A project has a 0.8 chance of doubling your investment in a year and a 0.2 chance of halving your investment in a year. Required: What is the standard deviation of the rate of return on this investment? (Round your answer to 2 decimal places. Omit the "%" sign in your response.) Standard deviation %,4.00% was not the correct answer. Below is the chapter reference for this problem: 6.6 Risk of Long-Term Investments p. 175 So far we have envisioned portfolio investment for one period. We have not made any explicit assumptions about the duration of that period, so one might take it to be of any length, and thus our analysis would seem to apply as well to long-term investments. Yet investors are frequently advised that stock investments for the long run are not as risky as it might appear from the statistics presented in this chapter and the previous one. To understand this widespread misconception, we must first understand how the argument goes. Are Stock Returns Less Risky in the Long Run? Advocates of the notion that investment risk is lower over longer horizons apply the logic of diversification across many risky assets to an investment in a risky portfolio over many years. Because stock returns in successive years are almost uncorrelated, they conclude that (1) the annual standard deviation of an investment in stocks falls with the investment horizon and, hence, (2) investment risk in a stock portfolio declines with the investment horizon. To be concrete, consider a two-year investment for which the rate of return in each year is normally distributed with an identical standard deviation of ?, and for which the returns in different years are uncorrelated with each other, so that Cov(r1, r2) = 0. The total rate of return over the two years7 is: (2 years) = r1 + r2. The variance of the total return over the two years equals The standard deviation is the square root of the variance, so Thus, the variance of the total two-year return is double that of the one-year return, and the standard deviation is higher by a multiple of Generalizing to an investment horizon of n years, the variance and standard deviation of the total return over n years will grow to: To put the standard deviation of total return on a per-year or annualized basis, we divide the standard deviation by the number of years, n, to obtain: In fact, this result seems identical to the annual standard deviation of an equally weighted portfolio diversified across n uncorrelated stocks, all with a common standard deviation, ?. To illustrate, consider a portfolio of two identical, uncorrelated stocks. Since the stocks are identical, the efficient portfolio will be equally weighted. Applying Equation 6.6 with weights of wA = wB = ?, p. 176 If each stock has identical standard deviation, then ?A = ?B = ?, and if they are uncorrelated, then ?AB = 0. In this case, therefore, Similarly for n stocks, with portfolio weights of 1/n in each stock, In fact, we used Equation 6.24 to draw Figure 6.1A illustrating diversification with uncorrelated stocks. Since the annual standard deviation of a portfolio diversified across n identical, uncorrelated stocks in Equation 6.24 is similar to the annualized standard deviation of a stock portfolio invested over n years (Equation 6.22), there is a temptation (to which many financial advisers have succumbed) to interpret the latter as evidence of ?time diversification? and conclude that risk over the long haul declines with investment horizon. By this reasoning, Figure 6.1A would seem to apply to time diversification as well, if you replace the number of stocks on the horizontal axis with the number of years. If this were true, time diversification would be very comforting to the many long-term investors who should, by this logic, replace safe investments with risky investments in stocks. Unfortunately, however, the logic is flawed. The Fly in the ?Time Diversification? Ointment (or More Accurately, the Snake Oil) The flaw in the logic is the use of the annualized standard deviation to gauge the risk of a long-term investment. Annualized standard deviation is an appropriate measure of risk only for annual horizon portfolios! It cannot serve to measure risk when comparing investments of different horizons and different scales. To illustrate with a simple example, suppose that investors can invest in safe bonds, and that the rate of return on all bonds is zero. The value of a stock portfolio in any year will either double or fall by one-half with equal probability. Our investor considers two strategies: A. Short-term risky strategy: Invest the entire budget in stocks for one year, then liquidate and invest the proceeds in a safe bond for the second year. B. Long-term risky strategy: Invest the entire budget in stocks for two years. The possible outcomes to this strategy are: quadrupling of value (doubling in each year), unchanged value (doubling in one year and halving in the other), or value falling by a factor of 1/4 (halving in each year). The following table compares the probability distributions of final outcomes of the two investments. p. 177 Since risk aversion makes investors concerned with downside risk, you can see that the two strategies cannot be compared on the basis of standard deviation of annualized returns. Surely a risk-averse investor will consider the two-year investment (for which value can decline by 75%) riskier and will reject outright the notion that the two-year stock investment is less risky. Time diversification advocates will say: ?But the probability of a loss is smaller, only 25%.? This argument implies that, somehow, probability of loss is a valid measure of risk. The fact of the matter is that probability of loss alone is not a legitimate measure of risk any more than is the size of the loss alone. The correct comparison is based on risk of the total (end of horizon) return, which accounts for both magnitudes as well as probabilities of possible losses. The variance of the total rate of return, which accounts for both, grows in direct proportion to the number of years, and the standard deviation grows in proportion to as in Equation 6.21. While the average risk per year may be smaller with longer horizons as in Equation 6.22, that risk compounds for a greater number of years, which certainly makes your cumulative investment outcome riskier, as Equation 6.21 makes clear. Empirical evidence on this debate is provided by the actual cost of portfolio insurance. Such insurance is common and we can observe the actual cost of insurance for various horizons and loss coverage. Suppose that for the two-year stock portfolio in our example, we purchase portfolio insurance against an investment loss that exceeds 50%. Such a policy will pay us 25? per dollar invested if the portfolio value falls by 75%, thereby equating the maximum possible loss of the two strategies. The expected loss to the insurer, per dollar of coverage, is: .25 ? 25 = 6.25?. But we observe that in capital markets, such insurance costs much more for longer horizons, which contradicts any notion that the long-term risky investment is safer than shorter-term one. Time diversification advocates consistently ignore this unshakable fact. 7To account for compounding of rates over the years, these rates must be viewed as continuously compounded returns, as explained in Chapter 5.,After entering 4% as my solution and hitting the check solution button, it was marked incorrect. Could you please provide me with the correct solution?
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