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# P1.T1.20.8. Modern portfolio theory (MPT)

#### Elizabeth_Babalola

##### Member
This does not relate to this thread. Unfortunately, i could not find a suitable one. Taking a look at the second bullet point under the relationship between Sharpe and Jensen; are we dividing by $\sigma p\;$ or $\sigma m\;$

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Elizabeth_Babalola dividing by σ(P) because Jensen's alpha says E(Rp) - Rf = α(P) + β*E(Rm - Rf) and dividing by α(P) gives [E(Rp) - Rf]/α(P) = α(P)/α(P) + β/α(P)*E(Rm - Rf), but if B ~ σ(P)/σ(M) then β/α(P) = σ(P)/σ(M) * 1/σ(P) = 1/σ(M) and α(P) = α(P)/α(P) + 1/σ(M)*E(Rm - Rf).

Append, mistake(s) above, I meant:
dividing by σ(P) because Jensen's alpha says E(Rp) - Rf = α(P) + β*E(Rm - Rf) and dividing by σ(P) gives [E(Rp) - Rf]/σ(P) = α(P)/σ(P) + β/σ(P)*E(Rm - Rf), but if B ~ σ(P)/σ(M) then β/σ(P) = σ(P)/σ(M) * 1/σ(P) = 1/σ(M) and [E(Rp) - Rf]/σ(P) = α(P)/σ(P) + 1/σ(M)*E(Rm - Rf).

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#### Elizabeth_Babalola

##### Member
Thanks @David.

The last equation is not clear though

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Elizabeth_Babalola sorry, in haste i wrote α(P) instead of σ(P). Fixed above. Should be as follows:

E(Rp) - Rf = α(P) + β*E(Rm - Rf); divide by sides by σ(P):
[E(Rp) - Rf]/σ(P) = [α(P) + β*E(Rm - Rf)] / σ(P) = α(P)/σ(P) + β*E(Rm - Rf)/σ(P). But β ~= σ(P)/σ(M), so
[E(Rp) - Rf]/σ(P) = α(P)/σ(P) + [σ(P)/σ(M)]*E(Rm - Rf)/σ(P); and we can cancel σ(P)/σ(P):
[E(Rp) - Rf]/σ(P) = α(P)/σ(P) + [σ(P)/σ(M)]*E(Rm - Rf)/σ(P) = α(P)/σ(P) + *E(Rm - Rf)/σ(M)

#### Elizabeth_Babalola

##### Member
Duly noted. Thanks