investments and portfolio analysis this lecture: real vs nominal interest rate risk & return, and...

INVESTMENTS AND PORTFOLIO ANALYSIS This lecture: Real vs Nominal Interest Rate Risk & Return, and Portfolio Mechanics BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 1

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

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

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

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

EQUILIBRIUM REAL RATE OF INTEREST Determined by: Supply Demand Government actions Expected rate of inflation BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 6

FIGURE 5.1 DETERMINATION OF THE EQUILIBRIUM REAL RATE OF INTEREST BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 7

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

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

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

Ending Price =48 Beginning Price = 40 Dividend = 2 HPR = (48 - 40 + 2 )/ (40) = 25% RATES OF RETURN: SINGLE PERIOD EXAMPLE

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

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

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

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

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

FUTURE VALUE OF MONEY BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 17

FUTURE VALUE AND INTEREST EARNED Future Value and Interest Earned BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 18

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

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

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

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

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

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

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

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

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

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

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

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

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

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

STYLIZED FACTS OF ASSET RETURNS: SKEWNESS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 33

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

STYLIZED FACTS OF ASSET RETURNS: KURTOSIS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 35

DESCRIPTIVE STATISTICS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 36

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

STANDARD DEVIATIONS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 38

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

AUTOCORRELATION FUNCTIONS: GE BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 40

ACF: MSFT BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 41

ACF: IBM BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 42

ACF: S&P 500 BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 43

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

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

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

MONTHLY RETURNS BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 47

UNCONDITIONAL DISTRIBUTION DAILY RETURNS S&P500 BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 48

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

EXPECTED UTILITY FUNCTION Mean-Variance Optimization: Mean-Variance-Skewness Optimization BAHATTIN BUYUKSAHIN, JHU INVESTMENT AND PORTFOLIO ANALYSIS 65

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

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

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

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

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

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

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

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

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

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

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

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

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

RISK AND RISK AVERSION Speculation Considerable risk Sufficient to affect the decision Commensurate gain Gamble Bet or wager on an uncertain outcome

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

TABLE 6.1 AVAILABLE RISKY PORTFOLIOS (RISK-FREE RATE = 5%)

UTILITY FUNCTION Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion = variance of returns

TABLE 6.2 UTILITY SCORES OF ALTERNATIVE PORTFOLIOS FOR INVESTORS WITH VARYING DEGREE OF RISK AVERSION

FIGURE 6.1 THE TRADE-OFF BETWEEN RISK AND RETURNS OF A POTENTIAL INVESTMENT PORTFOLIO, P

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

FIGURE 6.2 THE INDIFFERENCE CURVE

TABLE 6.3 UTILITY VALUES OF POSSIBLE PORTFOLIOS FOR AN INVESTOR WITH RISK AVERSION, A = 4

TABLE 6.4 INVESTORS WILLINGNESS TO PAY FOR CATASTROPHE INSURANCE

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

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%

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

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

FIGURE 6.3 SPREAD BETWEEN 3-MONTH CD AND T-BILL RATES

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

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

r c = complete or combined portfolio For example, y =.75 E(r c ) =.75(.15) +.25(.07) =.13 or 13% EXPECTED RETURNS FOR COMBINATIONS

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

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

FIGURE 6.4 THE INVESTMENT OPPORTUNITY SET WITH A RISKY ASSET AND A RISK-FREE ASSET IN THE EXPECTED RETURN-STANDARD DEVIATION PLANE

FIGURE 6.5 THE OPPORTUNITY SET WITH DIFFERENTIAL BORROWING AND LENDING RATES

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:

TABLE 6.5 UTILITY LEVELS FOR VARIOUS POSITIONS IN RISKY ASSETS (Y) FOR AN INVESTOR WITH RISK AVERSION A = 4

FIGURE 6.6 UTILITY AS A FUNCTION OF ALLOCATION TO THE RISKY ASSET, Y

TABLE 6.6 SPREADSHEET CALCULATIONS OF INDIFFERENCE CURVES

FIGURE 6.7 INDIFFERENCE CURVES FOR U =.05 AND U =.09 WITH A = 2 AND A = 4

FIGURE 6.8 FINDING THE OPTIMAL COMPLETE PORTFOLIO USING INDIFFERENCE CURVES

TABLE 6.7 EXPECTED RETURNS ON FOUR INDIFFERENCE CURVES AND THE CAL

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

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

TABLE 6.8 AVERAGE ANNUAL RETURN ON STOCKS AND 1-MONTH T-BILLS; STANDARD DEVIATION AND REWARD-TO-VARIABILITY RATIO OF STOCKS OVER TIME

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