Principles of Investment

Solutions to Assignment 6

Portfolio Returns

  1. What is a portfolio of assets?

A portfolio is a collection of financial assets (such as stocks and bonds) in which each asset represents a specific percentage, or weight, of the wealth invested in the portfolio.

  1. Suppose you have 3 assets in a portfolio; Asset A, Asset B, and Asset C. Suppose the weight of Asset A is 0.5, the weight of Asset B is 0.3, and the weight of Asset C is 0.2. If the return on A is 10%, the return on B is 15%, and the return on C is 5%, what is the return on the portfolio?
  1. Suppose again you have 3 assets in a portfolio; Asset A, Asset B, and Asset C. Write down the general equation for the return on the portfolio.
  1. Now suppose there are n assets. Write down the general equation for the return on the portfolio.

Or

  1. Using the result of number 4, write down the equation for the expected return on a portfolio of n assets.

Portfolio Risk

  1. Assume there are n assets to choose from. Also assume that these assets have identical risk as measured by the standard deviation, and the standard deviation is 0.30. Also assume the returns of these assets are not correlated. Calculate the risk in the portfolio if there is

Use the formula

  1. 1 asset in the portfolio
  1. 2 assets in the portfolio
  1. 5 assets in the portfolio
  1. 10 assets in the portfolio
  1. What can you conclude about the importance of diversifying your portfolio?

Diversifying the portfolio substantially reduces risk in the portfolio.

  1. If you add an asset to a portfolio, what is the effect on portfolio risk if
  1. the asset has a positive correlation to the other assets in the portfolio?

Risk tends to rise due to correlation effect, but tends to fall due to diversification effect. Thus overall risk may rise or fall.

  1. the asset has a zero correlation to the other assets in the portfolio?

Risk is unchanged due to correlation effect, but tends to fall due to diversification effect. Hence overall risk must fall.

  1. the asset has a negative correlation to the other assets in the portfolio?

Risk tends to fall due to correlation effect,and tends to fall due to diversification effect. Hence, overall risk must fall.

  1. Suppose there are two assets in your portfolio; A and B. Suppose the weight of A is 0.6 and the weight of B is 0.4. Also suppose that the standard deviation of A is 12% and the standard deviation of B is 20%. Compute the variance of the portfolio under the following cases:

The formula we need is

  1. the correlation coefficient between A and B is 0.3

σ2 = 0.1331

  1. the correlation coefficient between A and B is 0.7

σ2 = 0.01561

  1. Write down the general formula for computing the risk in a portfolio with n assets.

Efficient Portfolios

  1. What is an efficient portfolio? Suppose there are three portfolios. Each has expected return of 15%. However, Portfolio 1 has a standard deviation of 20%, Portfolio 2 has a standard deviation of 30%, and Portfolio 3 has a standard deviation of 15%. Which is the efficient portfolio from these three?

An efficient portfolio is defined to be a portfolio that for a given level of expected return, the variance of the portfolio is minimized. Alternatively, an efficient portfolio is defined to be a portfolio that for a given level of variance, the expected return of the portfolio is maximized.

  1. Draw a set of efficient portfolios in a graph with expected return on the vertical axis and risk on the horizontal axis.

Portfolio Selection among the Set of Efficient Portfolios

  1. Suppose an individual’s preferences can be described by utility, which increases with expected return and falls with risk. An indifference curve shows the combinations of expected return and risk that provide the same utility. Using a graph with expected return on the vertical axis and risk on the horizontal axis draw a person’s indifference curve.
  1. Using indifference curves, demonstrate the optimal portfolio from the set of efficient portfolios.
  1. Your answer in number 2 proves an optimal portfolio exists. What is the problem with using this method to select the optimal portfolio? In particular, what problem exists in finding the set of efficient portfolios?

The problem is it can be difficult to operationalize. The reason is in order to calculate the risk in even one portfolio requires a massive amount of computation if there are a large number of assets in the portfolio, since one must find the correlation between every pair of assets. But this does not have to be done for only one portfolio, but it must be done for a very large number of portfolios to create the efficient set of portfolios.

  1. The single index model can be defined by the equation

Define each term in the equation and explain the single-index model.

  • Ri is the return on asset i.
  • ai is the part of the return that is unique to asset i.
  • RM is the return on the market portfolio.
  • Βi is a coefficient that measures how sensitive is the return of asset i to the rest of the market. Hence the term βRM is the part of the return of asset i that is determined by its connection to the rest of the market. It is through this part that correlation between the assets arises.
  • ei is an error term. It is a statistical concept meant to capture the inability to perfectly predict returns. The E(ei) = 0.
  1. How does the single-index model address the problem identified in number 3?

The benefit of the single-index model is in the reduced computation. In the Markowitz one must estimate every covariance between every possible pair. In the Single-Index model one only estimates the β coefficients for each asset. This greatly reduces the computation.

  1. What is the major assumption of the single-index model (hint: it relates to the term ei)?

The major assumption is that the error term of one asset is not correlated to the errors in other assets. That is, E(eiej) = 0. This is what makes the single-index model work. Because of this assumption, this means that all the correlation between different assets is captured by the β coefficients.

  1. What is the covariance between any two assets?
  1. Write down the equation for asset risk under the single index model. Which part corresponds to market risk, and which part corresponds to company specific risk?

The first part measures the risk coming from market risk, and the second part is the risk in the error term.

  1. Write down the equation for portfolio risk under the single index model. Which part corresponds to systematic risk, and which part corresponds to non-systematic risk?

The portfolio risk is given by

The first term measures the risk in the portfolio owing to its connection to market risk. This is referred to as systematic risk. Systematic risk is simply due to the fact that the portfolio’s assets are part of the overall economy and is thus subject to the same risk as we have in the rest of the economy. It cannot be diversified away. The second part of portfolio risk is unique to the portfolio and is called non-systematic risk.

  1. What problems exist with the single index model? How does the multiple index model overcome these problems?

The problem with the single-index model is that the β coefficients may not capture all the correlation between assets. The reason is that if two assets are from the same sector of the economy they would have a greater correlation than would be implied by their connection to the overall market.

To correct for this problem, a multiple-index model can be used. One index connects the return on an asset to the overall market, and another index connects the return on an asset to a specific sector of the economy. This gives rise to the following equation for return on asset i.

where RS is the return on an index of stock from a particular sector, and γi is a coefficient measuring how sensitive is the return of asset i to that sector. Hence the term γRS is the part of the return of asset i that is determined by its connection to that sector.