1. Asset Allocation
Week 4

2. Asset Allocation: Risky vs. Riskless Asset
Consider the allocation between the risky and riskless asset.
Rf = expected return on riskfree asset.
Rp= expected return on risky asset portfolio.
Volatility of riskfree asset = 0.
W1 = proportion in riskfree asset.
W2 = proportion in risky asset.
Is there an optimal w1, w2?
We shall show that the choice of w1, w2 is individual-specific, and will depend on the individual’s risk aversion and objectives. Thus, there is no one optimal portfolio.

3. Portfolio of Risky + Riskless Asset
To calculate the portfolio return and portfolio variance when we combine the risky asset and riskless asset, we can use the usual formulas, noting that the volatility of the riskfree rate is zero.
Portfolio Return = w1 Rf + w2 Rp.
Portfolio Variance = (w1)^2 (0) + (w2 )^2 (vol of risky asset)^2 + 2 (correlation) (w1 )(w2 ) (0)(vol of risky asset).
Portfolio Volatility = w2 *(vol of risky asset).
This simplification in the formula for the portfolio volatility occurs because the vol of the riskfree asset is zero.
To understand the tradeoff between risk and return, we can graph the portfolio return vs the the portfolio volatility.
The following graph shows this graph for the case when the mean return for the riskfree asset is 5%, the mean return for the risky asset is 12%, and the volatility of the risky asset is 15%.

4. Riskfree Return=0.05, Risky Return=0.12, Vol of Risky Asset=0.15

5. Portfolio Return vs. Portfolio Volatility

6. Capital Allocation Line (CAL)
The graph from the previous slide is called the capital allocation line (CAL).
For the special case when one of the two assets is the riskfree asset, the CAL is a straight line, with a slope of (Rp-Rf)/(Vol of risky portfolio).
This slope equals the increase in return of the portfolio for a unit increase in volatility. Therefore, it is also called the reward-to-variability ratio. We will also refer to this ratio as the Sharpe ratio.
The greater the slope the greater the reward for taking risk. Ideally, you want to achieve the highest return per unit risk, so that you choose a risky portfolio that gives you the steepest slope.
Note that this tradeoff will be essentially determined by the mean return and volatility of the risky portfolio.

7. How to allocate between the riskfree asset and the risky stock portfolio.
The conclusion we draw from the straight-line graph is that: when we combine a riskfree asset with the risky stock portfolio, all portfolios have the same Sharpe ratio.
Therefore, it is not possible to make a decision on allocation between the riskfree asset and the risky stock portfolio based solely on the Sharpe ratio. Instead, we will have to take into account individual-specific considerations. There is no single allocation here that is best for all investors.
Your decision to allocate between the risky asset and the riskfree asset will be determined by your level of risk aversion and your objectives, depending on factors like your age, wealth, horizon, etc. The more risk averse you are, the less you will invest in the risky asset.
Although different investors may differ in the level of risk they take, they are also alike in that each investor faces exactly the same risk-return tradeoff.

8. A Digression into “Market Timing”
Why not actively manage the allocation between the riskfree asset and the risky stock portfolio?
There are funds that actively manage the decision to allocate between the risky/riskless asset for the investor: these funds are typically called “market allocation” funds.
Typically, the funds actively manage a mix of stocks, bonds and money market securities, and they may change the fraction of their holding in each of these assets, depending on what they think is “optimal” at that time.
Such a trading strategy is also called market timing. The objective of market timing is to be invested in stocks in a bull market, and to be invested in bonds/cash in a bear market.

9. Returns to Market Timing
Here is an example that illustrates how you could do if you were a good/bad market timer. If you could time the market, using the S&P 500, what would your returns be over the period Jan Dec 2002? We start with $1 on January 1, 1950, and ask how much we would have on December 31, 2002.
1. Buy and hold strategy: $51.60 (average return=7.72%).
2. Perfect timer: $238,203 (26.31%) (!!).
3. Occasional timer (miss the worst 10 months): $200 (10.52%).
4. Mis-timer (miss the best 10 months): $16.87 (5.48%).
5. Miss both best/worst 10 months: $65.49 (8.21%).
Moral of the story: time the market only if you have a good crystal ball.
But its tempting to keep trying even when one doesn’t have a crystal ball.

10. The Optimal Risky Stock Portfolio
We discussed the allocation between the risky (stock) portfolio and the riskless (cash) portfolio.
Now we will consider the other decision that an investor must make: how should the risky stock portfolio be constructed?
Once again we will assume that investors want to maximize the Sharpe ratio (so that investors want the best tradeoff between return and volatility).

11. Determining the Optimal Portfolio
If we can plot the portfolio return vs. Portfolio volatility for all possible allocations (weights), then we can easily locate the optimal portfolio with the highest Sharpe ratio of (Rp - Rf)/(Vol of risky portfolio).
When we only have two risky assets, as in this case, it is easy to construct this graph by simply calculating the portfolio returns for all possible weights.
When we have more than 2 assets, it becomes more difficult to represent all possible portfolios, and instead we will only graph only a subset of portfolios. Here, we will choose only those portfolios that have the minimum volatility for a given return. We will call this graph the variance-return frontier.
Once we solve for this minimum variance frontier, we will show that there exists one portfolio on this frontier that has the highest Sharpe ratio, and thus is the optimal stock portfolio.
Because there exists one specific portfolio with the highest Sharpe ratio, all investors will want to invest in that portfolio. Thus, the weights that make up this portfolio determines the optimal allocation between the risky assets for all investors.

12. Frontier with KO and PEP
As an example, consider a portfolio of KO and PEP. What should be the optimal combination of KO and PEP?
Refer to excel file on web page.
As we only have two assets here, we can easily tabulate the Sharpe ratio for a range of portfolio weights, and check which portfolio has the highest Sharpe ratio.
The next slide shows the results. In the calculation of the Sharpe ratio, it is assumed that the riskfree rate is constant (which is not strictly true). The portfolio mean and portfolio return are calculated with the usual formulae over the 10-year sample period , with monthly data.
As can be seen, the optimal weight for a portfolio (to get the maximum Sharpe ratio) appears to be in the range of 0.6 in KO. If the exact answer is required, we can easily solve for it using the excel solver.
It can also be observed that, amongst these 11 portfolios, the portfolio with the minimum volatility is one that invests 50% in each of the two stocks. This is the minimum variance portfolio. The minimum variance portfolio may be different from the portfolio with the highest Sharpe ratio.

13. The Sharpe Ratio: KO + PEP

14. Volatility-Return Frontier
Consider the graph of the portfolio return vs. Portfolio volatility.
Graphically, the optimal portfolio (with the highest Sharpe ratio) is the portfolio that lies on a tangent to the graph. This tangent is drawn so that it has the riskfree rate as its intercept.
This is because the slope of the line that passes connects the riskfree asset and the risky portfolio is equal to the Sharpe ratio. Thus, the steeper the line, the higher the Sharpe ratio. The tangent to the graph has the steepest slope, and thus the portfolio that lies on this tangent is the optimal portfolio (having the highest Sharpe ratio).
This tangent is now the capital allocation line. All investments represented on this line are optimal (and will comprise of combination of the riskfree asset and risky stock portfolio).

15. Portfolio Return-Volatility Frontier

16. Creating the mean variance frontier
How to use a spreadsheet to calculate the frontier when there are more than 2 assets

17. The Minimum Variance Frontier
With two assets, as we saw, we can construct the frontier by brute force - by listing almost all possible portfolios.
When we have more than 2 assets, its gets difficult to consider all possible portfolio combinations. Instead, we will make the process simpler by considering only a subset of portfolios: those portfolios that have the minimum volatility for a given return.
When we plot the return and volatilities of these portfolios, the resultant graph will be known as the minimum variance (or volatility) frontier.
We will use Excel’s “Solver” for these calculations (look under Tools. If it is not there, then add it into the menu through Add-in).

18. The Steps We will implement the procedure in three steps:
1. For each asset (and for the time period that you have chosen), calculate the mean return, volatility and the correlation matrix.
2. Set up the spreadsheet so that the Solver can be used. See the sample spreadsheet. Your objective here is to determine the weights of the portfolio that will allow you to achieve a specified required rate of return with the lowest possible volatility.
3. Repeat 2 for a range of returns, and plot the frontier (return vs. volatility).

19. Step 1: Assembling the Data
A. Fix the time period for the analysis. You want a sufficiently long period so that your estimates of the mean return, volatility and correlation are accurate. But you don’t want a period too long, because the data may not be valid.
B. Estimate the mean return and volatility for each of your assets. Next, calculate the correlation between each pair of assets. If there are N assets, you will have to calculate N(N-1) correlations.

20. Step 2: Setting up the spreadsheet to use the Solver (1/4)
The objective here is to set up the spreadsheet in a manner that is easy to use with the solver.
The estimates of the return, volatility and the correlation matrix are used to set up a matrix for covariances, which is then used to calculate the portfolio volatility for a given set of weights.
To create the frontier, you will ask the solver to find you the weights that gives you the minium volatility for a required return.

21. Step 2: Using the Solver (2/4)
1.Target Cell: When you call the solver, it will ask you to specify the objective or the “target cell”. Your objective is to minimize the volatility - so in this case, you will specify the cell that calculates the portfolio volatility [$B$25]. As you want to minimize the volatility, you click the “Min”.
2. Constraints: You will have to specify the constraints under which the optimization must work. There are two constraints that hold, and a third which will usually also apply.

22. Using the Solver: Constraints on the Optimization (3/4)
1. First, the sum of the weights must add up to 1.
2. Second, you have to specify the required rate of return for which you want the portfolio of least volatility. For each level of return, you will solve for the weights that give you the minimum volatility. To construct the frontier, you will vary this required return over a range. Thus, you will have to change this constraint every time you change the required return.
Third, if there are constraints to short-selling, you will have to specify that each portfolio weight is positive.

23. Step 2: (4/4)
Finally, you specify the arguments that need to be optimized. In this case, you are searching for the optimal weights, so you will have to specify the range in the spreadsheet where the portfolio weights used [A20, A21, A22].

24. Step 3
The final step is to simply repeat step 2, until you have a sufficiently large data set so that the minimum variance frontier can be plotted. .

25. The Optimal Allocation
We can now use the graph of the minimum variance frontier to figure out the portfolio with the highest Sharpe Ratio. This portfolio will be the portfolio such that the CAL passing through it is tangent to the minimum variance frontier.
The weights of this portfolio determines the optimal allocation within the assets that make up the “risky portfolio”. All investors should opt for this allocation.
The portfolio will always be on the upper portion of the frontier, above the portfolio with the lowest volatility - this portion is called the efficient frontier.

26. Diversification (1/6)
We have observed that by combining stocks into portfolios, we can create an asset with a better risk-return tradeoff.
The reduction of risk in a portfolio occurs because of diversification. By combining different assets into a portfolio, we can diversify risk and reduce the overall volatility of the portfolio.
Let us review the factors that affect how risk can be diversified. Here we will ignore the issue of allocation (as we have already considered it), and instead assume that our portfolio is equally weighted.

27. Factors that affect diversification in an equally weighted portfolio (2/6)
There are two main factors that affect the extent to which volatility can be reduced: the number of assets in the portfolio, and the correlation between the assets.
Increasing the number of assets reduces the volatility of the portfolio.
Adding an asset with a low correlation with the existing assets of a portfolio also helps to reduce the volatility of the portfolio.

28. (3/6)
To examine the effect of correlation and the number of assets, lets assume, for simplicity, that each of the assets have the same volatility (say, 40%) and the same average correlation with each other.
The portfolio volatility can then be calculated by the usual formula, and we can examine the reduction in volatility of the portfolio as we change the number of assets, or the correlation.

29. Sample spreadsheet (4/6)

30. Some Conclusions (5/6)
By changing N=number of stocks in portfolio, and the correlation, we can examine how the portfolio volatility decreases.
We can make the following observations:
1. For all positive correlation, there is a threshold beyond which we cannot reduce the portfolio volatility. This threshold depends on the magnitude of the correlation. If the correlation is zero or less than zero, then it is possible to bring down the portfolio volatility to zero by having a large number of assets. This threshold represents the undiversifiable or the systematic risk of the portfolio.

31. Some Conclusions (6/6)
2. As the correlation decreases, the more we can reduce the portfolio volatility. However, it takes more assets to bring down the portfolio volatility to its theoretical minimum.
Example: if the correlation is 0.9 and the average volatility of each stock in the portfolio is 40%, then the lowest portfolio volatility that is possible is about 37.95%. We can reach within 0.5% of this minimum volatility by creating a portfolio of only 4 assets. Suppose instead that the average correlation is Then the lowest possible portfolio volatility is 28.28%; however, to reach within 0.5% of this value, we need as many as 30 stocks.

32. In Summary (1/2)
1. The optimal allocation is determined in two steps. First, we decide the allocation between the risky portfolio, and the riskless asset. Second, we determine the allocation between the assets that comprise the risky portfolio.
2. As every portfolio of the risky assets and the riskless asset has the same Sharpe ratio, there is not one optimal portfolio for all investors. Instead, the allocation will be determined by individual-specific factors like risk aversion and the objectives of the investor, taking into account factors like the investor’s horizon, wealth, etc.
3. When we are considering the allocation between different classes of risky assets, it is possible to create a portfolio that has the highest Sharpe Ratio. The weights of the risky assets in this portfolio will determine the optimal allocation between various risky assets. This portfolio can be determined graphically by drawing the capital allocation line (CAL) such that it is tangent to the minimum variance frontier. This portfolio will always lie on the upper part of the frontier (or on the efficient part of the frontier).

33. In Summary (2/2)
4. The extent to which you can decrease the volatility of the portfolio depends also on the correlation. The lower the average correlation of the stocks in your portfolio, the lower you can decrease the volatility of your portfolio.
5. The homework provides you with an exercise to determine the optimal allocations.