Simple and optimal alpha strategy selection and risk budgeting or Goodbye to 91.5 per cent and all that

Abstract

A simple measure is developed that can determine if investment efficiency is increased by including an alpha strategy. If the correlation between alpha and beta is lower than the ratio of information to Sharpe ratios, the strategy should be pursued. A combined alpha and beta Sharpe ratio measure is developed and used to determine a simple but optimal strategy for an alpha-beta risk budget. When alpha-beta correlation is zero, the risk budget is optimal at the ratio of the information to Sharpe ratios. An optimal risk budget for non-zero correlation is also addressed.

Appendices

Appendix A

Equation (7) in the main body:

rearranging (1),

cleaning up (2),

factoring out the left-hand side produces:

dividing through by β,

replacing σ/β with the inverse of the Sharpe ratio,

Squaring both sides,

Factoring out the left-hand side,

Subtracting σ² from both sides gives:

Dividing both sides by TE produces:

Subtracting TE from both sides and rearranging the left-hand term, replacing α/TE with IR:

Dividing through by 2σ

Cleaning up:

Rearranging and subtracting the third term from the first produces:

If we define beta as:

then replacing beta in (A14) with (A15) gives:

removing the 2σ from the second term,

If we define alpha as:

Then replacing (A18) into (A17) gives:

We can define the risk budget, or ratio of tracking error to beta risk (σ) as:

Then substituting (A20) into (A19) gives:

as tracking error approaches zero, (A21) becomes

Appendix B

Rearranging (B1) and squaring both sides:

Expanding the left-hand side of (B2):

Extracting TE from the right-hand side of (B3)

Dividing (B4) by TE:

Replacing TE/σ with λ

Replacing α/TE with I (information ratio):

Replacing β with σS (S representing the Sharpe ratio):

Dividing through by σ:

Cleaning up:

Replacing α with IxTE

Replacing TE/σ with λ:

Cleaning up:

Rearranging (B13):

Taking the square root of both sides:

This can be simplified by multiplying numerator and denominator by λ/λ to:

Appendix C

As established in Appendix B, the total Sharpe ratio of the investment (incorporating both α and β) is:

To find the maximum of T, the approach is to set the first derivative of the function to zero. This is equivalent to setting T² to zero as well.

using the power rule where the derivative of f(x), which is the ratio of two functions, g(x) and h(x) can be determined as:

results in the derivative of (C1) being:

To find the maximum Sharpe ratio, the derivative is set to zero with respect to λ. Splitting the terms results in:

which reduces to:

removing negative exponents and rearranging:

multiplying both sides by (λ+1/λ+2ρ)²:

multiplying terms through results in:

Subtracting λI², adding S²/λ³ and rationalising the lambdas in numerators and denominators gives:

Rearranging gives:

Dividing both sides by 2 and rearranging gives:

Multiplying both sides by λ² gives:

Rearranging to equate to zero:

Factoring for lambdas results in:

Using the quadratic formula to solve for lambda where:

and:

Inserting the terms for a, b and c into (C17) gives:

which expands to:

and can be subsequently shortened to:

The square root in the numerator can now be removed:

which can be shortened to:

and finally dividing numerator and denominator by -2I

and when correlation between alpha and beta is zero, this shortens to

Equations (26) and (27) denote the optimal risk budget, or since the total Sharpe ratio (T) is at a maximum.