# How can we think about the tradeoff between risk and return?

Discussion Questions

· Exhibit 3 shows the summary of assumptions for the analysis (both tabular and graphically) for the five asset classes: US Equity, Foreign Equity, Bonds, REITS, and Commodities.

· If you could only invest in one of those classes, which would you select? Why?

· How can we think about the tradeoff between risk and return?

· The Excel file Partners Healthcare_Portfolios of 3-5 Assets (which is macro enabled) is available as a Material. It uses the Excel plug-in “Solver” to calculate efficient portfolios given inputs of means, standard deviations, and correlations (See Exhibit 3 in the Partner’s Healthcare Case). The spreadsheet has 3 tabs: One for a portfolio of three assets (US equities, Foreign Equities, and Bonds (corresponding to exhibits 4 and 5 in the Partner’s Healthcare case); One for four assets (Adding REITS to the other 3 (Exhibit 6); One with all five asset classes (Exhibit 8).

· Using the “3 Assets” tab, you can create different portfolio combinations by selecting varying weights in each of the three asset classes (US Equities, Foreign Equities, and Bonds). Remember that the sum of the weights needs to add to 100% for the portfolio to be legitimate. Exhibit 4a has some arbitrary combinations of the three asset classes. Enter some of the combinations into the spreadsheet in the yellow boxes and you should get the same portfolio return and portfolio standard deviation in the Exhibit (subject to rounding).

Review the Tip: Portfolio is simply the weighted average of the individual returns.

· Consider this information: Exhibit 5a gives different efficient portfolios for the combination of 3 assets.

· Enter the weights for portfolio 12 (47.9% USE, 49.7% FE, 2.4% Bonds), which should give you an expected return of 12.5% and a standard deviation of 13.06%.

· Now try to find a “better” portfolio which gives you a 12.5% expected return by varying the weights (but they must still sum to 1). A portfolio will be better if it has a 12.50% expected return but a lower standard deviation than 13.06%.

TIP: Portfolio is simply the weighted average of the individual returns.

Recall that the return of a portfolio is simply the weighted average of the individual returns.

Rp =∑wR ii

i

For our three asset portfolio, this means that the return is

R=wR+wR+wR 3assets US _ equity US _ equity Foreign _ equity Foreign _ equity US _ Bonds US _ Bonds

This is precisely how the return is calculated in cell E10 on the “3 assets” tab.

The variance of a portfolio is more complicated because it is NOT just a weighted average of the individual standard deviations. As shown in the appendix to the Partner’s Healthcare Case, we have:

σ p2 = ∑ ∑ w w σ σ ρ i j i j ij

ij

Where wi and wj stand for the weights in assets i and j, σi and σj are the standard deviations of assets i and j, and ρi,j is the correlation coefficient between asset i and asset j. Note that when i=j, this just means the variance of the particular asset. Moreover, since the combination i,j is the same as j,i, this means that there will be 2 of each combination where i≠j. While this formula seems really complicated, it is easier than it appears. For our 3 asset combination, we would have (USE means US equities, FE means foreign equities, and Bond is bonds)

1

σ2 =w2σ2+w2σ2+w2σ2+

3 Assets

USE USE FE FE Bond Bond 2wwσσρ +

4assets σ2

USE USE

FE FE Bond Bond REIT REIT

4 Assets

=w2 σ2 USE USE

+w2σ2 +w2 σ2 FE FE Bond Bond

+w2 σ2 + REIT REIT

USE FE USE FE USE,FE 2wwσσρ +

USE Bond USE Bond USE , Bond 2wFEwBondσFEσBond ρFE,Bond

Notice that the first part (line 1) is just the weight of each asset squared multiplied by its own variance. The remaining parts (lines 2-4) are the pairwise combinations of each of the groups of assets. The standard deviation of the portfolio is simply the square root of the above formula.

As we add an additional asset to the portfolio (i.e. REITS), notice that the variance formula would include the term for the REIT variance, but would also include ANOTHER 3 covariance terms because we need to account for how REITS covary with each of USE, FE, and Bond. You can see this in the formula for cell F10 on the “4 assets” tab.

For completeness, below are the formulas for the 4 asset portfolio

R=wR+wR+wR+wR

2wwσσρ + USE FE USE FE USE ,FE

2wwσσρ + USE Bond USE Bond USE , Bond

2wFEwBondσFEσBond ρFE,Bond + 2wwσσρ +

USE REIT USE REIT USE , REIT 2wFEwREITσFEσREIT ρFE,REIT + 2wBond wREITσBondσREIT ρBond,REIT

2

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