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Investments and Portfolio Analysis

Investments and Portfolio Analysis. This lecture: Real vs Nominal Interest Rate Risk & Return, and Portfolio Mechanics. Real vs. Nominal Rates and Risk. Intuitively real rate = nominal rate - expected inflation Formally Rate guarantees nominal or real? expectations vs . realizations

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Investments and Portfolio Analysis

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  1. Investments and Portfolio Analysis • This lecture: • Real vs Nominal Interest Rate • Risk & Return, and Portfolio Mechanics BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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

  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

  4. Real vs. Nominal Risk • Risk • volatility vs. downside • Risk-free rate • Risk premium for asset i • E[Ri] - Rf • Excess return • Ri - Rf Bahattin Buyuksahin, JHU Investment and Portfolio Analysis

  5. Real vs. Nominal Rate Determinants • Determinants of the real rate • supply of funds by savers • demand of funds by businesses • government’s 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

  6. Equilibrium Real Rate of Interest • Determined by: • Supply • Demand • Government actions • Expected rate of inflation Bahattin Buyuksahin, JHU Investment and Portfolio Analysis

  7. Figure 5.1 Determination of the Equilibrium Real Rate of Interest Bahattin Buyuksahin, JHU Investment and Portfolio Analysis

  8. Equilibrium Nominal Rate of Interest Bahattin Buyuksahin, JHU Investment and Portfolio Analysis 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:

  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

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  11. Ending Price = 48 Beginning Price = 40 Dividend = 2 HPR = (48 - 40 + 2 )/ (40) = 25% Rates of Return: Single Period Example

  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. BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  14. Continuous Compounding(Page 77) • In the limit as we compound more and more frequently we obtain continuously compounded interest rates • $100 grows to $100eRTwhen invested at a continuously compounded rateRfor time T • $100 received at time Tdiscounts to $100e-RTat time zero when the continuously compounded discount rate is R BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  17. Future Value of Money BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  18. Future Value and Interest Earned • Future Value and Interest Earned BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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. BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  20. Conversion Formulas • What if we want to find the equivalent interest rate, if compounding is done continuously? Define Rc: continuously compounded rate Rm: equivalent rate with compounding m times per year BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  21. Expected Return • Expected return formulas • expected return on individual asset • 1 period considered with a number of statesdenoted s • expected return based on time seriesfrom t=1 to T • expected return on portfolio of N assets BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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) 0.4 0.6 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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) BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  24. Asset Risk • Risk formulas • variance of return on individual asset • 1 period considered with a number of statesdenoteds • expected return based on time seriesfrom t=1 to T • risk of portfolio of N assets each with weight wi BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 SOW’s (states of the world) • state: sbadgood • wealth: W(s) $80 $150 • probability: p(s) 0.4 0.6 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  26. Asset Risk 2 • Computing risk in practice • example 2:stdev. of several managers’ portf. Returns • average across observations from a sample BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  27. Asset Risk & Return: Historical data • 1926-2005 (BKM7 Table 5.3) • security: small stocks large stocks LT 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% BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 st= ln[St] st = st-1 + t t ~ (,2) st = Rt = t • If the distribution of t is constant over time t (and Rt) are independently and identically distributed (i.i.d.) BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 (Rt, Rt-i) > 0 movements in one direction one day are followed by movements in the same direction  trend if (Rt, Rt-i) < 0 movements in one direction one day are followed by movements in the opposite direction  mean reversion BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  31. Stylized facts of asset returns:autocorrelation • Daily returns have very little autocorrelation (Rt, Rt-i)  0 for i = 1,2,3, … T  Returns are impossible to predict from their own past  Market efficiency!!! BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  33. Stylized facts of asset returns:skewness BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  35. Stylized facts of asset returns:kurtosis BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  36. Descriptive Statistics BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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. BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  38. Standard Deviations BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  39. Stylized facts of asset returns:squared returns • Squared returns  variance 2 = E(x2) – [E(x)]2 st = Rt = t t ~ (0,2) E(Rt) = 0  2 = E(x2) • 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. (R2t, R2t-i) > 0 for i = 1,2,3, … T BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  40. AutoCorrelation Functions: GE BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  41. ACF: MSFT BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  42. ACF: IBM BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  43. ACF: S&P 500 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  44. 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) BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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(Ri,t, Rj,t) = E(Ri,t, Rj,t) Covariance between asset returns may be estimated by the product of the returns BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  47. Monthly returns BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  48. Unconditional distribution daily returns S&P500 BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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.  BahattinBuyuksahin, JHU Investment and Portfolio Analysis

  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: "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) Bahattin Buyuksahin, JHU Investment and Portfolio Analysis

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