The Treynor and Black (1973) model offers a simple procedure to add a stock with a significant alpha to the passive market portfolio. The main ingredients of the model are the stock alpha (abnormal return), the stock beta, and the stock residual variance. Additional statistics are the market risk premium, and the market variance.
The case where there is only one stock, it is easy since it requires only few steps.
Suppose you have estimated a simple index model, where you regressed a stock excess return on a constant and the market excess return. Suppose both the constant and the beta are significant coefficients. Your alpha = 1.75%, beta = 1.75, residual standard deviation is 30%, the market standard deviation is 25%, the risk-free rate (rf) is 3%, and the market return (rm) is 14%.
The first step is to compute the ratio of alpha to the stock variance. This ratio is the base of the model, and it is referred to as the information variance ratio (IR).
IR = alpha/residual variance = 1.75%/30%^2 = 0.1944
The second step is to compute the ratio of the market risk premium (MRP) to the market variance, which is referred to as the market price of risk (MPR)
MRP = rm - rf = 14% - 3% = 11%
MPR = MRP/market variance = 11%/25%^2 = 1.76
The third step is to compute the ratio of IR to mpr (W0), which gives the risk-unadjusted investment proportion in the stock.
W0 = IR/MPR = 0.1944/1.76 = 0.11048
The fourth step is to adjust W0 for the beta risk. This is the case when beta is different from 1.
W1 = W0/(1+W0(1-beta)) = 0.11048/(1+0.11048(1-1.75)) = 0.12046.
W2 = 1 - W1 = 1 - 0.12046 = 0.87954
W1 is the investment proportion in the stock, whereas W2 is the investment proportion in the passive market portfolio.
The last step to obtain the performance of the active portfolio (AP) is simplified, where the optimal investment proportions do not appear.
AP = (SM^2 + alpha^2/residual variance)^0.5, where SM is the market Sharpe ratio
SM = 11%/25% = 0.44
AP = (0.44^2 + 1.75^2/30%^2)^0.5 = 0.44385
An improvement of 0.0583 over the passive market portfolio
***
The same steps apply when dealing with a number of stocks with significant alphas. Suppose the following:
alpha = [0.001, 0.0005, 0.021, 0.00135]', where alpha is a transposed row vector
sigma = [0.15, 0.08, 0.3, 0.18]', where sigma is a transposed row of standard deviations
ones = [1, 1, 1, 1]', where ones is a transposed row of ones
beta = [1.7, 1.1, 1.95,0.95]', where beta is a transposed row of betas
rho = [0.2, 0.35, 0.7, 0.8, -0.35]', where rho gives the correlation for r12, r13, r14, r23, and r34.
To obtain W1 for each of the stock, we use Excel. Let me give the formula in two parts. Start by making a 4 by 4 variance covariance matrix (VCM), and write in the cell the following
a = minverse(mmult(mmult(transpose(alpha),inverse(VCM)),ones), and take ctrl+shift+enter
a is a scalar. For b select 4 cells in a column and write in the first of the selected cells
b = mmult(inverse(VCM),alpha), and take ctrl+shift+enter
c = mmult(a,b), and take+shift+enter giving W*
bp = mmult(transpose(W*),beta), and take ctrl+shift+enter, where bp is the beta portfolio
ap = mmult(transpose(W*),alpha), and take ctrl+shift+enter, where ap is the alpha portfolio
vp = mmult(mmult(transpose(W*),VCM),W*), where vp is the portfolio variance
W0* = (ap/vp)/MPR = 0.2423
W1* = W0*/(1+W0*(1-bp)) = 0.2367
W2* = 1 - W1* = 0.7633
AP* = (SM^2 + ap^2/vp)^0.5 = 0.447903
An improvement of 0.0838 over the passive market portfolio
***
The case where there is only one stock, it is easy since it requires only few steps.
Suppose you have estimated a simple index model, where you regressed a stock excess return on a constant and the market excess return. Suppose both the constant and the beta are significant coefficients. Your alpha = 1.75%, beta = 1.75, residual standard deviation is 30%, the market standard deviation is 25%, the risk-free rate (rf) is 3%, and the market return (rm) is 14%.
The first step is to compute the ratio of alpha to the stock variance. This ratio is the base of the model, and it is referred to as the information variance ratio (IR).
IR = alpha/residual variance = 1.75%/30%^2 = 0.1944
The second step is to compute the ratio of the market risk premium (MRP) to the market variance, which is referred to as the market price of risk (MPR)
MRP = rm - rf = 14% - 3% = 11%
MPR = MRP/market variance = 11%/25%^2 = 1.76
The third step is to compute the ratio of IR to mpr (W0), which gives the risk-unadjusted investment proportion in the stock.
W0 = IR/MPR = 0.1944/1.76 = 0.11048
The fourth step is to adjust W0 for the beta risk. This is the case when beta is different from 1.
W1 = W0/(1+W0(1-beta)) = 0.11048/(1+0.11048(1-1.75)) = 0.12046.
W2 = 1 - W1 = 1 - 0.12046 = 0.87954
W1 is the investment proportion in the stock, whereas W2 is the investment proportion in the passive market portfolio.
The last step to obtain the performance of the active portfolio (AP) is simplified, where the optimal investment proportions do not appear.
AP = (SM^2 + alpha^2/residual variance)^0.5, where SM is the market Sharpe ratio
SM = 11%/25% = 0.44
AP = (0.44^2 + 1.75^2/30%^2)^0.5 = 0.44385
An improvement of 0.0583 over the passive market portfolio
***
The same steps apply when dealing with a number of stocks with significant alphas. Suppose the following:
alpha = [0.001, 0.0005, 0.021, 0.00135]', where alpha is a transposed row vector
sigma = [0.15, 0.08, 0.3, 0.18]', where sigma is a transposed row of standard deviations
ones = [1, 1, 1, 1]', where ones is a transposed row of ones
beta = [1.7, 1.1, 1.95,0.95]', where beta is a transposed row of betas
rho = [0.2, 0.35, 0.7, 0.8, -0.35]', where rho gives the correlation for r12, r13, r14, r23, and r34.
To obtain W1 for each of the stock, we use Excel. Let me give the formula in two parts. Start by making a 4 by 4 variance covariance matrix (VCM), and write in the cell the following
a = minverse(mmult(mmult(transpose(alpha),inverse(VCM)),ones), and take ctrl+shift+enter
a is a scalar. For b select 4 cells in a column and write in the first of the selected cells
b = mmult(inverse(VCM),alpha), and take ctrl+shift+enter
c = mmult(a,b), and take+shift+enter giving W*
bp = mmult(transpose(W*),beta), and take ctrl+shift+enter, where bp is the beta portfolio
ap = mmult(transpose(W*),alpha), and take ctrl+shift+enter, where ap is the alpha portfolio
vp = mmult(mmult(transpose(W*),VCM),W*), where vp is the portfolio variance
W0* = (ap/vp)/MPR = 0.2423
W1* = W0*/(1+W0*(1-bp)) = 0.2367
W2* = 1 - W1* = 0.7633
AP* = (SM^2 + ap^2/vp)^0.5 = 0.447903
An improvement of 0.0838 over the passive market portfolio
***
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