Financial Investments

Practice exercises

These are multiple choice questions that should be used to practice your level of mastery of the material.

They are sample exercises and should not be interpreted as a complete set of exercises that can be created for this chapter material.

Please note that doing the required readings is essential.

Question 1

Consider two risky securities, A and B, with a correlation of 0.039. Security A has an expected rate of return of 14% and a standard deviation of return of 19.5%. B has an expected rate of return 8.2% and a standard deviation of return of 11.8%. If an investor wants to create a portfolio with these two assets, with a standard deviation of 13.5%, what proportion should she invest in asset A?

Considere dois activos A e B com uma correlação de 0.039. O activo A tem uma rendibilidade esperada de 14% e um desvio padrão de 19.5%. O activo B tem uma rendibilidade esperada de 8.2% e um desvio padrão de 11.8%. Se um investidor pretender criar uma carteira com esses dois activos com um desvio padrão de 13.5%, qual a proporção que recomendaria que investisse no activo A?

Make sure you find an efficient portfolio.

cov(a,b) = \rho_{a,b} \sigma_a \sigma_b\\ cov(a,b) = 0.039 \times 0.195 \times 0.118\\ cov(a,b) = 0.0009

We also know that w_a + w_b = 1

\sigma_p^2 = w_a^2 \sigma_a^2 + w_b^2 \sigma_b^2 + 2 w_a w_b cov(a,b) \\ 0.135^2 = w_a^2 \sigma_a^2 + (1-w_a)^2 \sigma_b^2 + 2 w_a(1-w_a) cov(a,b)\\ 0.135^2 = w_a^2 0.195^2 + (1-w_a)^2 0.118^2 + 2 w_a(1-w_a) 0.0009\\

Now solve for w_a: w_a = -0.1317 \;V\; w_a = 0.65116

We need to find which of the two solutions correspont to an efficient portfolio. We do this, by calculating the expected return for each portfolio:

If w_a = -0.1317 then: E[r_p] = w_a E[r_a] + w_b E[r_b]\\ E[r_p] = -0.1317 \times 0.14 + (1+0.1317) 0.082\\ E[r_p] = 0.07436

If w_a = 0.65116 then: E[r_p] = w_a E[r_a] + w_b E[r_b]\\ E[r_p] = 0.65116 \times 0.14 + (1-0.65116) 0.082\\ E[r_p] = 0.11977

The solution is then: w_a = 0.65116

Question 2

A portfolio is composed of two stocks, A and B. Stock A has a standard deviation of return of 12.8% while stock B has a standard deviation of return of 29.2%. The correlation coefficient between the returns on A and B is 0.04. Stock A comprises 58.8% of the portfolio while stock B comprises 41.2% of the portfolio. The variance of return on the portfolio is:

Uma carteira é composta pelos activos A e B. As rendibilidades do activo A têm um desvio padrão de 12.8%, e as do activo B um desvio padrão de 29.2%. A correlação entre as rendibilidades dos dois activos é de 0.04. O peso do activo A na carteira é de 58.8% e o do activo B é de 41.2%. A variância das rendibilidades da carteira é então de:

cov(a,b) = \rho_{a,b} \sigma_a \sigma_b\\ cov(a,b) = 0.04 \times 0.128 \times 0.292\\ cov(a,b) = 0.0015

We also know that w_a= 0.588 and w_b = 0.412

\sigma_p^2 = w_a^2 \sigma_a^2 + w_b^2 \sigma_b^2 + 2 w_a w_b cov(a,b) \\ \sigma_p^2 = 0.588^2 \times 0.128^2 + 0.412^2 \times 0.292^2 + 2 \times 0.588 \times 0.412 \times 0.0015\\ \sigma_p^2 = 0.02086

Question 3

The standard deviation of return on investment A is 9.1% while the standard deviation of return on investment B is 14.5%. If the correlation coefficient between the returns on A and B is 0.63, the covariance of returns on A and B is:

O desvio padrão das rendibilidades de um investimento em A é de 9.1%, enquanto o desvio padrão das rendibilidades de um investimento em B é de 14.5%. Se o coeficiente de correlação entre A e B for de 0.63, a covariância entre as rendibilidades de A e B é de:

cov(a,b) = \rho_{a,b} \sigma_a \sigma_b\\ cov(a,b) = 0.63 \times 0.091 \times 0.145\\ cov(a,b) = 0.00831

Question 4

Consider two assets A and B. The return correlation between the two assets is 0.04. Security A has an expected rate of return of 14% and a standard deviation of return of 19.5%. B has an expected rate of return of 14% and a standard deviation of return of 11.8%. The weight of security B in the global minimum variance portfolio is:

Considere dois activos A e B com uma correlação de 0.04. O activo A tem uma rendibilidade esperada de 14% e um desvio padrão de 19.5%. O activo B tem uma rendibilidade esperada de 22.2% e um desvio padrão de 11.8%. O peso do activo B na carteira de variância mínima é:

You have to find the weights that minimize the variance of the portfolio built using these two assets. Be aware that you must provide the weight of asset B in that portfolio.

The mininum variance portfolio is found by setting the partial derivative of the portfolio variance with respect to w_a, to zero: \frac{\partial \sigma_p^2}{\partial w_a} = 0\\ w_a = \frac{\sigma_b^2 - \rho\sigma_a\sigma_b}{\sigma_b^2 - 2\rho\sigma_a\sigma_b+\sigma_a^2}\\ w_a = 0.25951 Therefore the weight of B in the portfolio is 1-w_a= 0.74049.

Question 5

Which of the following is the most likely reward to variability ratio for a capital allocation line that is optimal, assuming all ratios are generated from the same set of potential assets?

Qual das seguintes opções é mais provável que corresponda ao rácio de prémio de risco por unidade de risco de uma linha do mercado de capitais que é optima, assumindo que todos os rácios foram gerados a partir do mesmo conjunto de activos?

Consider the characteristics of the optimal portfolio of risky assets.

The optimal portfolio of risky assets is the one with the highest reward-to-variability ratio possible.