Sample exam Financial Analysis and Investor Behavior
Question 1 (35 points)
Zinedine manages a large fund called ‘Galacticos’ investing in the largest 100 European stocks.
Zinedine’s strategy is to invest in each stock according to the stock’s weight in the index (comprising
of exactly those 100 stocks), unless he receives views on any of the stocks from the battery of analysts
working for his investment firm
a. Explain in a bullet point list why the Black-Litterman model is an appropriate tool for Zinedine.
1. The portfolio slightly tilts toward views
2. You can add as many or as few views as you want
3. You can assign a confidence level to each view
4. Views can be absolute and relative
5. The Black-Litterman optimal weights are stable and intuitive
b. A good characterization of the variance-covariance matrix of the returns on the 100 stocks is a
key step in a successful tactical asset allocation model that Zinedine tries to develop. He will
rebalance his portfolio every month, so he needs a new estimate of the expected variance-
covariance matrix every month as well. A former Tilburg University Master in Finance graduate
that works now as a consultant to Zinedine’s fund advises him to use a simple factor structure
to model the conditional variance-covariance matrix. He wrote the following equations on
paper:
rm,t = μm + εm,t
εm,t ~N(0, hm,t )
rj,t = μj + βj εm,t + εj,t
εj,t ~N(0, hj ) for j = 1, … , 100
With rm,t is the return on the world market, hm,t the conditional market variance, and hj the
constant firm-specific variances. As a proxy for next period’s market variance, Zinedine plans
to use the volatility of daily returns over the last 3 months. Derive symbiotically the model-
implied conditional variance for firm 𝑘 , as well as its correlation with the market and with some
other firm z, with k, zϵ{1, … ,100}, k ≠ z.
Answer: We are interested in obtaining the variance of stock 𝑘 and the correlations with the
market (benchmark) and firm 𝑧. For the one-factor model, the market portfolio is also used as
the baseline portfolio. The expected returns stock 𝑘 is:
𝑟𝑘,𝑡 = 𝜇𝑘 + 𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡
The model-implied variance is then derived as follows:
𝑉𝑎𝑟(𝑟𝑘 , 𝑡) = 𝑉𝑎𝑟(𝜇𝑘 + 𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡 ) = ℎ𝑘 = 𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘
where ℎ𝜀𝑘 represents the idiosyncratic risk, ℎ𝑚 represents the variance of the market, and ℎ𝑘
represents the variance of returns on asset 𝑘. We can derive the volatility of stock 𝑘 by taking
the square root of the variance ℎ𝑘 .
, The correlation of firm 𝑘 with the market and firm z is calculated using the following formula:
𝐶𝑜𝑣(𝑟𝑘 , 𝑟𝑧 )
𝜌𝑘𝑧 =
√ℎ𝑘 ℎ𝑧
Hence, to determine the correlation we first need to calculate the covariance of stock 𝑘 with
stock 𝑧 and of stock 𝑘 with the market. Assume that 𝐶𝑜𝑣(𝜀𝑖 , 𝜀𝑗 ) = 0 for 𝑖 ≠ 𝑗 and
𝐶𝑜𝑣(𝜀𝑖 , 𝑅𝑚 ) = 0. Hence, the covariances are:
𝐶𝑜𝑣(𝑅𝑘 , 𝑅𝑚 ) = 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡 , 𝑅𝑚 ) =
𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝑅𝑚 ) + 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝜀𝑘,𝑡 ) + 𝐶𝑜𝑣(𝜀𝑘,𝑡 , 𝑅𝑚 ) = 𝜷𝒌 𝒉𝒎
𝐶𝑜𝑣(𝑅𝑘 , 𝑅𝑧 ) = 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 + 𝜀𝑧,𝑡 ) =
𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 ) + 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝜀𝑘,𝑡 ) + 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝜀𝑧,𝑡 ) +
𝐶𝑜𝑣(𝜀𝑘,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 ) + 𝐶𝑜𝑣(𝜀𝑧,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 ) + 𝐶𝑜𝑣(𝜀𝑧,𝑡 , 𝜀𝑘,𝑡 ) = 𝜷𝒌 𝜷𝒛 𝒉𝒎
By plugging in the covariances and the volatilities, we can find the correlation of stock 𝑘 with
the market and with firm 𝑧.
𝐶𝑜𝑣(𝑟𝑘 , 𝑟𝑧 ) 𝛽𝑘 𝛽𝑧 ℎ𝑚
𝜌𝑘𝑧 = =
√ℎ𝑘 ℎ𝑧 √𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘,𝑡 √𝛽𝑧2 ℎ𝑚 + ℎ𝜀𝑧,𝑡
𝐶𝑜𝑣(𝑟𝑘 , 𝑟𝑚 ) 𝛽𝑘 ℎ𝑚
𝜌𝑘𝑚 = =
√ℎ𝑘 ℎ𝑚 √𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘,𝑡
The variance-covariance matrix is built on variances on the diagonal axis and covariances in the
remainder. Hence, based on the above information we can derive the one-factor variance-
covariance matrix as follows:
𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘,𝑡 𝛽𝑘 𝛽𝑧 ℎ𝑚 𝛽𝑘 ℎ𝑚
𝑉=[ 2
𝛽𝑘 𝛽𝑧 ℎ𝑚 𝛽𝑧 ℎ𝑚 + ℎ𝜀𝑧,𝑡 𝛽𝑧 ℎ𝑚 ]
𝛽𝑘 ℎ𝑚 𝛽𝑧 ℎ𝑚 ℎ𝑚
Question 1 (35 points)
Zinedine manages a large fund called ‘Galacticos’ investing in the largest 100 European stocks.
Zinedine’s strategy is to invest in each stock according to the stock’s weight in the index (comprising
of exactly those 100 stocks), unless he receives views on any of the stocks from the battery of analysts
working for his investment firm
a. Explain in a bullet point list why the Black-Litterman model is an appropriate tool for Zinedine.
1. The portfolio slightly tilts toward views
2. You can add as many or as few views as you want
3. You can assign a confidence level to each view
4. Views can be absolute and relative
5. The Black-Litterman optimal weights are stable and intuitive
b. A good characterization of the variance-covariance matrix of the returns on the 100 stocks is a
key step in a successful tactical asset allocation model that Zinedine tries to develop. He will
rebalance his portfolio every month, so he needs a new estimate of the expected variance-
covariance matrix every month as well. A former Tilburg University Master in Finance graduate
that works now as a consultant to Zinedine’s fund advises him to use a simple factor structure
to model the conditional variance-covariance matrix. He wrote the following equations on
paper:
rm,t = μm + εm,t
εm,t ~N(0, hm,t )
rj,t = μj + βj εm,t + εj,t
εj,t ~N(0, hj ) for j = 1, … , 100
With rm,t is the return on the world market, hm,t the conditional market variance, and hj the
constant firm-specific variances. As a proxy for next period’s market variance, Zinedine plans
to use the volatility of daily returns over the last 3 months. Derive symbiotically the model-
implied conditional variance for firm 𝑘 , as well as its correlation with the market and with some
other firm z, with k, zϵ{1, … ,100}, k ≠ z.
Answer: We are interested in obtaining the variance of stock 𝑘 and the correlations with the
market (benchmark) and firm 𝑧. For the one-factor model, the market portfolio is also used as
the baseline portfolio. The expected returns stock 𝑘 is:
𝑟𝑘,𝑡 = 𝜇𝑘 + 𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡
The model-implied variance is then derived as follows:
𝑉𝑎𝑟(𝑟𝑘 , 𝑡) = 𝑉𝑎𝑟(𝜇𝑘 + 𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡 ) = ℎ𝑘 = 𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘
where ℎ𝜀𝑘 represents the idiosyncratic risk, ℎ𝑚 represents the variance of the market, and ℎ𝑘
represents the variance of returns on asset 𝑘. We can derive the volatility of stock 𝑘 by taking
the square root of the variance ℎ𝑘 .
, The correlation of firm 𝑘 with the market and firm z is calculated using the following formula:
𝐶𝑜𝑣(𝑟𝑘 , 𝑟𝑧 )
𝜌𝑘𝑧 =
√ℎ𝑘 ℎ𝑧
Hence, to determine the correlation we first need to calculate the covariance of stock 𝑘 with
stock 𝑧 and of stock 𝑘 with the market. Assume that 𝐶𝑜𝑣(𝜀𝑖 , 𝜀𝑗 ) = 0 for 𝑖 ≠ 𝑗 and
𝐶𝑜𝑣(𝜀𝑖 , 𝑅𝑚 ) = 0. Hence, the covariances are:
𝐶𝑜𝑣(𝑅𝑘 , 𝑅𝑚 ) = 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡 , 𝑅𝑚 ) =
𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝑅𝑚 ) + 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝜀𝑘,𝑡 ) + 𝐶𝑜𝑣(𝜀𝑘,𝑡 , 𝑅𝑚 ) = 𝜷𝒌 𝒉𝒎
𝐶𝑜𝑣(𝑅𝑘 , 𝑅𝑧 ) = 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 + 𝜀𝑘,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 + 𝜀𝑧,𝑡 ) =
𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 ) + 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝜀𝑘,𝑡 ) + 𝐶𝑜𝑣(𝛽𝑘 𝜀𝑚,𝑡 , 𝜀𝑧,𝑡 ) +
𝐶𝑜𝑣(𝜀𝑘,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 ) + 𝐶𝑜𝑣(𝜀𝑧,𝑡 , 𝛽𝑧 𝜀𝑚,𝑡 ) + 𝐶𝑜𝑣(𝜀𝑧,𝑡 , 𝜀𝑘,𝑡 ) = 𝜷𝒌 𝜷𝒛 𝒉𝒎
By plugging in the covariances and the volatilities, we can find the correlation of stock 𝑘 with
the market and with firm 𝑧.
𝐶𝑜𝑣(𝑟𝑘 , 𝑟𝑧 ) 𝛽𝑘 𝛽𝑧 ℎ𝑚
𝜌𝑘𝑧 = =
√ℎ𝑘 ℎ𝑧 √𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘,𝑡 √𝛽𝑧2 ℎ𝑚 + ℎ𝜀𝑧,𝑡
𝐶𝑜𝑣(𝑟𝑘 , 𝑟𝑚 ) 𝛽𝑘 ℎ𝑚
𝜌𝑘𝑚 = =
√ℎ𝑘 ℎ𝑚 √𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘,𝑡
The variance-covariance matrix is built on variances on the diagonal axis and covariances in the
remainder. Hence, based on the above information we can derive the one-factor variance-
covariance matrix as follows:
𝛽𝑘2 ℎ𝑚 + ℎ𝜀𝑘,𝑡 𝛽𝑘 𝛽𝑧 ℎ𝑚 𝛽𝑘 ℎ𝑚
𝑉=[ 2
𝛽𝑘 𝛽𝑧 ℎ𝑚 𝛽𝑧 ℎ𝑚 + ℎ𝜀𝑧,𝑡 𝛽𝑧 ℎ𝑚 ]
𝛽𝑘 ℎ𝑚 𝛽𝑧 ℎ𝑚 ℎ𝑚
