Apostila - Teoria de Carteiras

Prévia do material em texto

https://www.youtube.com/watch?v=PfVaOn9W3mo
https://www.youtube.com/watch?v=PfVaOn9W3mo
https://www.youtube.com/watch?v=PfVaOn9W3mo
https://www.youtube.com/watch?v=PfVaOn9W3mo
https://www.youtube.com/watch?v=PfVaOn9W3mo
https://www.youtube.com/watch?v=PfVaOn9W3mo
http://www.planalto.gov.br/ccivil_03/constituicao/constituicaocompilado.htm
http://www.planalto.gov.br/ccivil_03/constituicao/constituicaocompilado.htm
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Retornos dos ativos
Tempo
Ativo 1 Ativo 2
 
 
 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Retornos dos ativos
Tempo
Ativo 1 Ativo 2
 
 
 
𝜌𝑋,𝑌 =
𝑐𝑜𝑣(𝑋, 𝑌)
𝑆(𝑋). 𝑆(𝑌)
 𝑐𝑜𝑣(𝑋, 𝑌)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Retornos dos ativos
Tempo
Ativo 1 Ativo 2
 
 
 
 𝑆(𝑋)
 𝑆(𝑌)
𝑅𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎 = 𝑤1𝑅𝐴𝑡𝑖𝑣𝑜1 + 𝑤2𝑅𝐴𝑡𝑖𝑣𝑜2
 𝑤1
 𝑤2
 𝑅𝐴𝑡𝑖𝑣𝑜1
 𝑅𝐴𝑡𝑖𝑣𝑜2
�̃�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎 = 𝑤1�̃�𝐴𝑡𝑖𝑣𝑜1 + 𝑤2�̃�𝐴𝑡𝑖𝑣𝑜2
𝐸[�̃�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎]
 
 
 
�̅�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎 = 𝐸[�̃�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎]
�̅�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎
�̅�𝐶
𝐸[�̃�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎] = 𝐸[𝑤1�̃�𝐴𝑡𝑖𝑣𝑜1 + 𝑤2�̃�𝐴𝑡𝑖𝑣𝑜2]
𝐸[�̃�𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎] = 𝑤1𝐸[�̃�𝐴𝑡𝑖𝑣𝑜1] + 𝑤2𝐸[�̃�𝐴𝑡𝑖𝑣𝑜2]
�̅�𝐶 = 𝑤1�̅�𝐴𝑡𝑖𝑣𝑜1 + 𝑤2�̅�𝐴𝑡𝑖𝑣𝑜2
 �̅�𝐴𝑡𝑖𝑣𝑜1
 
 
 
 �̅�𝐴𝑡𝑖𝑣𝑜2
𝑆𝐶
2 = 𝑆2(�̃�𝐶) = 𝑆
2(𝑤1�̃�𝐴𝑡𝑖𝑣𝑜1 + 𝑤2�̃�𝐴𝑡𝑖𝑣𝑜2)
�̃�𝐴𝑡𝑖𝑣𝑜1 �̃�1 �̃�𝐴𝑡𝑖𝑣𝑜2 �̃�2
𝑆𝐶
2 = 𝑆2(𝑤1�̃�1 + 𝑤2�̃�2)
𝑆𝐶
2 = 𝑆2(𝑤1�̃�1) + 𝑆
2(𝑤2�̃�2) + 2𝑐𝑜𝑣(𝑤1�̃�1, 𝑤2�̃�2)
𝑆𝐶
2 = 𝑤1
2𝑆2(�̃�1) + 𝑤2
2𝑆2(�̃�2) + 2𝑤1𝑤2𝑐𝑜𝑣(�̃�1, �̃�2)
𝑆2(�̃�1) 𝑆1
2 𝑆2(�̃�2) 𝑆2
2
𝑆𝐶
2 = 𝑤1
2𝑆1
2 + 𝑤2
2𝑆2
2 + 2𝑤1𝑤2𝑐𝑜𝑣(�̃�1, �̃�2)
 
 
 
𝑆𝐶
2 = (𝑤1𝑆1)
2 + (𝑤2𝑆2)
2 + 2𝑤1𝑤2𝑐𝑜𝑣(�̃�1, �̃�2)
𝑆𝐶
2 = (𝑤1𝑆1)
2 + (𝑤2𝑆2)
2 + 2(𝑤1𝑆1)(𝑤2𝑆2)𝜌1,2
𝜌1,2
𝑆𝐶
2 = (𝑤1𝑆1)
2 + (𝑤2𝑆2)
2 + 2(𝑤1𝑆1)(𝑤2𝑆2)1
𝑤1𝑆1 𝑤2𝑆2
𝑆𝐶
2 = 𝑋2 + 𝑌2 + 2𝑋𝑌
𝑆𝐶
2 = (𝑤1𝑆1 + 𝑤2𝑆2)
2
 
 
 
𝑆𝐶 = 𝑤1𝑆1 + 𝑤2𝑆2
𝜌1,2
𝑆𝐶
2 = (𝑤1𝑆1)
2 + (𝑤2𝑆2)
2 − 2(𝑤1𝑆1)(𝑤2𝑆2)
𝑤1𝑆1 𝑤2𝑆2
𝑆𝐶
2 = 𝑋2 + 𝑌2 − 2𝑋𝑌
𝑆𝐶
2 = (𝑤1𝑆1 − 𝑤2𝑆2)
2
 
 
 
𝑆𝐶 = |𝑤1𝑆1 − 𝑤2𝑆2|
 
 
 
 
𝑆𝐶
2 = (𝑤𝐴𝑆𝐴)
2 + (𝑤𝑟𝑓𝑆𝑟𝑓)
2
+ 2(𝑤𝐴𝑆𝐴)(𝑤𝑟𝑓𝑆𝑟𝑓)𝜌𝐴,𝑟𝑓
𝑆𝐶
2 = (𝑤𝐴𝑆𝐴)
2 + (𝑤𝑟𝑓0)
2
+ 2(𝑤𝐴𝑆𝐴)(𝑤𝑟𝑓0)0
𝑆𝐶
2 = (𝑤𝐴𝑆𝐴)
2 + 0 + 0
𝑆𝐶 = 𝑤𝐴𝑆𝐴
�̅�𝐶 = 𝑤𝐴�̅�𝐴 + 𝑤𝑟𝑓𝐼𝑓
 𝐼𝑓
𝑤𝐴 + 𝑤𝑟𝑓 = 1
𝑤𝑟𝑓 = 1 − 𝑤𝐴
�̅�𝐶 = 𝑤𝐴�̅�𝐴 + (1 − 𝑤𝐴)𝐼𝑓
 
 
 
𝑤𝐴 =
𝑆𝐶
𝑆𝐴
�̅�𝐶 =
𝑆𝐶
𝑆𝐴
�̅�𝐴 + (1 −
𝑆𝐶
𝑆𝐴
) 𝐼𝑓
�̅�𝐶 =
𝑆𝐶
𝑆𝐴
�̅�𝐴 + 𝐼𝑓 −
𝑆𝐶
𝑆𝐴
𝐼𝑓
�̅�𝐶 = 𝐼𝑓 +
𝑆𝐶
𝑆𝐴
(�̅�𝐴 − 𝐼𝑓)
�̅�𝐶 = 𝐼𝑓 +
(�̅�𝐴 − 𝐼𝑓)
𝑆𝐴
𝑆𝐶
�̅�𝐶
𝐼𝑓
𝑆𝐶
(�̅�𝐴−𝐼𝑓)
𝑆𝐴
�̅�𝐶 𝑆𝐶
 
 
 
�̅�𝐶 𝐼𝑓
 
 
 
 
 
 
0,00%
0,10%
0,20%
0,30%
0,40%
0,50%
0,60%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50%
Retorno Esperado
Risco (desvio padrão)
Ativo A
Ativo livre de 
risco
If
Carteiras alavancadas 
(wA > 100%; wIf < 0%)
 
 
 
0,00%
0,10%
0,20%
0,30%
0,40%
0,50%
0,60%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50%
Retorno Esperado
Risco (desvio padrão)
Ativo A
Ativo livre de 
risco
If
Carteiras alavancadas 
(wA < 100%; wIf > 0%)
 
 
 
 
 
 
 
 
 
 
Ativo 1
Ativo 2
Ativo 3
0,00%
0,10%
0,20%
0,30%
0,40%
0,50%
0,60%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50%
Retorno Esperado
Risco (desvio padrão)
 
 
 
Ativo 1
Ativo 2
Ativo 3
0,00%
0,10%
0,20%
0,30%
0,40%
0,50%
0,60%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50%
Retorno Esperado
Risco (desvio padrão)
 
 
 
Ativo 1
Ativo 2
Ativo 3
0,00%
0,10%
0,20%
0,30%
0,40%
0,50%
0,60%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50%
Retorno Esperado
Risco (desvio padrão)
 
 
 
Ativo 1
Ativo 2
Ativo 3
0,00%
0,10%
0,20%
0,30%
0,40%
0,50%
0,60%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50%
Retorno Esperado
Risco (desvio padrão)
 
 
 
 
 
 
𝐸[�̃�𝐶] = 𝑤1𝐸[�̃�1] + 𝑤2𝐸[�̃�2] + 𝑤3𝐸[�̃�3]
o �̃�𝐶
o �̃�1
o �̃�2
o �̃�3
o 𝑤1
o 𝑤2
o 𝑤3
�̅�𝐶 = 𝑤1�̅�1 + 𝑤2�̅�2 + 𝑤3�̅�3
o �̅�𝐶
o �̅�1
 
 
 
o �̅�2
o �̅�3
𝑆𝐶
2 = 𝑆2(�̃�𝐶) = 𝑆
2(𝑤1�̃�1 + 𝑤2�̃�2 + 𝑤3�̃�3)
𝑆𝐶
2 = 𝑆2(𝑤1�̃�1) + 𝑆
2(𝑤2�̃�2) + 𝑆
2(𝑤3�̃�3) + 2𝑐𝑜𝑣(𝑤1�̃�1, 𝑤2�̃�2)
+ 2𝑐𝑜𝑣(𝑤1�̃�1, 𝑤3�̃�3) + 2𝑐𝑜𝑣(𝑤2�̃�2, 𝑤3�̃�3)
𝑆𝐶
2 = 𝑤1
2𝑆2(�̃�1) + 𝑤2
2𝑆2(�̃�2) + 𝑤3
2𝑆2(�̃�3) + 2𝑤1𝑤2𝑐𝑜𝑣(�̃�1, �̃�2)
+ 2𝑤1𝑤3𝑐𝑜𝑣(�̃�1, �̃�3) + 2𝑤2𝑤3𝑐𝑜𝑣(�̃�2, �̃�3)
𝑆2(�̃�𝑋) 𝑆𝑋
2
𝑐𝑜𝑣(�̃�𝑋 , �̃�𝑌) = 𝜌𝑋,𝑌𝑆𝑋𝑆𝑌
𝑆𝐶
2 = 𝑤1
2𝑆1
2 + 𝑤2
2𝑆2
2 + 𝑤3
2𝑆3
2 + 2𝑤1𝑤2𝜌1,2𝑆1𝑆2 + 2𝑤1𝑤3𝜌1,3𝑆1𝑆3
+ 2𝑤2𝑤3𝜌2,3𝑆2𝑆3
 
 
 
𝑆𝐶
2 = (𝑤1𝑆1)
2 + (𝑤2𝑆2)
2 + (𝑤3𝑆3)
2 + 2(𝑤1𝑆1)(𝑤2𝑆2)𝜌1,2
+ 2(𝑤1𝑆1)(𝑤3𝑆3)𝜌1,3 + 2(𝑤2𝑆2)(𝑤3𝑆3)𝜌2,3
�̅�𝐶 = 𝑤1�̅�1 + 𝑤2�̅�2 + 𝑤3�̅�3
= [𝑤1 𝑤2 𝑤3] [
�̅�1
�̅�2
�̅�3
]
𝑆𝐶
2 = (𝑤1𝑆1)
2 + (𝑤2𝑆2)
2 + (𝑤3𝑆3)
2 + 2(𝑤1𝑆1)(𝑤2𝑆2)𝜌1,2
+ 2(𝑤1𝑆1)(𝑤3𝑆3)𝜌1,3 + 2(𝑤2𝑆2)(𝑤3𝑆3)𝜌2,3 =
𝑆𝐶
2 = [𝑤1𝑆1 𝑤2𝑆2 𝑤3𝑆3] [
𝜌1,1 𝜌1,2 𝜌1,3
𝜌2,1 𝜌2,2 𝜌2,3
𝜌3,1 𝜌3,2 𝜌3,3
] [
𝑤1𝑆1
𝑤2𝑆2
𝑤3𝑆3
]
 
 
 
�̅�𝐶 = 𝑤1�̅�1 + 𝑤2�̅�2 + 𝑤3�̅�3 + ⋯+ 𝑤𝑛�̅�𝑛 = [𝑤1 𝑤2 𝑤3 … 𝑤𝑛]
[
 
 
 
 
�̅�1
�̅�2
�̅�3
…
�̅�𝑛]
 
 
 
 
𝑆𝐶
2
= [𝑤1𝑆1 𝑤2𝑆2 𝑤3𝑆3 … 𝑤𝑛𝑆𝑛]
[
 
 
 
 
𝜌1,1 𝜌1,2 𝜌1,3 … 𝜌1,𝑛
𝜌2,1 𝜌2,2 𝜌2,3 … 𝜌2,𝑛
𝜌3,1 𝜌3,2 𝜌3,3 … 𝜌3,𝑛
… … … … …
𝜌𝑛,1 𝜌𝑛,2 𝜌𝑛,3 … 𝜌𝑛,𝑛]
 
 
 
 
[
 
 
 
 
𝑤1𝑆1
𝑤2𝑆2
𝑤3𝑆3
…
𝑤𝑛𝑆𝑛]
 
 
 
 
𝜌𝑗,𝑘
 
 
 
 
 
 
 
Fronteira para 4 ativos 
com risco
Fronteira para 3 ativos 
com risco
 
 
 
Fronteira para 5 
ativos com risco
 
 
 
 
 
 
Fronteira Eficiente 
(ponto de vista dos 
ativos / investimento)
Carteira de 
mínimo risco 
Fronteira Eficiente (ponto 
de vista dos passivos / 
financiamento)
 
 
 
 
 
 
 
 
 
 
P
a1
1
 
 
 
𝑦 = 𝑎 + 𝑏𝑥
𝑦 = 𝑎1 + 𝑏1𝑥
P
a1
1
a2
2
 
 
 
𝑦 = 𝑎2 + 𝑏2𝑥
P
a1
1
a2
2
at
3
 
 
 
𝑦 = 𝑎𝑡 + 𝑏𝑡𝑥 = 𝑟𝑡
𝑎𝑡 𝑎𝑡
�̅�𝐶 = 𝑎 + 𝑏𝑉
 �̅�𝐶
 𝑎
 𝑏
 𝑉
𝑎1 𝑎2
𝑎𝑡
𝑎 = �̅�𝐶 − 𝑏𝑉
𝑎
 
 
 
𝑤1 + 𝑤2 + 𝑤3 = 100%
min𝑎 = min(�̅�𝐶 − 𝑏𝑉)
𝑎
(�̅�𝐶 − 𝑏𝑉) = λ(w1 + w2 + w3
− 1)
λ
𝐹(w1, w2, w3, λ) = �̅�𝐶 − 𝑏𝑉 − 𝑎
w1 w2 w3 λ �̅�𝐶
 
 
 
𝐹(w1, w2, w3, λ)
= 𝑤1�̅�1 + 𝑤2�̅�2 + 𝑤3�̅�3
− 𝑏[𝑤1
2𝑆1
2 + 𝑤2
2𝑆2
2 + 𝑤3
2𝑆3
2 + 2𝑤1𝑤2𝑐𝑜𝑣(�̃�1, �̃�2)
+ 2𝑤1𝑤3𝑐𝑜𝑣(�̃�1, �̃�3) + 2𝑤2𝑤3𝑐𝑜𝑣(�̃�2, �̃�3)]
− λ(w1 + w2 + w3 − 1)
𝜕𝐹
𝜕𝑤1
= 0 = �̅�1 − 𝑏[2𝑤1𝑆1
2 + 2𝑤2𝑐𝑜𝑣(�̃�1, �̃�2) + 2𝑤3𝑐𝑜𝑣(�̃�1, �̃�3)] − λ
𝜕𝐹
𝜕𝑤2
= 0 = �̅�2 − 𝑏[2𝑤2𝑆2
2 + 2𝑤1𝑐𝑜𝑣(�̃�1, �̃�2) + 2𝑤3𝑐𝑜𝑣(�̃�2, �̃�3)] − λ
𝜕𝐹
𝜕𝑤3
= 0 = �̅�3 − 𝑏[2𝑤3𝑆3
2 + 2𝑤1𝑐𝑜𝑣(�̃�1, �̃�3) + 2𝑤2𝑐𝑜𝑣(�̃�2, �̃�3)] − λ
𝜕𝐹
𝜕λ
= 0 = −(w1 + w2 + w3 − 1)
𝑏 w1 w2 w3 λ
 
 
 
�̅�1
𝑏𝑡
= 2𝑆1
2𝑤1
∗ + 2𝑐𝑜𝑣(�̃�1, �̃�2)𝑤2
∗ + 2𝑐𝑜𝑣(�̃�1, �̃�3)𝑤3
∗ +
λ∗
𝑏𝑡
�̅�2
𝑏𝑡
= 2𝑐𝑜𝑣(�̃�1, �̃�2)𝑤1
∗ + 2𝑆2
2𝑤2
∗ + 2𝑐𝑜𝑣(�̃�2, �̃�3)𝑤3
∗ +
λ∗
𝑏𝑡
�̅�3
𝑏𝑡
= 2𝑐𝑜𝑣(�̃�1, �̃�3)𝑤1
∗ + 2𝑐𝑜𝑣(�̃�2, �̃�3)𝑤2
∗ + 2𝑆3
2𝑤3
∗ +
λ∗
𝑏𝑡
1 = 1𝑤1
∗ + 1𝑤2
∗ + 1𝑤3
∗ +
λ
𝑏𝑡
0
[
 
 
 
�̅�1 𝑏𝑡⁄
�̅�2 𝑏𝑡⁄
�̅�3 𝑏𝑡⁄
1 ]
 
 
 
=
[
 
 
 
 
2𝑆1
2 2𝑐𝑜𝑣(�̃�1, �̃�2) 2𝑐𝑜𝑣(�̃�1, �̃�3) 1
2𝑐𝑜𝑣(�̃�2, �̃�1) 2𝑆2
2 2𝑐𝑜𝑣(�̃�2, �̃�3) 1
2𝑐𝑜𝑣(�̃�3, �̃�1) 2𝑐𝑜𝑣(�̃�3, �̃�2) 2𝑆3
2 1
1 1 1 0]
 
 
 
 
[
 
 
 
𝑤1
∗
𝑤2
∗
𝑤3
∗
λ∗ 𝑏𝑡⁄ ]
 
 
 
𝑤1
∗ 𝑤2
∗ 𝑤3
∗
[
 
 
 
𝑤1
∗
𝑤2
∗
𝑤3
∗
λ∗ 𝑏𝑡⁄ ]
 
 
 
=
[
 
 
 
 
2𝑆1
2 2𝑐𝑜𝑣(�̃�1, �̃�2) 2𝑐𝑜𝑣(�̃�1, �̃�3) 1
2𝑐𝑜𝑣(�̃�2, �̃�1) 2𝑆2
2 2𝑐𝑜𝑣(�̃�2, �̃�3) 1
2𝑐𝑜𝑣(�̃�3, �̃�1) 2𝑐𝑜𝑣(�̃�3, �̃�2) 2𝑆3
2 1
1 1 1 0]
 
 
 
 
−1
𝑥
[
 
 
 
�̅�1 𝑏𝑡⁄
�̅�2 𝑏𝑡⁄
�̅�3 𝑏𝑡⁄
1 ]
 
 
 
w1 + w2 + w3 = 100%
 
 
 
w1 w2 w3
𝑤1
∗ 𝑤2
∗ 𝑤3
∗
C(w1,w2,w3)
 
 
 
�̅�𝐶 = [𝑤1 𝑤2 𝑤3 … 𝑤𝑛]
[
 
 
 
 
�̅�1
�̅�2
�̅�3
…
�̅�𝑛]
 
 
 
 
= [𝑤][�̅�𝑗]
𝑇
𝑆𝐶
2 = [𝑤1𝑆1 𝑤2𝑆2 … 𝑤𝑛𝑆𝑛] [
𝜌1,1 𝜌1,2 … 𝜌1,𝑛
𝜌2,1 𝜌2,2… 𝜌2,𝑛
… … … …
𝜌𝑛,1 𝜌𝑛,2 … 𝜌𝑛,𝑛
] [
𝑤1𝑆1
𝑤2𝑆2
…
𝑤𝑛𝑆𝑛
] = [𝑤𝑆][𝜌𝑗,𝑘][𝑤𝑆]
𝑇
𝐶(𝑤1
∗, 𝑤2
∗, 𝑤3
∗) 
C(w1,w2,w3)
 
 
 
[
 
 
 
 
 
 
𝑤1
∗
𝑤2
∗
𝑤3
∗
…
𝑤𝑛
∗
λ∗ 𝑏𝑡⁄ ]
 
 
 
 
 
 
=
[
 
 
 
 
 
 
2𝑆1
2 2𝑐𝑜𝑣(�̃�1, �̃�2) 2𝑐𝑜𝑣(�̃�1, �̃�3) … 2𝑐𝑜𝑣(�̃�1, �̃�𝑛) 1
2𝑐𝑜𝑣(�̃�2, �̃�1) 2𝑆2
2 2𝑐𝑜𝑣(�̃�2, �̃�3) … 2𝑐𝑜𝑣(�̃�2, �̃�𝑛) 1
2𝑐𝑜𝑣(�̃�3, �̃�1) 2𝑐𝑜𝑣(�̃�3, �̃�2) 2𝑆3
2 … 2𝑐𝑜𝑣(�̃�3, �̃�𝑛) 1
… … … … … 1
2𝑐𝑜𝑣(�̃�𝑛, �̃�1) 2𝑐𝑜𝑣(�̃�𝑛, �̃�2) 2𝑐𝑜𝑣(�̃�𝑛, �̃�3) … 2𝑆𝑛
2 1
1 1 1 1 1 0]
 
 
 
 
 
 
−1
𝑥
[
 
 
 
 
 
�̅�1 𝑏𝑡⁄
�̅�2 𝑏𝑡⁄
�̅�3 𝑏𝑡⁄
…
�̅�𝑛 𝑏𝑡⁄
1 ]
 
 
 
 
 
Ativo 1
Ativo 2
Ativo 3
0%
5%
10%
15%
20%
25%
30%
35%
40%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%
Retorno Esperado 
Risco (desvio padrão)
 
 
 
𝑐𝑜𝑣(�̃�1, �̃�2)
𝑐𝑜𝑣(�̃�1, �̃�3)
𝑐𝑜𝑣(�̃�2, �̃�3)
[
 
 
 
𝑤1
∗
𝑤2
∗
𝑤3
∗
λ∗ 𝑏𝑡⁄ ]
 
 
 
=
[
 
 
 
2(0,05)2 2.0,00245 2.0,001 1
2.0,00245 2(0,07)2 2.0,0035 1
2.0,001 2.0,0035 2(0,10)2 1
1 1 1 0]
 
 
 
−1
𝑥 [
0,15 𝑏𝑡⁄
0,25 𝑏𝑡⁄
0,35 𝑏𝑡⁄
1
]
 
 
 
[
 
 
 
𝑤1
∗
𝑤2
∗
𝑤3
∗
λ∗ 𝑏𝑡⁄ ]
 
 
 
= [
0,9836 200,0253 −201,2917 1,2639
−0,1671 −201,2913 265,8564 −64,5651
0,1835 1,2659 −64,5651 63,2991
−0,0044 0,9836 0,1671 0,1834
]
−1
𝑥 [
1
0,15 𝑏𝑡⁄
0,25 𝑏𝑡⁄
0,35 𝑏𝑡⁄
]
𝑤1
∗
𝑤1
∗ = 0,9836 +
200,0253(0,15)
𝑏𝑡
+
−201,2917(0,25)
𝑏𝑡
+
1,2639(0,35)
𝑏𝑡
𝑤1
∗ = 0,9836 −
19,876
𝑏𝑡
𝑤2
∗ = −0,1671 +
13,6726
𝑏𝑡
𝑤3
∗ = 0,1835 +
6,2034
𝑏𝑡
𝑏𝑡
𝑏𝑡
 
 
 
𝑏𝑡
𝐶 = (98,36% 𝑒𝑚 𝐴1, −16,71% 𝑒𝑚 𝐴2, 18,35% 𝑒𝑚 𝐴3)
Ativo 1
Ativo 2
Ativo 3
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%
Retorno Esperado 
Risco (desvio padrão)
Carteira de mínima variância
Reta em que bt é infinito
 
 
 
𝑅𝐶∗ = (0,9836 −
19,876
𝑏𝑡
)0,15 + (−0,1671 +
13,6726
𝑏𝑡
) 0,25 + (0,1835 +
6,2034
𝑏𝑡
) 0,35
𝑏𝑡
𝑅𝐶∗ = (0,9836)0,15 + (−0,1671)0,25 + (0,1835)0,35 = 17,00%
𝑆𝐶
2 = [(0,9836 −
19,876
𝑏𝑡
) 0,05]
2
+ [(−0,1671 +
13,6726
𝑏𝑡
) 0,07]
2
+ [(0,1835 +
6,2034
𝑏𝑡
) 0,10]
2
+ 2 [(0,9836 −
19,876
𝑏𝑡
) (−0,1671 +
13,6726
𝑏𝑡
)] 0,00245
+ 2 [(0,9836 −
19,876
𝑏𝑡
) (0,1835 +
6,2034
𝑏𝑡
)] 0,001
+ 2 [(−0,1671 +
13,6726
𝑏𝑡
) (0,1835 +
6,2034
𝑏𝑡
)] 0,0035
 
 
 
𝑏𝑡 = ∞
𝑆𝐶∗
2 = [(0,9836)0,05]2 + [(−0,1671)0,07]2 + [(0,1835)0,10]2
+ 2[(0,9836)(−0,1671)]0,00245 + 2[(0,9836)(0,1835)]0,001
+ 2[(−0,1671)(0,1835)]0,0035
𝑆𝐶∗
2 = 0,002233
𝑆𝐶∗ = 4,73%
 
 
 
1 = (
𝑋 − 𝑋0
𝐴
)
2
− (
𝑌 − 𝑌0
𝐵
)
2
1 = (
𝑆𝐶 − 0
0,0473
)
2
− (
𝑅𝐶 − 0,17
0,1079
)
2
Ativo 1
Ativo 2
Ativo 3
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%
Retorno Esperado 
Risco (desvio padrão)
Carteira de mínima variância
4,73%
17%
 
 
 
 
𝑐𝑜𝑣(𝑅1, 𝑅𝑚)
𝑐𝑜𝑣(𝑅2, 𝑅𝑚)
𝑐𝑜𝑣(𝑅3, 𝑅𝑚) 𝑐𝑜𝑣(𝑅2, 𝑅3)
 
 
 
𝑐𝑜𝑣(𝑅𝑛, 𝑅𝑚)
-3,00%
-2,00%
-1,00%
0,00%
1,00%
2,00%
3,00%
4,00%
-3,00% -2,00% -1,00% 0,00% 1,00% 2,00% 3,00% 4,00% 5,00%
Retorno do Ativo 
Estudado
Retorno do Ativo Padrão
 
 
 
𝑦 = 𝑎 + 𝑏𝑥
�̃�1 = 𝑎1 + 𝑏1�̃�𝑀 + �̃�1
 �̃�1
 𝑎1
 𝑏1
 �̃�𝑀
 �̃�1
𝐸[�̃�1] = 𝑎1 + 𝑏1𝐸[�̃�𝑀] + 𝐸[�̃�1]
𝑎1
𝑏1
 
 
 
�̅�1 = 𝑎1 + 𝑏1�̅�𝑀
𝑏1�̅�𝑀
𝑎1
𝑆2(�̃�1) = 𝑆1
2 = 𝑆2(𝑎1) + 𝑏1
2𝑆2(�̃�𝑚) + 𝑆
2(�̃�1) + 2𝑐𝑜𝑣(𝑎1, 𝑏1�̃�𝑚)
+ 2𝑐𝑜𝑣(𝑎1, �̃�1) + 2𝑐𝑜𝑣(𝑏1�̃�𝑚, �̃�1)
𝑐𝑜𝑣(𝑎1, �̃�1) 𝑐𝑜𝑣(𝑏1�̃�𝑚, �̃�1)
𝑎1
𝑎1 𝑏1�̃�𝑚
𝑆1
2 = 𝑏1
2𝑆2(�̃�𝑚) + 𝑆
2(�̃�1)
𝑆1
2 = 𝑆𝑒1
2 + 𝑏1
2𝑆𝑀
2
 
 
 
 𝑆𝑒1
2
 𝑏1
2𝑆𝑀
2
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝐸[�̃�𝑗, �̃�𝑘] − 𝐸[�̃�𝑗]𝐸[�̃�𝑘]
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝐸[(𝑎𝑗 + 𝑏𝑗�̃�𝑀 + �̃�𝑗)(𝑎𝑘 + 𝑏𝑘�̃�𝑀 + �̃�𝑘)]
− (𝑎𝑗 + 𝑏𝑗�̃�𝑀)(𝑎𝑘 + 𝑏𝑘�̃�𝑀)
 
 
 
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝐸[𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̃�𝑀 + �̃�𝑗𝑎𝑘 + 𝑏𝑗�̃�𝑀𝑎𝑘 + 𝑏𝑗𝑏𝑘�̃�𝑀
2 + 𝑏𝑗�̃�𝑀�̃�𝑘
+ �̃�𝑗𝑎𝑘 + 𝑏𝑘�̃�𝑗�̃�𝑀 + �̃�𝑘�̃�𝑗]
− (𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̅�𝑀 + 𝑏𝑗�̅�𝑀𝑎𝑘 + 𝑏𝑗𝑏𝑘�̅�𝑀
2 )
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝐸[𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̃�𝑀 + 𝑏𝑗�̃�𝑀𝑎𝑘 + 𝑏𝑗𝑏𝑘�̃�𝑀
2 ]
− (𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̅�𝑀 + 𝑏𝑗�̅�𝑀𝑎𝑘 + 𝑏𝑗𝑏𝑘�̅�𝑀
2 )
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘𝐸[�̃�𝑀] + 𝑏𝑗𝑎𝑘𝐸[�̃�𝑀] + 𝑏𝑗𝑏𝑘𝐸[�̃�𝑀
2 ]
− (𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̅�𝑀 + 𝑏𝑗�̅�𝑀𝑎𝑘 + 𝑏𝑗𝑏𝑘�̅�𝑀
2 )
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̅�𝑀 + 𝑏𝑗𝑎𝑘�̅�𝑀 + 𝑏𝑗𝑏𝑘𝐸[�̃�𝑀
2 ]
− (𝑎𝑗𝑎𝑘 + 𝑎𝑗𝑏𝑘�̅�𝑀 + 𝑏𝑗�̅�𝑀𝑎𝑘 + 𝑏𝑗𝑏𝑘�̅�𝑀
2 )
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝑏𝑗𝑏𝑘𝐸[�̃�𝑀
2 ] − (𝑏𝑗𝑏𝑘�̅�𝑀
2 )
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝑏𝑗𝑏𝑘(𝐸[�̃�𝑀
2 ] − �̅�𝑀
2 )
 
 
 
𝑆2(�̃�𝑀) = 𝐸[�̃�𝑀
2 ] − �̅�𝑀
2
𝑐𝑜𝑣(�̃�𝑗 , �̃�𝑘) = 𝑏𝑗𝑏𝑘𝑆
2(�̃�𝑀)
 
 
 
 
𝑎1
𝑏1�̅�𝑀
 
 
 
𝑆𝑒1
2
𝑏1
2𝑆𝑀
2
�̅�𝐶 = [𝑤1 𝑤2 𝑤3] [
�̅�1
�̅�2
�̅�3
] = [𝑤1 𝑤2 𝑤3] [
𝑎1 + 𝑏1�̅�1
𝑎2 + 𝑏2�̅�2
𝑎3 + 𝑏3�̅�3
]
�̅�𝐶 = [𝑤1 𝑤2 𝑤3] [[
𝑎1
𝑎2
𝑎3
] + [
𝑏1
𝑏2
𝑏3
] �̅�𝑀]
�̅�𝐶 = [𝑤1 𝑤2 𝑤3] [
𝑎1
𝑎2
𝑎3
] + [𝑤1 𝑤2 𝑤3] [
𝑏1
𝑏2
𝑏3
] �̅�𝑀
𝑆𝐶
2 = [𝑤][𝑐𝑜𝑣][𝑤]𝑇
 
 
 
𝑆𝐶
2 = [𝑤1𝑤2𝑤3] [
𝑐𝑜𝑣(�̃�1, �̃�1) 𝑐𝑜𝑣(�̃�2, �̃�1) 𝑐𝑜𝑣(�̃�3, �̃�1)
𝑐𝑜𝑣(�̃�1, �̃�2) 𝑐𝑜𝑣(�̃�2, �̃�2) 𝑐𝑜𝑣(�̃�3, �̃�2)
𝑐𝑜𝑣(�̃�1, �̃�3) 𝑐𝑜𝑣(�̃�2, �̃�3) 𝑐𝑜𝑣(�̃�3, �̃�3)
] [
𝑤1
𝑤2
𝑤3
]
𝑆𝐶
2 = [𝑤1𝑤2𝑤3] [
𝑏1
2 𝑏2𝑏1 𝑏3𝑏1
𝑏1𝑏2 𝑏2
2 𝑏3𝑏2
𝑏1𝑏3 𝑏2𝑏3 𝑏3
2
] [
𝑤1
𝑤2
𝑤3
] 𝑆𝑀
2
𝑆𝐶
2
𝑆𝑀
2
 
 
 
 
 
 
 
 
 
 
𝑖𝑛𝑒𝑔ó𝑐𝑖𝑜/𝑎𝑡𝑖𝑣𝑜 = 𝑖𝑓 + 𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜/𝑎𝑡𝑖𝑣𝑜(�̅�𝑀 − 𝑖𝑓)
 𝑖𝑛𝑒𝑔ó𝑐𝑖𝑜/𝑎𝑡𝑖𝑣𝑜
 𝑖𝑓
 𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜/𝑎𝑡𝑖𝑣𝑜
 �̅�𝑀
i = 0
i = rf
ilanchonete
iconstrutora
isoftware house
i = rmercado
rmercado-rf
 
 
 
𝑖𝑛𝑒𝑔ó𝑐𝑖𝑜 = 𝑎𝑛𝑒𝑔ó𝑐𝑖𝑜 + 𝑏𝑛𝑒𝑔ó𝑐𝑖𝑜�̅�𝑀
0,00%
1,00%
2,00%
3,00%
4,00%
5,00%
6,00%
0,00% 0,50% 1,00% 1,50% 2,00% 2,50% 3,00% 3,50%
Retorno do Ativo 
Estudado
Retorno do Mercado
 
 
 
𝑎𝑎𝑡𝑖𝑣𝑜
𝑖𝑛𝑒𝑔ó𝑐𝑖𝑜 = 𝑖𝑓 + 𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜�̅�𝑀 − 𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜𝑖𝑓
𝑖𝑛𝑒𝑔ó𝑐𝑖𝑜 = (𝑖𝑓 − 𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜𝑖𝑓) + 𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜�̅�𝑀
𝐾𝑛𝑒𝑔ó𝑐𝑖𝑜 𝑏𝑛𝑒𝑔ó𝑐𝑖𝑜
𝑏𝑛𝑒𝑔ó𝑐𝑖𝑜
𝑖𝑛𝑒𝑔ó𝑐𝑖𝑜/𝑎𝑡𝑖𝑣𝑜 = 𝑖𝑓 + 𝑏𝑛𝑒𝑔ó𝑐𝑖𝑜/𝑎𝑡𝑖𝑣𝑜(�̅�𝑀 − 𝑖𝑓)
 
 
 
 
𝐶𝑎𝑟𝑡𝑒𝑖𝑟𝑎 = {𝑤1𝑅𝑀, 𝑤2𝐴}
𝑤1 𝑤 𝑤2 1 − 𝑤
�̅�𝐶 = 𝑤�̅�𝑀 + (1 − 𝑤)�̅�𝐴
𝑆𝐶
2 = 𝑤2𝑆𝑀
2 + (1 − 𝑤)2𝑆𝐴
2 + 2𝑤(1 − 𝑤)𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
lim
∆𝑤→0
∆𝑅
∆𝑤
=
𝜕𝑅
𝜕𝑤
lim
∆𝑤→0
∆𝑆
∆𝑤
=
𝜕𝑆
𝜕𝑤
 
 
 
𝜕
𝜕𝑅
𝜕𝑤
= �̅�𝑀 − �̅�𝐴
𝜕𝑆
𝜕𝑤
=
2𝑤𝑆𝑀
2 − 2(1 − 𝑤)𝑆𝐴
2 ∓ 2𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴) − 4𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
2√𝑤2𝑆𝑀
2 + (1 − 𝑤)2𝑆𝐴
2 + 2𝑤(1 − 𝑤)𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
∆𝑅
∆𝑤
∆𝑆
∆𝑤
=
∆𝑅
∆𝑆
 
 
 
𝜕𝑅
𝜕𝑤
𝜕𝑆
𝜕𝑤
=
�̅�𝑀 − �̅�𝐴
𝑤𝑆𝑀
2 − (1 − 𝑤)𝑆𝐴
2 − 𝑐𝑜𝑣(�̃�𝑀 , �̃�𝐴)
√𝑤2𝑆𝑀
2 + (1 − 𝑤)2𝑆𝐴
2 + 2𝑤(1 − 𝑤)𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
𝐶 𝐶∗
𝐶 𝐶∗
0,0%
5,0%
10,0%
15,0%
20,0%
25,0%
30,0%
35,0%
40,0%
45,0%
0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 0,11 0,12 0,13 0,14
Retorno Esperado 
Risco (desvio padrão)
RA Ativo A
C
RM Mercado
C*
 
 
 
𝑤𝐴 𝑤𝐶
𝐶
𝜕𝑅
𝜕𝑤
𝜕𝑆
𝜕𝑤
=
�̅�𝑀 − �̅�𝐴
𝑆𝑀
2 − 𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
√𝑆𝑀
2
=
�̅�𝑀 − �̅�𝐴
𝑆𝑀
2 − 𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
𝑆𝑀
�̅�𝐶 − 𝐼𝐹
𝑆𝐶
𝜕𝑅𝑅̅̅ ̅̅
𝜕𝑤
= 0
 
 
 
𝜕𝑅𝑅̅̅ ̅̅
𝜕𝑤
=
𝜕𝑅
𝜕𝑤
𝑆𝐶 −
𝜕𝑆𝐶
𝜕𝑤
(�̅�𝐶 − 𝐼𝐹)
𝑆𝐶
2
𝜕𝑅
𝜕𝑤
𝑆𝐶 =
𝜕𝑆𝐶
𝜕𝑤
(�̅�𝐶 − 𝐼𝐹)
𝜕𝑅
𝜕𝑤
𝜕𝑆𝐶
𝜕𝑤
=
�̅�𝐶 − 𝐼𝐹
𝑆𝐶
𝜕𝑅
𝜕𝑤
𝜕𝑆𝐶
𝜕𝑤
| =
�̅�𝑀 − 𝐼𝐹
𝑆𝑀
 
 
 
𝐼𝐹
�̅�𝑀 − �̅�𝐴
𝑆𝑀
2 − 𝑐𝑜𝑣(�̃�𝑀 , �̃�𝐴)
𝑆𝑀
=
�̅�𝑀 − 𝐼𝐹
𝑆𝑀
�̅�𝑀 − �̅�𝐴 = (
�̅�𝑀 − 𝐼𝐹
𝑆𝑀
)(
𝑆𝑀
2 − 𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
𝑆𝑀
)
�̅�𝑀 − �̅�𝐴 = (�̅�𝑀 − 𝐼𝐹) − [
(�̅�𝑀 − 𝐼𝐹)𝑐𝑜𝑣(�̃�𝑀, �̃�𝐴)
𝑆𝑀
2 ]
�̅�𝐴 = 𝐼𝐹 + [
(�̅�𝑀 − 𝐼𝐹)𝑏𝐴𝑆𝑀
2
𝑆𝑀
2 ]
�̅�𝐴 = 𝐼𝐹 + 𝑏𝐴(�̅�𝑀 − 𝐼𝐹)
 
 
 
𝑏𝐴
𝑏𝐴