Portfolio Mathematics

Quantitative Methods

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Learning Module 5: Portfolio Mathematics

Expected Return on Portfolio

\[ E(R_{P})=w_{1} E(R_{1})+w_{2} E(R_{2})+\cdots+w_{n} E(R_{n}) \tag{1} \]

Where:

\(E(R_{P})\): expected return on the portfolio
\(w_{i}\): proportion of portfolio invested in asset \(i\)
\(E(R_{i})\): expected return on asset \(i\)
\(n\): number of assets in the portfolio

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## Expected Return on Portfolio

$$
E(R_{P})=w_{1} E(R_{1})+w_{2} E(R_{2})+\cdots+w_{n} E(R_{n}) \tag{1}
$$

Where:

* $E(R_{P})$: expected return on the portfolio
* $w_{i}$: proportion of portfolio invested in asset $i$
* $E(R_{i})$: expected return on asset $i$
* $n$: number of assets in the portfolio

Portfolio Variance

\[ \sigma^2(R_p) = E\{[R_p - E(R_p)]^2\} \tag{2} \]

Where:

\(\sigma^2(R_p)\): portfolio variance
\(R_p\): portfolio return
\(E(R_p)\): expected portfolio return

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## Portfolio Variance

$$
\sigma^2(R_p) = E\{[R_p - E(R_p)]^2\} \tag{2}
$$

Where:

* $\sigma^2(R_p)$: portfolio variance
* $R_p$: portfolio return
* $E(R_p)$: expected portfolio return

Covariance of Returns

\[ Cov(R_{i}, R_{j}) =E[(R_{i}-ER_{i})(R_{j}-ER_{j})] \tag{3} \]

Where:

\(Cov(R_{i}, R_{j})\): covariance between returns on assets \(i\) and \(j\)
\(R_{i}\): return on asset \(i\)
\(R_{j}\): return on asset \(j\)
\(E(R_{i})\): expected return on asset \(i\)
\(E(R_{j})\): expected return on asset \(j\)
Equation 3 states that the covariance between two random variables is the probability-weighted average of the cross-products of each random variable’s deviation from its own expected value.

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## Covariance of Returns

$$
Cov(R_{i}, R_{j}) =E[(R_{i}-ER_{i})(R_{j}-ER_{j})] \tag{3}
$$

Where:

* $Cov(R_{i}, R_{j})$: covariance between returns on assets $i$ and $j$
* $R_{i}$: return on asset $i$
* $R_{j}$: return on asset $j$
* $E(R_{i})$: expected return on asset $i$
* $E(R_{j})$: expected return on asset $j$
* Equation 3 states that the covariance between two random variables is
  the probability-weighted average of the cross-products of each random
  variable’s deviation from its own expected value.

Sample Covariance of Returns

\[ Cov(R_i, R_j) = \frac{\sum_{n=1}^{n} (R_{i,t} - \bar{R}_i)(R_{j,t} - E\bar{R}_j)}{(n - 1)} \tag{4} \]

Where:

\(Cov(R_i, R_j)\): sample covariance between returns on assets \(i\) and \(j\)
\(R_{i,t}\): return on asset \(i\) at time \(t\)
\(R_{j,t}\): return on asset \(j\) at time \(t\)
\(\bar{R}_i\): sample mean return on asset \(i\)
\(\bar{R}_j\): sample mean return on asset \(j\)
\(n\): sample of past data of size \(n\)

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## Sample Covariance of Returns

$$
Cov(R_i, R_j) = \frac{\sum_{n=1}^{n} (R_{i,t} - \bar{R}_i)(R_{j,t} - E\bar{R}_j)}{(n - 1)} \tag{4}
$$

Where:

* $Cov(R_i, R_j)$: sample covariance between returns on assets $i$ and $j$
* $R_{i,t}$: return on asset $i$ at time $t$
* $R_{j,t}$: return on asset $j$ at time $t$
* $\bar{R}_i$: sample mean return on asset $i$
* $\bar{R}_j$: sample mean return on asset $j$
* $n$: sample of past data of size $n$

Portfolio Variance Decomposition (Three-Asset Portfolio)

\[ \sigma^2(R_p) = E[(R_p - ER_p)^2] \]

\[ = E\{[w_1 R_1 + w_2 R_2 + w_3 R_3 - E(w_1 R_1 + w_2 R_2 + w_3 R_3)]^2\} \tag{5} \]

\[ = E\{[w_1 R_1 + w_2 R_2 + w_3 R_3 - w_1 ER_1 - w_2 ER_2 - w_3 ER_3]^2\} \]

\[ = w_1^2 \sigma^2(R_1) + w_1 w_2 \text{Cov}(R_1, R_2) + w_1 w_3 \text{Cov}(R_1, R_3) \]

\[ \quad + w_1 w_2 \text{Cov}(R_1, R_2) + w_2^2 \sigma^2(R_2) + w_2 w_3 \text{Cov}(R_2, R_3) \]

\[ \quad + w_1 w_3 \text{Cov}(R_1, R_3) + w_2 w_3 \text{Cov}(R_2, R_3) + w_2^3 \sigma^2(R_3) \]

The most compact way to state Equation 5 is

\[ \sigma^2(R_p) = \sum_{i=1}^{3} \sum_{j=1}^{3} w_i w_j \text{Cov}(R_i, R_j) \]

Moreover, this expression generalizes for a portfolio of any size n to

\[ \sigma^2(R_p) = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \text{Cov}(R_i, R_j) \tag{6} \]

Where:

\(\sigma^2(R_p)\): variance of portfolio return
\(w_i\): proportion of portfolio invested in asset \(i\)
\(w_j\): proportion of portfolio invested in asset \(j\)
\(\sigma^2(R_i)\): variance of return on asset \(i\)
\(\text{Cov}(R_i, R_j)\): covariance between returns on assets \(i\) and \(j\)

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## Portfolio Variance Decomposition (Three-Asset Portfolio)

$$
\sigma^2(R_p) = E[(R_p - ER_p)^2]
$$

$$
= E\{[w_1 R_1 + w_2 R_2 + w_3 R_3 - E(w_1 R_1 + w_2 R_2 + w_3 R_3)]^2\} \tag{5}
$$

$$
= E\{[w_1 R_1 + w_2 R_2 + w_3 R_3 - w_1 ER_1 - w_2 ER_2 - w_3 ER_3]^2\}
$$

$$
= w_1^2 \sigma^2(R_1) + w_1 w_2 \text{Cov}(R_1, R_2) + w_1 w_3 \text{Cov}(R_1, R_3)
$$

$$
\quad + w_1 w_2 \text{Cov}(R_1, R_2) + w_2^2 \sigma^2(R_2) + w_2 w_3 \text{Cov}(R_2, R_3)
$$

$$
\quad + w_1 w_3 \text{Cov}(R_1, R_3) + w_2 w_3 \text{Cov}(R_2, R_3) + w_2^3 \sigma^2(R_3)
$$

---

The most compact way to state Equation 5 is

$$
\sigma^2(R_p) = \sum_{i=1}^{3} \sum_{j=1}^{3} w_i w_j \text{Cov}(R_i, R_j)
$$

Moreover, this expression generalizes for a portfolio of any size n to

$$
\sigma^2(R_p) = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \text{Cov}(R_i, R_j) \tag{6}
$$

Where:

* $\sigma^2(R_p)$: variance of portfolio return
* $w_i$: proportion of portfolio invested in asset $i$
* $w_j$: proportion of portfolio invested in asset $j$
* $\sigma^2(R_i)$: variance of return on asset $i$
* $\text{Cov}(R_i, R_j)$: covariance between returns on assets $i$ and $j$

Correlation between two random variables

\[ \rho(R_i, R_j) = \frac{\text{Cov}(R_i, R_j)}{[\sigma(R_i)\sigma(R_j)]} \tag{7} \]

Where:

\(\rho(R_i, R_j)\): correlation coefficient between returns on assets \(i\) and \(j\)
- Alternative notations are \(\text{Corr}(R_i, R_j)\) and \(p_{ij}\)
\(\text{Cov}(R_i, R_j)\): covariance between returns on assets \(i\) and \(j\)
\(\sigma(R_i)\): standard deviation of returns on asset \(i\)
\(\sigma(R_j)\): standard deviation of returns on asset \(j\)

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## Correlation  between two random variables

$$
\rho(R_i, R_j) = \frac{\text{Cov}(R_i, R_j)}{[\sigma(R_i)\sigma(R_j)]} \tag{7}
$$

Where:

* $\rho(R_i, R_j)$: correlation coefficient between returns on assets $i$ and $j$
  * Alternative notations are $\text{Corr}(R_i, R_j)$ and $p_{ij}$
* $\text{Cov}(R_i, R_j)$: covariance between returns on assets $i$ and $j$
* $\sigma(R_i)$: standard deviation of returns on asset $i$
* $\sigma(R_j)$: standard deviation of returns on asset $j$

Covariance Given a Joint Probability Function

\[ \operatorname{Cov}(R_{A}, R_{B})=\sum_{i} \sum_{j} P(R_{A, i}, R_{B, j}) (R_{A, i}-ER_{A}) (R_{B, j}-ER_{B}) \tag{8} \]

Where:

\(\operatorname{Cov}(R_{A}, R_{B})\): covariance between random variables \(R_A\) and \(R_B\)
\(P(R_{A, i}, R_{B, j})\): joint probability of returns \(R_{A, i}\) and \(R_{B, j}\)
\(R_{A, i}\): return \(i\) on asset \(A\)
\(R_{B, j}\): return \(j\) on asset \(B\)
\(E(R_{A})\): expected return on asset \(A\)
\(E(R_{B})\): expected return on asset \(B\)

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## Covariance Given a Joint Probability Function

$$
\operatorname{Cov}(R_{A}, R_{B})=\sum_{i} \sum_{j} P(R_{A, i}, R_{B, j}) (R_{A, i}-ER_{A}) (R_{B, j}-ER_{B}) \tag{8}
$$

Where:

* $\operatorname{Cov}(R_{A}, R_{B})$: covariance between random variables $R_A$ and $R_B$
* $P(R_{A, i}, R_{B, j})$: joint probability of returns $R_{A, i}$ and $R_{B, j}$
* $R_{A, i}$: return $i$ on asset $A$
* $R_{B, j}$: return $j$ on asset $B$
* $E(R_{A})$: expected return on asset $A$
* $E(R_{B})$: expected return on asset $B$

Safety-First Ratio

\[ \text { SFRatio }=\frac{E(R_{P})-R_{L}}{\sigma_{P}} \tag{9} \]

Where:

\(\text{SFRatio}\): safety-first ratio
\(E(R_{P})\): expected portfolio return
\(R_{L}\): threshold level (minimum acceptable return)
\(\sigma_{P}\): portfolio standard deviation

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## Safety-First Ratio

$$
\text { SFRatio }=\frac{E(R_{P})-R_{L}}{\sigma_{P}} \tag{9}
$$

Where:

* $\text{SFRatio}$: safety-first ratio
* $E(R_{P})$: expected portfolio return
* $R_{L}$: threshold level (minimum acceptable return)
* $\sigma_{P}$: portfolio standard deviation