Suppose you are forming an equally weighted portfolio of stocks and
many stocks have the same beta of 0.84 for factor 1 and the same beta
of 1.69 for factor 2.  And all stocks have the same expected return of
11 percent. Assuming a two factor model describe the returns on each
of these stocks.
Write an equation of the returns on the portfolio if you place only
five stocks in it and write an equation of the returns on the
portfolio if there is a very large number of stocks placed with the
same expected returns and the same beta.

Hi!!

- Equation of the returns on the portfolio if you place only five stocks in it :

In general we have that:
Rp = SUM(Xi*Ri)  (i= 1 to 5)

Where:
Rp = return of the portfolio
Ri = actual return observed on asset i (i=1 to 5)
Xi = proportion in asset i (i=1 to 5)

Because this is a two factors model:
Ri = Ei + beta_1*F1 + beta_2*F2 + e_i 

Where:
Ei = expected return on asset i (i=1 to 5)
F1 and F2 denote the factors in this model
e_i = non-systematic or residual risk for asset i (i=1 to 5)


For this problem for each stock we have that:

Ri = 0.11 + 0.84*F1 + 1.69*F2 + e_i  (for i=1 to 5)

and (since it is an equally weighted portfolio of stocks):
Xi = 1/5 for all i

Then:

Rp = SUM(Xi*Ri) =         (i=1 to 5) 
   = SUM(1/5*Ri) =
   = 1/5*SUM(0.11 + 0.84*F1 + 1.69*F2 + e_i) =
   = 1/5*5*(0.11 + 0.84*F1 + 1.69*F2) + 1/5*SUM(e_i) = 
   = 0.11 + 0.84*F1 + 1.69*F2 + SUM(e_i)/5

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- Equation of the returns on the portfolio if there is a very large
number of stocks placed with the same expected returns and the same
beta:

If you place in this portfolio a very large number N of stocks that
all have the same expected returns and the same betas you will have
that:

Xi = 1/N for all i
Ei = E for all i

Rp = SUM(Xi*Ri) =  (i= 1 to N)
   = 1/N * SUM(Ri) =
   = 1/N * SUM(E + beta_1*F1 + beta_2*F2 + e_i) =
   = E + beta_1*F1 + beta_2*F2 + 1/N * SUM(e_i) =
   
Here we can ignore the term 1/N * SUM(e_i) because N is too large, and we obtain:

Rp = E + beta_1*F1 + beta_2*F2    

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I hope that this helps you. Feel free to request for a clarification
if you need it.

Best regards.
livioflores-ga