8. International Financial Allocation

Transcription

1 8. International Financial Allocation An Example and Definitions... 1 Expected eturn, Variance, and Standard Deviation.... S&P 500 Example... The S&P 500 and Treasury bill Portfolio... 8.S. 10-Year Note eturns S&P 500, 10 Year Treasury Note, and Treasury Bill Portfolios Covariance Correlation International Diversification... 1 Optimal Portfolio... 7 No Short Sales Incremental Project Choice Note eturn Appendix eferences Table 1 Correlation Matrix ( )... Table Covariance Matrix ( )... 3 Figure 1 SampleAverage Normal Distribution... 5 Figure S&P 500 and Treasury bill Portfolio Figure 3 Portfolio Standard Deviation eduction with Diversification... 6 Figure 4 Efficient Portfolio Frontier with Short Sales... 8 Figure 5 - Efficient Frontier with iskless Borrowing and Lending Figure 6 - Efficient Portfolio Frontier without Short Sales

2 8. International Financial Allocation Classnotes Jim Bodurtha Over recent years, it has become relatively easy to invest internationally. International mutual funds have proliferated. Financial markets are becoming more and more integrated. We fast approach 4 hour markets centered in New York, Tokyo and London. A major reason behind this change has been the erosion of financial market regulatory barriers. In many cases, this growth has been beneficial to investors. A likely source of these benefits is the increased diversification and risk sharing available across many countries' markets. If the investor can earn the same expected return on an international portfolio as a domestic one, but with lower risk, that investor should pursue the benefits of international diversification. Nonetheless, the partial elimination of market barriers may not benefit all investors, or even benefit investors in aggregate. Potential for monopoly, oligopoly, and potential fraud heighten these concerns, we put aside the concerns to see how the investor's risk-return trade-off is affected by additional international investment opportunities. The rising tide of world financial market integration seems inexorable. It is no longer a question of whether or not to deal in these markets, but how. We now turn to develop some vocabulary and tools for this analysis. An Example and Definitions As is common in finance, we will assume that investors are risk averse. This assumption implies that a trade-off exists between expected return and risk. To identify this trade-off, we need to estimate expected return and risk. We will use the sample mean and variance over a historical period for this purpose. Importantly, this sampling should be done so that the return and risk manifest in the historical sample period are consistent with what is reasonably expected to occur in the future. For example, should the world move to fixed exchange rates, sampling estimates of expected -1-

3 exchange rate movements and their risk during a floating rate period will generally be non-sensical. The economic characteristics of the sample period should not greatly differ a priori from the characteristics of the economy as it is expected to evolve. Otherwise, some adjustments must be made to the estimates of expected return and risk Expected eturn, Variance, and Standard Deviation. Putting aside the problem of identifying the appropriate sampling period, we identify the return on the i th asset in period t as it. We have a T returns in the estimation period. The return is the proportional change in price of the i th asset, Pit, from period t-1 to period t, or it P P P it it1 it1 As the estimate of the i th asset expected return, we will use the sample average.. T i t1 Sample variance is the weighted sum of the squared deviations of the realized returns from their sample average. We will also be concerned with the sample standard deviation of returns, σi. The sample standard deviation is just the square root of the sample variance. The formula for the sample variance is the following: 1 T it T i it i t 1 T S&P 500 Example The calculation of these statistics can be illustrated for the index level of the Standard & Poor's 500 (S&P 500). This index is made up by value weighting the prices of five hundred of the largest nited States' corporations in terms of value. This index represents a well-diversified portfolio of.s. and multinational 1 This estimate, which is scaled by the total observation count, T, is the maximum likelihood estimate. An alternative unbiased estimate divides the sum of squared return deviations from the average by T-1. --

4 equities. We will assume that the S&P 500 is a portfolio that can be bought. With the development of exchangetraded funds, ETF, this assumption is quite tenable. As a portfolio, the S&P 500 is often used as the benchmark for evaluating.s. mutual fund performance. Generally, few mutual fund managers outperform the S & P 500 portfolio consistently. As we move on in our evaluation of international diversification, this portfolio will be one of our first investments. To apply the estimators we have defined in making an investment decision, let us suppose that it is December 015. We also assume that we expect 016 S&P 500 returns to be similar to those earned over the last twelve years. To see how the sample mean, variance of the sample mean, and variance of returns are calculated, consider the following series of S&P 500 index values for (including dividends). The subscript (for.s.) will replace the general subscript i to indicate prices, returns, the mean, variance, and standard deviation of the S&P 500 index portfolio. Date Value t t 1/31/ /31/ /30/ /9/ /31/ /31/ /31/ /31/ /31/ /31/ /31/ /31/ /31/ = ˆ ˆ If we expect 016 to be, on average, like the first twelve years of the decade, then buying -3-

5 a portfolio of the S&P 500 stocks should average an annual return of 6.69%. A $1000 investment would grow to $ on average. However, this average is not equal to the true expected value. Like the next period return, the average has sampling error. The variance of the average return estimate is always smaller than the variance of the returns themselves. The variance of the sample expected return decreases as the number of observations increases. Particularly, the sample expected return variance is ˆ ˆ T, and i i ˆ ˆ T /1 = The true expected return will not be equal to the sample average. However, statistics can give us a confidence interval for how far the sample average is from the actual expected return. As the number of observations increase, sample averages converge relatively quickly to the familiar bell shaped normal distribution about the actual expected value. For roughly 50 or more return observations, we can state with 95% confidence that the sample average is within two standard deviations of its expectation or true value. With only 1 observations, about.15 sample standard deviations should bracket the 95% confidence interval. The sample standard deviation is the square root of the sample expected return variance, and for the S&P 500, The true expected return should be between and with 95% certainty. If the number of observations we had used to estimate our sample average and variance were greater than 1, the number of sample standard deviations bracketing the 95% interval would drop toward two. The sample average standard deviation should also likely decrease in this case. -4-

6 9 Figure 1 Sample Average - Normal Density nfortunately, our problem of international investment does not provide us with an investment opportunity in the average S&P 500 portfolio return. Our investment will yield the 016 return itself. To see how confident we should be in our forecast of the 016 return, an additional assumption will have to be made about the distribution of the time series of returns. We will assume that these returns are also drawn from the bell-shaped normal distribution, and that the only information of relevance in past returns is the average and variance. This second assumption implies that a particular sequence of up and down returns carries no information on the expected return beyond what is summarized in the average. However, our potential returns distribution will have two sources of error. The first source is the variability of the returns themselves as represented by the variance or standard deviation. With only 1 observations on returns from 004 on, we cannot be very confident that the sample average is the true -5-

7 expected value. Therefore, the potential error in our 016 return forecast has two parts. The two parts are the variance of returns,, and the variance of the sample average, forecast variance. For an i th asset, this variance is defined as the following: 11 T f i i i T i ˆ. This risk measure is called the As the number of returns observations on returns increases, the forecast variance decreases. Its lower limit is the variance of returns themselves, and, and as T grows large, f i i f i i The standard deviation of the forecasted return will just be the square root of the forecast variance. For the S&P 500 portfolio indexed by the subscript, we have: T = , f f Based on the assumption of normally distributed returns, and this estimate of the forecast standard deviation, the % returns confidence interval for the sample mean of 6.69% is the following: -30.0% < realized return < 43.58% In turn, if we invested $1000 in the S&P 500 portfolio under our scenario for 016, we should be 95% sure that we will end up with cash in the following range: $ < realized cash < $ For an actual investment in the S&P 500 portfolio, it may be unreasonable to assume that the period is representative of what to expect for 016. It is probably more reasonable to use a longer time period to estimate the annual risk and expected return on this, or any, investment. As an alternative, we will use S&P 500 portfolio returns to estimate expected risk and return. This sample period contains 55 S&P 500 portfolio returns. The average, variance, and standard deviation -6-

8 estimates for this period are the following: = f ˆ f ˆ elative to these sample statistics for , we can see that the sub-period included relatively worse years in terms of average return, and, marginally, in terms of variance. The longer sample period may give fairly good estimates of expected return and risk. This likelihood will be greater if any cycles in returns that are fractions of the 55-year sample duration were to exist. Our new estimates could, effectively, average out the shorter-term cyclical changes. However, longer term cycles would not be captured. Furthermore, if cycles do exist which we can estimate, we should condition our expected risk and return on the point that we are in in the cycle. Equity returns do not seem to follow obvious cycles, and we will not treat their potential existence. Exchange rates may follow cycles, and interest rates can be related over time. The implications of this more complicated situation for estimation and risk won t be discussed. More importantly, and as we ve addressed in discussing the representative sample period, the average return may not be representative of what is likely to occur. To address this situation, we estimate the random component of returns as the excess return over the riskless Treasury bill return. Since the average Treasury bill return from 1961 to 015 was 4.86%, the average S&P 500 excess return is 5.98%. As the excess return is the difference of two returns, the associated variance estimate will also differ. We have = f ˆ f ˆ 16.31% At any point in time, the expected return estimate is simply the current riskless rate for the investment period plus the excess expected return on the investment. Since the riskless rate at the end of 015 was 0.61%, the 016 S&P 500 expected annual return estimate would be 6.59%. We use the sample return average and Though are estimates include estimation risk, f ˆ, we omit the forecast symbol in ˆ. -7-

9 standard deviation as our estimates of expected return and risk, respectively. Therefore, our 95% confidence interval for 016 return forecast will be roughly bounded by two sample standard deviations around the average return. The 95% confidence range around the 6.59% expected return is -6.03% < realized return < 39.0% The 95% confidence interval for a $1000 investment in this portfolio is: $739.7 < realized cash < $ Another way to interpret this 95% confidence interval is to recognize that if we are 95% sure that the $1000 investment will net between $739.7 and $139.03, then we face a.5% chance of netting less than $739.7, and a.5% chance that we net more than $ From a risk management perspective the chance of netting more than $ is not troubling. However, the.5% chance of having less than $739.7 is troubling. The S&P 500 and Treasury bill Portfolio Let's assume that we are only willing to bear a.5% of having less than 90% of our investment in a year. This situation is equivalent to allowing a return below -10% with.5% chance. On a $1,000 investment, we want only a.5% chance of losing more than $100, and having less than $900. What can we do to lower our risk from what we have by only investing in the S&P 500 portfolio? We can diversify. Some alternatives that are available are nominally riskless.s. Treasury bills, other equities and bonds. We will first allow diversification into nominally riskless.s. Treasury bills and.s. Treasury notes. These investment opportunities will allow us to better our S&P 500 investment riskreturn trade-off. Subsequently, we will admit investments in international equity indices and bonds. These investment opportunities will further generate diversification gains. At the end of 015, the one year Treasury bill yield was 0.61%. The T-bill yield will be -8-

10 represented by the symbol. A $1000 T-bill purchase will net approximately $ The $900 maximum loss investment value is met by this investment. However, anyone who is not infinitely risk averse is willing to bear some risk. More risk is borne to attain a higher expected return stocks. We try 1/5 of our money in T-Bills. We will define w to be the proportion of our total investment that is in invested the S&P 500 portfolio, w = 0.. The amount invested in T-bills is the remainder, 1-w = 0.8. This leads us to define the portfolio expected return, w 1 w p p, to be the following: With the 0% investment in the S&P 500 portfolio, the expected portfolio return for 016 will be: Expected Portfolio eturn 0.8*0.61% 0.*6.59% % The $1000 investment is expected to net: Expected Cash Flow $ $ To see if we have achieved our risk target, this portfolio investment's variance is lowered relative to buying the S&P 500 portfolio. However, this investment is obviously more risky than T-bills. The formula for the variance of a portfolio made up of one risky asset and one riskless asset is w * = 0. * = , 3.6% f p f f p For this portfolio the 95% confidence intervals are % < realized return < 8.330% $95.84 < realized cash flow < $ We see that this portfolio has less than a.5% chance of having less than $ However, we can still bear still more risk in order to get a little higher expected return. To show how to attain the target risk level with the highest level of expected return in one calculation, remember that the $900 safety level earned on a $1000 investment is equivalent to a -10% return. Therefore, we want to ensure that returns 3 Though bills are priced on a discount basis, we simplify analysis slightly by using a simple interest basis. -9-

11 less than -10% have only a.5% probability. The -10% return should be two standard deviations below the expected portfolio return to attain the target. Algebraically, we can state this requirement as the following: p p 0.10 L.5%, the loss allowable with.5% To get the best risk-return trade off on the portfolio investment, we want this relationship to hold exactly. Substituting for the portfolio average return and standard deviation, we have the following: w 1w w ˆ L f.5% L % w ˆ u f * The expected return on this portfolio is.99%. Therefore, we should invest $ in Treasury bills, and $ in the S&P 500 portfolio. Given a risky portfolio and a riskless asset to invest in, this safety-first rule can be used to generate the maximum expected return for a given level of risk. This risk was represented by the return associated with a.5% chance of losing 10% or more on the $1000 investment. Other levels of confidence are treated similarly. An alternative way to look at the risk management problem is to recognize that a direct relationship exists between portfolio risk and expected return. Based on the definition of expected return and standard deviation, we have: w p, and, or k, k p p p p A direct linear relationship exists between portfolio risk and expected return when the two potential investments are a risky portfolio, and the riskless Treasury bill. Figure portrays the associated risk-return trade-off for our choice. The way to interpret the graph is to view it as representing feasible choices. Any point on the line one can be chosen. However, more expected return can only be had with -10-