Optimal Portfolios With Restrictions On The Riskfree Asset

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The availability of a risk-free asset greatly simplifies the portfolio decision. When all investors can borrow and lend at that risk-free rate, we are led to a unique optimal risky portfolio that is appropriate for all investors given a common input list. This portfolio maximizes the reward-to-variability ratio. All investors use the same risky portfolio and differ only in the proportion they invest in it versus in the risk-free asset.

What if a risk-free asset is not available? Although T-bills are risk-free assets in nominal terms, their real returns are uncertain. Without a risk-free asset, there is no tangency portfolio that is best for all investors. In this case investors have to choose a portfolio from the efficient frontier of risky assets redrawn in Figure 8.15.

Each investor will now choose an optimal risky portfolio by superimposing a personal set of indifference curves on the efficient frontier as in Figure 8.15. An investor with indifference curves marked U', U", and U" in Figure 8.15 will choose Portfolio P. More risk-averse investors with steeper indifference curves will choose portfolios with lower means and smaller standard deviations such as Portfolio Q, while more risk-tolerant investors will choose portfolios with higher means and greater risk, such as Portfolio S. The common feature of all these investors is that each chooses portfolios on the efficient frontier.

Even if virtually risk-free lending opportunities are available, many investors do face borrowing restrictions. They may be unable to borrow altogether, or, more realistically, they may face a borrowing rate that is significantly greater than the lending rate.

When a risk-free investment is available, but an investor cannot borrow, a CAL exists but is limited to the line FP as in Figure 8.16. Any investors whose preferences are represented by indifference curves with tangency portfolios on the portion FP of the CAL, such as Portfolio A, are unaffected by the borrowing restriction. Such investors are net lenders at rate f

Aggressive or more risk-tolerant investors, who would choose Portfolio B in the absence of the borrowing restriction, are affected, however. Such investors will be driven to portfolios such as Portfolio Q, which are on the efficient frontier of risky assets. These investors will not invest in the risk-free asset.

In more realistic scenarios, individuals who wish to borrow to invest in a risky portfolio will have to pay an interest rate higher than the T-bill rate. For example, the call money rate charged by brokers on margin accounts is higher than the T-bill rate.

Investors who face a borrowing rate greater than the lending rate confront a three-part CAL such as in Figure 8.17. CAL1, which is relevant in the range FP1, represents the efficient

CHAPTER 8 Optimal Risky Portfolios

Figure 8.15 Portfolio selection without a risk-free asset.

Expected return

Portfolio Selection
Standard deviation

Figure 8.16 Portfolio selection with risk-free lending but no borrowing.

Portfolio Theory Risk Free Borrowing

238 PART II Portfolio Theory

Figure 8.17 The investment opportunity set with differential rates for borrowing and lending.

Portfolio Frontier Borrowing

Figure 8.18 The optimal portfolio of defensive investors with differential borrowing and lending rates.

Investor Optimal Portfolio

portfolio set for defensive (risk-averse) investors. These investors invest part of their funds in T-bills at rate f They find that the tangency Portfolio is P1, and they choose a complete portfolio such as Portfolio A in Figure 8.18.

CALp, which is relevant in a range to the right of Portfolio Pp, represents the efficient portfolio set for more aggressive, or risk-tolerant, investors. This line starts at the borrowing rate rf, but it is unavailable in the range rfP p, because lending (investing in T-bills) is available only at the risk-free rate r, which is less than rf.

Investors who are willing to borrow at the higher rate, rf, to invest in an optimal risky portfolio will choose Portfolio P2 as the risky investment vehicle. Such a case is depicted

Figure 8.19 The optimal portfolio of aggressive investors with differential borrowing and lending rates.

CHAPTER 8 Optimal Risky Portfolios 239

CHAPTER 8 Optimal Risky Portfolios 239

Figure 8.19 The optimal portfolio of aggressive investors with differential borrowing and lending rates.

Capm Restricted Borrowing
Figure 8.20 The optimal portfolio of moderately risk-tolerant investors with differential borrowing and lending rates.
Markowitz And Indifference Curve

in Figure 8.19, which superimposes a relatively risk-tolerant investor's indifference curve on CAL2. The investor with the indifference curve in Figure 8.19 chooses Portfolio P2 as the optimal risky portfolio and borrows to invest in it, arriving at the complete Portfolio B.

Investors in the middle range, neither defensive enough to invest in T-bills nor aggressive enough to borrow, choose a risky portfolio from the efficient frontier in the range P1P2. This case is depicted in Figure 8.20. The indifference curve representing the investor in Figure 8.20 leads to a tangency portfolio on the efficient frontier, Portfolio C.

PART II Portfolio Theory


With differential lending and borrowing rates, only investors with about average degrees of risk aversion will choose a portfolio in the range P1P2 in Figure 8.18. Other investors will choose a portfolio on CAL1 if they are more risk averse, or on CAL2 if they are more risk tolerant.

a. Does this mean that investors with average risk aversion are more dependent on the quality of the forecasts that generate the efficient frontier?

b. Describe the trade-off between expected return and standard deviation for portfolios between P1 and P2 in Figure 8.18 compared with portfolios on CAL2 beyond P2.


1. The expected return of a portfolio is the weighted average of the component security expected returns with the investment proportions as weights.

2. The variance of a portfolio is the weighted sum of the elements of the covariance matrix with the product of the investment proportions as weights. Thus the variance of each asset is weighted by the square of its investment proportion. Each covariance of any pair of assets appears twice in the covariance matrix; thus the portfolio variance includes twice each covariance weighted by the product of the investment proportions in each of the two assets.

3. Even if the covariances are positive, the portfolio standard deviation is less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated. Thus portfolio diversification is of value as long as assets are less than perfectly correlated.

4. The greater an asset's covariance with the other assets in the portfolio, the more it contributes to portfolio variance. An asset that is perfectly negatively correlated with a portfolio can serve as a perfect hedge. The perfect hedge asset can reduce the portfolio variance to zero.

5. The efficient frontier is the graphical representation of a set of portfolios that maximize expected return for each level of portfolio risk. Rational investors will choose a portfolio on the efficient frontier.

6. A portfolio manager identifies the efficient frontier by first establishing estimates for the asset expected returns and the covariance matrix. This input list is then fed into an optimization program that reports as outputs the investment proportions, expected returns, and standard deviations of the portfolios on the efficient frontier.

7. In general, portfolio managers will arrive at different efficient portfolios because of differences in methods and quality of security analysis. Managers compete on the quality of their security analysis relative to their management fees.

8. If a risk-free asset is available and input lists are identical, all investors will choose the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL. All investors with identical input lists will hold an identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio construction.

9. When a risk-free asset is not available, each investor chooses a risky portfolio on the efficient frontier. If a risk-free asset is available but borrowing is restricted, only aggressive investors will be affected. They will choose portfolios on the efficient frontier according to their degree of risk tolerance.

KEY TERMS diversification market risk nondiversifiable risk insurance principle systematic risk unique risk

CHAPTER 8 Optimal Risky Portfolios firm-specific risk nonsystematic risk diversifiable risk minimum-variance portfolio portfolio opportunity set reward-to-variability ratio optimal risky portfolio minimum-variance frontier efficient frontier of risky assets input list separation property


http://finance.yahoo.com can be used to find historical price information to be used in estimating returns, standard deviation of returns, and covariance of returns for individual securities. The information is available within the chart function for individual securities.

http://www.financialengines.com has risk measures that can be used to compare individual stocks to an average hypothetical portfolio.

http://www.portfolioscience.com uses historical information to calculate potential losses for individual securities or portfolios of securities. The risk measure is based on the concept of value at risk and includes some capabilities of stress testing.


The following data apply to problems 1 through 8:

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows:

Expected Return

Standard Deviation

Stock fund (S )



Bond fund (B )



The correlation between the fund returns is .10.

1. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return?

2. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock funds of zero to 100% in increments of 20%.

3. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio?

4. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio.

5. What is the reward-to-variability ratio of the best feasible CAL?

6. You require that your portfolio yield an expected return of 14%, and that it be efficient on the best feasible CAL.

a. What is the standard deviation of your portfolio?

b. What is the proportion invested in the T-bill fund and each of the two risky funds?

7. If you were to use only the two risky funds, and still require an expected return of 14%, what must be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in problem 6. What do you conclude?

8. Suppose that you face the same opportunity set, but you cannot borrow. You wish to construct a portfolio of only stocks and bonds with an expected return of 24%. What

Bodie-Kane-Marcus: Investments, Fifth Edition

II. Portfolio Theory

8. Optimal Risky Portfolio

© The McGraw-Hill Companies, 2001

PART II Portfolio Theory are the appropriate portfolio proportions and the resulting standard deviations? What reduction in standard deviation could you attain if you were allowed to borrow at the risk-free rate?

9. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%.

a. In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.

b. Given the data above, reanswer (a) with the additional assumption that the correlation coefficient between gold and stocks equals 1. Draw a graph illustrating why one would or would not hold gold in one's portfolio. Could this set of assumptions for expected returns, standard deviations, and correlation represent an equilibrium for the security market?

10. Suppose that there are many stocks in the security market and that the characteristics of Stocks A and B are given as follows:


Expected Return

Standard Deviation







Correlation = -1

Suppose that it is possible to borrow at the risk-free rate, r. What must be the value of the risk-free rate? (Hint: Think about constructing a risk-free portfolio from Stocks A and B.)

Assume that expected returns and standard deviations for all securities (including the risk-free rate for borrowing and lending) are known. In this case all investors will have the same optimal risky portfolio. (True or false?)

The standard deviation of the portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio. (True or false?) Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment? 14. Suppose that you have $1 million and the following two opportunities from which to construct a portfolio:

a. Risk-free asset earning 12% per year.

b. Risky asset earning 30% per year with a standard deviation of 40%.

If you construct a portfolio with a standard deviation of 30%, what will be the rate of return?

The following data apply to problems 15 through 17.

Hennessy & Associates manages a $30 million equity portfolio for the multimanager Wilstead Pension Fund. Jason Jones, financial vice president of Wilstead, noted that Hennessy had rather consistently achieved the best record among the Wilstead's six equity managers. Performance of the Hennessy portfolio had been clearly superior to that of the S&P 500 in four of the past five years. In the one less-favorable year, the shortfall was trivial.

Hennessy is a "bottom-up" manager. The firm largely avoids any attempt to "time the market." It also focuses on selection of individual stocks, rather than the weighting of favored industries.

Bodie-Kane-Marcus: Investments, Fifth Edition

CHAPTER 8 Optimal Risky Portfolios

There is no apparent conformity of style among the six equity managers. The five managers, other than Hennessy, manage portfolios aggregating $250 million made up of more than 150 individual issues.

Jones is convinced that Hennessy is able to apply superior skill to stock selection, but the favorable returns are limited by the high degree of diversification in the portfolio. Over the years, the portfolio generally held 40-50 stocks, with about 2%-3% of total funds committed to each issue. The reason Hennessy seemed to do well most years was because the firm was able to identify each year 10 or 12 issues which registered particularly large gains.

Based on this overview, Jones outlined the following plan to the Wilstead pension committee:

Let's tell Hennessy to limit the portfolio to no more than 20 stocks. Hennessy will double the commitments to the stocks that it really favors, and eliminate the remainder. Except for this one new restriction, Hennessy should be free to manage the portfolio exactly as before.

All the members of the pension committee generally supported Jones's proposal because all agreed that Hennessy had seemed to demonstrate superior skill in selecting stocks. Yet the proposal was a considerable departure from previous practice, and several committee members raised questions. Respond to each of the following questions.

15. a. Will the limitations of 20 stocks likely increase or decrease the risk of the portfolio?


b. Is there any way Hennessy could reduce the number of issues from 40 to 20 without significantly affecting risk? Explain.

16. One committee member was particularly enthusiastic concerning Jones's proposal. He suggested that Hennessy's performance might benefit further from reduction in the number of issues to 10. If the reduction to 20 could be expected to be advantageous, explain why reduction to 10 might be less likely to be advantageous. (Assume that Wilstead will evaluate the Hennessy portfolio independently of the other portfolios in the fund.)

17. Another committee member suggested that, rather than evaluate each managed portfolio independently of other portfolios, it might be better to consider the effects of a change in the Hennessy portfolio on the total fund. Explain how this broader point of view could affect the committee decision to limit the holdings in the Hennessy portfolio to either 10 or 20 issues.

The following data are for problems 18 through 20.

The correlation coefficients between pairs of stocks are as follows: Corr(A,B) = .85; Corr(A,C) = .60; Corr(A,D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%.

18. If your entire portfolio is now composed of Stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice):

d. Need more data.

19. Would the answer to problem 18 change for more risk-averse or risk-tolerant investors? Explain.

20. Suppose that in addition to investing in one more stock you can invest in T-bills as well. Would you change your answers to problems 18 and 19 if the T-bill rate is

Bodie-Kane-Marcus: I II. Portfolio Theory I 8. Optimal Risky Portfolio I I © The McGraw-Hill

Investments, Fifth Edition Companies, 2001

PART II Portfolio Theory

21. Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?


Expected Return (%)

Standard Deviation (%)













22. Which statement about portfolio diversification is correct?

a. Proper diversification can reduce or eliminate systematic risk.

b. Diversification reduces the portfolio's expected return because it reduces a portfolio's total risk.

c. As more securities are added to a portfolio, total risk typically would be expected to fall at a decreasing rate.

d. The risk-reducing benefits of diversification do not occur meaningfully until at least 30 individual securities are included in the portfolio.

23. The measure of risk for a security held in a diversified portfolio is:

a. Specific risk.

b. Standard deviation of returns.

c. Reinvestment risk.

d. Covariance.

24. Portfolio theory as described by Markowitz is most concerned with:

a. The elimination of systematic risk.

b. The effect of diversification on portfolio risk.

c. The identification of unsystematic risk.

d. Active portfolio management to enhance return.

25. Assume that a risk-averse investor owning stock in Miller Corporation decides to add the stock of either Mac or Green Corporation to her portfolio. All three stocks offer the same expected return and total risk. The covariance of return between Miller and Mac is -.05 and between Miller and Green is +.05. Portfolio risk is expected to:

a. Decline more when the investor buys Mac.

b. Decline more when the investor buys Green.

c. Increase when either Mac or Green is bought.

d. Decline or increase, depending on other factors.

26. Stocks A, B, and C have the same expected return and standard deviation. The following table shows the correlations between the returns on these stocks.

Stock A

Stock B

Stock C

Stock A

+ 1.0

Stock B


+ 1.0

Stock C



+ 1.0

Given these correlations, the portfolio constructed from these stocks having the lowest risk is a portfolio:

a. Equally invested in stocks A and B.

b. Equally invested in stocks A and C.

CHAPTER 8 Optimal Risky Portfolios c. Equally invested in stocks B and C.

d. Totally invested in stock C.

^ 27. Statistics for three stocks, A, B, and C, are shown in the following tables.

Standard Deviations of Returns





Standard deviation:




Correlations of Returns














Based only on the information provided in the tables, and given a choice between a portfolio made up of equal amounts of stocks A and B or a portfolio made up of equal amounts of stocks B and C, state which portfolio you would recommend. Justify your choice.

The following table of compound annual returns by decade applies to problems 28 and 29.









Small company stocks









Large company stocks









Long-term government









Intermediate-term government




















- 2.04







*Based on the period 1926-1929. Source: Data in Table 5.2.

*Based on the period 1926-1929. Source: Data in Table 5.2.

28. Input the data from the table into a spreadsheet. Compute the serial correlation in decade returns for each asset class and for inflation. Also find the correlation between the returns of various asset classes. What do the data indicate?

29. Convert the asset returns by decade presented in the table into real rates. Repeat the analysis of problem 28 for the real rates of return.

1. a. The first term will be wD X wD X oD, since this is the element in the top corner of the matrix (o D) times the term on the column border (wD) times the term on the row border (wD). Applying this rule to each term of the covariance matrix results in the sum w2Do2D + wDwECov(rBrD) + wEwDCov(rD, rE) + wEoE, which is the same as equation 8.2, since Cov(rE>rD) = Cov(rD, rE).


PART II Portfolio Theory


b. The bordered covariance matrix is









Co v(rY,rx)







There are nine terms in the covariance matrix. Portfolio variance is calculated from these nine terms:

+ wXWjCov(rX, rY) + wYwXCov(rY, rX) + wxwzCov^x, rz) + wzwXCov(rz, rX) + wYwZ Cov(rY, rZ) + wZwY Cov(rZ, rY)

+ 2wXwYCov(rX, rY) + 2wXwZCov(rX, rZ) + 2wYwZCov(rY, rZ)

2. The parameters of the opportunity set are E(rD) = 8%, E(rE) = 13%, jd = 12%, je = 20%, and p (D,E) = 25. From the standard deviations and the correlation coefficient we generate the covariance matrix:










The global minimum-variance portfolio is constructed so that jE - Cov(rp, rE)

wD aD + aE - 2 Cov(rfl, rE) = 400 - 60 (144 + 400) - (2 X 60) " wE = 1 - wD = .1981


Its expected return and standard deviation are

E(rP) = (.8019 X 8) + (.1981 X 13) = 8.99% ctp = [wWD + wE^E + lwDwECov(rD, r£)]1/2

= [(.80192 X 144) + (.19812 X 400) + (2 X .8019 X .1981 X 60)]1/2 = 11.29%

For the other points we simply increase wD from .10 to .90 in increments of .10; accordingly, wE ranges from .90 to .10 in the same increments. We substitute these portfolio proportions in the formulas for expected return and standard deviation. Note that when wE = 1.0, the portfolio parameters equal those of the stock fund; when wD = 1, the portfolio parameters equal those of the debt fund. We then generate the following table:

CHAPTER 8 Optimal Risky Portfolios




















































11.29 minimum variance portfolio

You can now draw your graph. 3. a. The computations of the opportunity set of the stock and risky bond funds are like those of question 2 and will not be shown here. You should perform these computations, however, in order to give a graphical solution to part a. Note that the covariance between the funds is

b. The proportions in the optimal risky portfolio are given by

= (10 - 5)602 - (30 - 5) (-240) Wa (10 - 5)602 + (30 - 5)202 - 30(-240) = .6818 wB = 1 - wA = .3182

The expected return and standard deviation of the optimal risky portfolio are

ctp = {(.68182 X 202) + (.31822 X 602) + [2 X .6818 X .3182(-240)]}1/2 = 21.13%

Note that in this case the standard deviation of the optimal risky portfolio is smaller than the standard deviation of stock A. Note also that portfolio P is not the global minimum-variance portfolio. The proportions of the latter are given by


With these proportions, the standard deviation of the minimum-variance portfolio is o-(min) = {(.85712 X 202) + (.14292 X 602) + [2 X .8571 X .1429 X (- 240)]}1 = 17.57%


which is smaller than that of the optimal risky portfolio.

PART II Portfolio Theory


c. The CAL is the line from the risk-free rate through the optimal risky portfolio. This line represents all efficient portfolios that combine T-bills with the optimal risky portfolio. The slope of the CAL is

op 21.l3

d. Given a degree of risk aversion, A, an investor will choose a proportion, y, in the optimal risky portfolio of

E(rp) - rf = 16.36 - 5 = y = .01 X Aup .01 X 5 X 21.132

This means that the optimal risky portfolio, with the given data, is attractive enough for an investor with A = 5 to invest 50.89% of his or her wealth in it. Since stock A makes up 68.18% of the risky portfolio and stock B makes up 31.82%, the investment proportions for this investor are

Stock A: .5089 X 68.18 = 34.70% Stock B: .5089 X 31.82 = 16.19%

Total 50.89%

4. Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on various securities and estimates of risk, that is, the covariance matrix. The forecasts themselves do not control outcomes. Thus preferring managers with rosier forecasts (northwesterly frontiers) is tantamount to rewarding the bearers of good news and punishing the bearers of bad news. What we should do is reward bearers of accurate news. Thus if you get a glimpse of the frontiers (forecasts) of portfolio managers on a regular basis, what you want to do is develop the track record of their forecasting accuracy and steer your advisees toward the more accurate forecaster. Their portfolio choices will, in the long run, outperform the field.

5. a. Portfolios that lie on the CAL are combinations of the tangency (optimal risky)

portfolio and the risk-free asset. Hence they are just as dependent on the accuracy of the efficient frontier as portfolios that are on the frontier itself. If we judge forecasting accuracy by the accuracy of the reward-to-variability ratio, then all portfolios on the CAL will be exactly as accurate as the tangency portfolio. b. All portfolios on CAL1 are combinations of portfolio P1 with lending (buying T-bills). This combination of one risky asset with a risk-free asset leads to a linear relationship between the portfolio expected return and its standard deviation:

The same applies to all portfolios on CAL2; just replace E(rpi), op in equation 5.b with E(rp2), up2.

An investor who wishes to have an expected return between E(rP ) and E(rP ) must find the appropriate portfolio on the efficient frontier of risky assets between p1 and p2 in the correct proportions.

CHAPTER 8 Optimal Risky Portfolios



Go to www.morningstar.com and select the tab entitled Funds. In the dialog box for selecting a particular fund, type Fidelity Select and hit the Go button. This will list all of the Fidelity Select funds. Select the Fidelity Select Multimedia Fund. Find the fund's top 25 individual holdings from the displayed information. The top holdings are found in the Style section. Identify the top five holdings using the ticker symbol.

Once you have obtained this information, go to www.financialengines.com. From the Site menu, select the Forecast and Analysis tab and then select the fund's Scorecard tab. You will find a dialog box that allows you to search for funds or individual stocks. You can enter the name or ticker for each of the individual stocks and the fund. Compare the risk rankings of the individual securities with the risk ranking of the fund. What factors are likely leading to the differences in the individual rankings and the overall fund ranking?

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  • Wilcome Brown
    How to draw a tangent from the riskfree rate?
    8 years ago
  • Sigismond
    What are the optimal weights of the five stocks for each of the risky portfolios?
    8 years ago
    What is expected return for a portfolio in risk free and risky asset with standard deviation?
    8 years ago
  • theobald
    What is the rewardtovolatility ratio of the best feasible cal?
    7 years ago
  • miranda
    Why include risk free asset in portfolio?
    7 years ago
  • adelaide
    Which one of the following portfolios cannot lie on the efficient frontier as described by markowitz?
    7 years ago
  • medhanie
    What must be the value of the risk free rate?
    7 years ago
  • laura
    Which of the following portfolios cannot lie onthe effiecient frontier as described by markowitz?
    7 years ago
  • filmon
    How does an investor choose his or her optimal portfolio from among the efficient set?
    7 years ago
  • demet
    What are the investment proportions in the minimumvariance portfolio?
    7 years ago
  • Kalevi
    Can Lend, but Cannot Borrow at the RiskFree Rate?
    7 years ago
  • letterio rizzo
    Where does an investor unable to borrow on capm diagram of?
    6 years ago
  • Martha
    What impact would the risk free rate and standard deviation on the call price?
    6 years ago
  • Anja
    How would you construct optimal portfilo?
    6 years ago
  • sancho fairbairn
    What does your graph show for the expected return and standard deviation of the optimal risky?
    6 years ago
  • tewelde
    How to tabulate and draw the optimal investment opportunities of two stocks?
    6 years ago
  • Kelsey
    Which asset cannot lie on the efficient frontier?
    6 years ago
  • mikolaj
    Can you construct a portfolio of stocks that has an expected return equal to the risk free rate?
    5 years ago
  • adriana
    Does lending at the risk free rate lower expected return?
    2 years ago

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