major contributions to economics more generally was
to show that if an agent has preferences defined over lotteries,
then there is a utility function U: (X) R that assigns a utility to
every lottery p (X) that represents these preferences. BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 58
Slide 59
EXPECTED UTILITY FUNCTION The study of decision making under
uncertainty is a vast subject However, financial applications
almost invariably proceed under the guise of the expected utility
hypothesis: people rank random prospects according to the expected
utility of those prospects. Analytically, this involves solving
problems requiring selecting choice variables to maximize an
expected utility function. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 59
Slide 60
EXPECTED UTILITY FUNCTION where EU(x) is the expected utility
of x; S is the number of possible future state of the world; p j is
the probability that state j occur; and U(x j ) is the utility
associated with the amount of x received in state j. Expected
utility function (E U) ranks risky prospects with an ordering that
is unique up to linear transformation. However, how are we going to
assign probabilities? Is it subjective or objective. CAPM model,
for example treats these probabilities as objective by assuming
that expectations and/or individuals are homogenous. BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 60
Slide 61
EXPECTED UTILITY FUNCTION Such an assumption might be
understandable in general equilibrium framework. But the decision
problems encountered are partial equilibrium. Consider the problem
of determining the cost of the risk. Economic agents choice is
between buying an insurance or investing in a risky capital
project. Let the expected value of one period ahead wealth be E(W
t+1 )= . Observe that is a parameter that permits the certainty
equivalent income of a risky prospect to be defined as -C, where C
is the cost of risk. It follows from the expected utility axioms
that the cost of risk can be calculated as the difference between
the expected value of the risky prospect and the associated
certainty equivalent income: BAHATTIN BUYUKSAHIN, JHU INVESTMENT
AND PORTFOLIO ANALYSIS 61
Slide 62
EXPECTED UTILITY FUNCTION We can estimate cost of risk by
expanding U[ -C] around the first order approximation is Similarly,
the second order approximation for the function U(W t+1 ) is
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 62
Slide 63
EXPECTED UTILITY FUNCTION Remember This gives us This shows us
that the cost of risk will vary across utility functions. This
results also provides theoretical measure of the risk. The measure
of absolute risk aversion, the relative risk aversion as well as
variance of interest rate have an effect on the cost of risk.
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 63
Slide 64
EXPECTED UTILITY FUNCTION Here using Taylor expansion, we
present mean-variance optimization problem. Utility depends
positively on the terminal wealth (maybe interpreted as return on
the asset) but negatively on the variance of terminal wealth. Risk
averse individual prefers higher return but lower risk. If we
introduce third moment preference (skewness), we will be talking
about mean-variance-skewness optimization problem. Question: What
will be third order approximation if you expand the function U(W
t+1 ) around the terminal wealth? Why do we require U >0, U 0?
What are the economic meanings of these derivatives? BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 64
Slide 65
EXPECTED UTILITY FUNCTION Mean-Variance Optimization:
Mean-Variance-Skewness Optimization BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 65
Slide 66
EXAMPLE: PORTFOLIO DIVERSIFICATION Consider now set of outcomes
as price of an asset, lotteries as different assets, which asset
class you will invest? It depends on several factors: Return on the
asset Riskiness of the asset When we speak of the riskiness of an
asset, we are speaking of the volatility of the control over
resources that is induced by holding that asset. From the
perspective of a consumer, concern focuses on how holding an asset
affects the consumer's purchasing power. There are many possible
sources of asset riskiness. For now we focus on currency risk. That
is, we focus on how currency denomination alone affects riskiness.
For example, we may think of debt issued in two different currency
denominations by the U.S. government, so that the only clear
difference in risk characteristics derives from the difference in
currency denomination. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 66
Slide 67
PORTFOLIO DIVERSIFICATION The basic sources of risk from
currency denomination are exchange rate risk and inflation risk.
Exchange rate risk is the risk of unanticipated changes in the rate
at which a currency trades against other currencies. Inflation risk
is the risk of unanticipated changes in the rate at which a
currency trades against goods priced in that currency. BAHATTIN
BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 67
Slide 68
PORTFOLIO DIVERSIFICATION If we consider the uncovered real
return from holding a foreign asset, it is rf = i* + s- So if s and
are highly correlated, the variance of the real return can be
small|in principle, even smaller than the variance of the return on
the domestic asset. Thus in countries with very unpredictable
inflation rates, we can see how holding foreign assets may be less
risky than holding domestic assets. This can be the basis of
capital flight capital outflows in response to increased
uncertainty about domestic conditions. Capital flight can simply be
the search for a hedge against uncertain domestic inflation.
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 68
Slide 69
PORTFOLIO DIVERSIFICATION The notion of the riskiness of an
asset is a bit tricky: it always depends on the portfolio to which
that asset will be added. Similarly, the risk of currency
denomination cannot be considered in isolation. That is, we cannot
simply select a currency and then determine its riskiness. We need
to know how the purchasing power of that currency is related to the
purchasing power of the rest of the assets we are holding. The
riskiness of holding a Euro denominated bond, say, cannot be
determined without knowing its correlation with the rest of my
portfolio. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO
ANALYSIS 69
Slide 70
PORTFOLIO DIVERSIFICATION We will use correlation as our
measure of relatedness. The correlation coefficient between two
variables is one way to characterize the tendency of these
variables to move together. An asset return is positively
correlated with my portfolio return if the asset tends to gain
purchasing power along with my portfolio. An asset that has a high
positive correlation with my portfolio is risky in the sense that
buying it will increase the variance of my purchasing power. Such
an asset must have a high expected rate of return for me to be
interested in holding it. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 70
Slide 71
PORTFOLIO DIVERSIFICATION In contrast, adding an asset that has
a low correlation with my portfolio can reduce the variance of my
purchasing power. For example, holding two equally variable assets
that are completely uncorrelated will give me a portfolio with half
the variability of holding either asset exclusively. When one asset
declines in value, the other has no tendency to follow suit. In
this case diversification pays, in the sense that it reduces the
riskiness of my portfolio. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 71
Slide 72
PORTFOLIO DIVERSIFICATION From the point of view of reducing
risk, an asset that is negatively correlated with my portfolio is
even better. In this case there is a tendency of the asset to
offset declines in the value of my portfolio. That is, when the
rest of my portfolio falls in value, this asset tends to rise in
value. If two assets are perfectly negatively correlated, we can
construct a riskless portfolio by holding equal amounts of each
asset: whenever one of the assets is falling in value, the other is
rising in value by an equal amount. In order to reduce the
riskiness of my portfolio, I may be willing to accept an inferior
rate of return on an asset in order to get its negative correlation
with my portfolio rate of return. BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 72
Slide 73
PORTFOLIO DIVERSIFICATION If we look at an asset in isolation,
we can determine its expected return and the variance of that
return. A high variance would seem on the face of it to be risky.
However we have seen that the currency risk and inflation risk of
an isolated asset are not very interesting to consider. We may be
interested in holding an asset denominated in a highly variable
foreign currency if doing so reduces the variance of our portfolio
rate of return. To determine whether the asset can do this, we must
consider its correlation with our current portfolio. A low
correlation offers an opportunity for diversification, and a
negative correlation allows even greater reductions in portfolio
risk. We are willing to pay extra for this reduction in risk, and
the risk premium is the amount extra we pay. If adding foreign
assets to our portfolio reduces its riskiness, then the risk
premium on domestic assets will be positive. BAHATTIN BUYUKSAHIN,
JHU INVESTMENT AND PORTFOLIO ANALYSIS 73
Slide 74
OPTIMAL DIVERSIFICATION Consider an investor who prefers higher
average returns but lower risk. We will capture these preferences
in a utility function, which depends positively on the average
return of the investors portfolio and negatively on its
variability, U(E(rp); var(rp)). We can think of portfolio choice as
a two stage procedure. First we determine the portfolio with the
lowest risk: the minimum-variance portfolio. Second, we decide how
far to deviate from the mimimum-variance portfolio based on the
rewards to risk bearing. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 74
Slide 75
Let us return to our investor who prefers higher average
returns but lower risk, as represented by the utility function
U(E(rp); var(rp)). Domestic assets pay r = i and foreign assets pay
rf = i* + s- as real returns to domestic residents. The total real
return on the portfolio rp will then be a weighted average of the
returns on the two assets, where the weight is just (the fraction
of the portfolio allocated to foreign assets). rp = rf+(1- )r
Therefore the expected value of the portfolio rate of return is
E(rp) = E(rf) + (1 - )E(r) BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 75
Slide 76
OPTIMAL DIVERSIFICATION And the variance of the portfolio
Var(rp) = 2 var(rf) + 2 (1 - )cov(r, rf ) + (1 - ) 2 var(r)
Consider how to maximize utility, which depends on the mean and
variance of the portfolio rate of return. The objective is to
choose to maximize utility. Max U( E(rf) + (1 - )E(r), 2 var(rf) +
2 (1 - )cov(r, rf ) + (1 - ) 2 var(r)) BAHATTIN BUYUKSAHIN, JHU
INVESTMENT AND PORTFOLIO ANALYSIS 76
Slide 77
OPTIMAL DIVERSIFICATION As long as this derivative is positive,
so that increasing produces and increase in utility, we want to
increase alpha. If this derivative is negative, we can increase
utility by reducing alpha. These considerations lead to the
first-order condition": the requirement that dU/d = 0 at a maximum.
We use the first-order condition to produce a solution for .
BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 77
Slide 78
OPTIMAL DIVERSIFICATION Here RRA = -2U 2 /U 1 (the coefficient
of relative risk aversion) and 2 = var(rf) + var(r)-2cov(r,rf ).
Recalling that E(rf)- E(r) = i* + s e - i = rp; we therefore have
Here is the that yields the minimum variance portfolio (Kouri
1978), so the rest can be considered the speculative portfolio
share. Investors can be thought of as initially investing entirely
in the minimum variance portfolio and then exchanging some of the
lower return asset for some of the higher return asset. They accept
some increase in risk for a higher average return. If the assets
have the same expected return, they will simply hold the minimum
variance portfolio. BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND
PORTFOLIO ANALYSIS 78
Slide 79
RISK AND RISK AVERSION Speculation Considerable risk Sufficient
to affect the decision Commensurate gain Gamble Bet or wager on an
uncertain outcome
Slide 80
RISK AVERSION AND UTILITY VALUES Risk averse investors reject
investment portfolios that are fair games or worse These investors
are willing to consider only risk-free or speculative prospects
with positive risk premiums Intuitively one would rank those
portfolios as more attractive with higher expected returns
Slide 81
TABLE 6.1 AVAILABLE RISKY PORTFOLIOS (RISK-FREE RATE = 5%)
Slide 82
UTILITY FUNCTION Where U = utility E ( r ) = expected return on
the asset or portfolio A = coefficient of risk aversion = variance
of returns
Slide 83
TABLE 6.2 UTILITY SCORES OF ALTERNATIVE PORTFOLIOS FOR
INVESTORS WITH VARYING DEGREE OF RISK AVERSION
Slide 84
FIGURE 6.1 THE TRADE-OFF BETWEEN RISK AND RETURNS OF A
POTENTIAL INVESTMENT PORTFOLIO, P
Slide 85
ESTIMATING RISK AVERSION Observe individuals decisions when
confronted with risk Observe how much people are willing to pay to
avoid risk Insurance against large losses
Slide 86
FIGURE 6.2 THE INDIFFERENCE CURVE
Slide 87
TABLE 6.3 UTILITY VALUES OF POSSIBLE PORTFOLIOS FOR AN INVESTOR
WITH RISK AVERSION, A = 4
Slide 88
TABLE 6.4 INVESTORS WILLINGNESS TO PAY FOR CATASTROPHE
INSURANCE
Slide 89
CAPITAL ALLOCATION ACROSS RISKY AND RISK-FREE PORTFOLIOS
Control risk Asset allocation choice Fraction of the portfolio
invested in Treasury bills or other safe money market
securities
Slide 90
THE RISKY ASSET EXAMPLE Total portfolio value = $300,000
Risk-free value = 90,000 Risky (Vanguard & Fidelity) = 210,000
Vanguard (V) = 54% Fidelity (F) = 46%
Slide 91
THE RISKY ASSET EXAMPLE CONTINUED Vanguard 113,400/300,000 =
0.378 Fidelity 96,600/300,000 = 0.322 Portfolio P 210,000/300,000 =
0.700 Risk-Free Assets F 90,000/300,000 = 0.300 Portfolio C
300,000/300,000 = 1.000
Slide 92
THE RISK-FREE ASSET Only the government can issue default-free
bonds Guaranteed real rate only if the duration of the bond is
identical to the investors desire holding period T-bills viewed as
the risk-free asset Less sensitive to interest rate
fluctuations
Slide 93
FIGURE 6.3 SPREAD BETWEEN 3-MONTH CD AND T-BILL RATES
Slide 94
Its possible to split investment funds between safe and risky
assets. Risk free asset: proxy; T-bills Risky asset: stock (or a
portfolio) PORTFOLIOS OF ONE RISKY ASSET AND A RISK-FREE ASSET
Slide 95
r f = 7% rf = 0% E(r p ) = 15% p = 22% y = % in p(1-y) = % in r
f EXAMPLE USING CHAPTER 6.4 NUMBERS
Slide 96
r c = complete or combined portfolio For example, y =.75 E(r c
) =.75(.15) +.25(.07) =.13 or 13% EXPECTED RETURNS FOR
COMBINATIONS
Slide 97
c =.75(.22) =.165 or 16.5% If y =.75, then c = 1(.22) =.22 or
22% If y = 1 c = (.22) =.00 or 0% If y = 0 COMBINATIONS WITHOUT
LEVERAGE
Slide 98
Borrow at the Risk-Free Rate and invest in stock. Using 50%
Leverage, r c = (-.5) (.07) + (1.5) (.15) =.19 c = (1.5) (.22) =.33
CAPITAL ALLOCATION LINE WITH LEVERAGE
Slide 99
FIGURE 6.4 THE INVESTMENT OPPORTUNITY SET WITH A RISKY ASSET
AND A RISK-FREE ASSET IN THE EXPECTED RETURN-STANDARD DEVIATION
PLANE
Slide 100
FIGURE 6.5 THE OPPORTUNITY SET WITH DIFFERENTIAL BORROWING AND
LENDING RATES
Slide 101
RISK TOLERANCE AND ASSET ALLOCATION The investor must choose
one optimal portfolio, C, from the set of feasible choices
Trade-off between risk and return Expected return of the complete
portfolio is given by: Variance is:
Slide 102
TABLE 6.5 UTILITY LEVELS FOR VARIOUS POSITIONS IN RISKY ASSETS
(Y) FOR AN INVESTOR WITH RISK AVERSION A = 4
Slide 103
FIGURE 6.6 UTILITY AS A FUNCTION OF ALLOCATION TO THE RISKY
ASSET, Y
Slide 104
TABLE 6.6 SPREADSHEET CALCULATIONS OF INDIFFERENCE CURVES
Slide 105
FIGURE 6.7 INDIFFERENCE CURVES FOR U =.05 AND U =.09 WITH A = 2
AND A = 4
Slide 106
FIGURE 6.8 FINDING THE OPTIMAL COMPLETE PORTFOLIO USING
INDIFFERENCE CURVES
Slide 107
TABLE 6.7 EXPECTED RETURNS ON FOUR INDIFFERENCE CURVES AND THE
CAL
Slide 108
PASSIVE STRATEGIES: THE CAPITAL MARKET LINE Passive strategy
involves a decision that avoids any direct or indirect security
analysis Supply and demand forces may make such a strategy a
reasonable choice for many investors
Slide 109
PASSIVE STRATEGIES: THE CAPITAL MARKET LINE CONTINUED A natural
candidate for a passively held risky asset would be a
well-diversified portfolio of common stocks Because a passive
strategy requires devoting no resources to acquiring information on
any individual stock or group we must follow a neutral
diversification strategy
Slide 110
TABLE 6.8 AVERAGE ANNUAL RETURN ON STOCKS AND 1-MONTH T-BILLS;
STANDARD DEVIATION AND REWARD-TO-VARIABILITY RATIO OF STOCKS OVER
TIME