Need an all-in-one list with the **Portfolio Management** formulas included in the CFA® Level 1 Exam? We have compiled them for you here. The relevant formulas have been organized and presented by chapter. In this section, we will consider **Portfolio Risk and Return **calculations.

**1. Portfolio Management: an Overview**

*Diversification ratio*

Diversification~ratio = \frac {σ~of~equally~weighted~portfolio~of~n~securities}{σ~of~single~security~selected~at~random}

σ = Volatility (Standard deviation)

*Net asset value per share*

Net~asset~value~per~share = \frac {Fund~Assets - Fund~Liabilities}{Number~of~Shares~Outstanding}

**2. Portfolio Risk and Return: Part I**

**Holding Period Return (HPR)** – No cash flows

HPR = \frac {Ending~value - Beginning~value}{Beginning~value}

**Holding Period Return (HPR)** – Cash flows occur at the end of the period

HPR = \frac {Ending~value-Beginning~value+Cash~flows~received}{Beginning~value}=\frac {P{_1} - P{_0} + D{_1}}{Beginning~value}

**Holding Period Return (HPR)** – Multiple years

HPR = [(1 + R{_1}) \times (1 + R{_2})] - 1

R{_1} = Holding period return in year 1

R{_2} = Holding period return in year 2

**Arithmetic mean return**

\overline {R{_i}}=\frac {R{_{i1}}+R{_{i2}}+…+R{_{iT-1}}+R{_{iT}}}{t}=\frac {1}{T}\displaystyle\sum_{t=1}^TR{_{it}}

\overline {R{_i}} = Arithmetic mean return

R{_{it}} – Return in period t

T = Total number of periods

**Geometric mean return**

\overline {R{_{Gi}}}=\sqrt {(1+{R{_{i1}})\times (1+{R{_{i2}}) \times…\times (1+R{_{i,T-1}}) \times (1+R{_{i,T})}}}-1}=\sqrt [T]{ \displaystyle\prod_{t=1}^T(1+R{_{it})-1}}

**Internal Rate of Return (IRR)**

\displaystyle\sum_{t=0}^N \frac {CF_t}{(1+IRR)^t}=0

t = Number of periods

CF_t = Cash flow at time t

**Time-weighted rate of return**

r{_{TW}} = \big[(1 + r_1) \times (1 + r_2) \times ... \times (1 + r_N)\big]^{\frac {1}{N}}-1

r_{N} = Holding period return in year n

**Annualized return**

r_{annual} = (1 + r_{period})^c - 1

r = Periodic return

c = Number of periods in a year

**Nominal rate of return**

(1 + r) = (1 + r_{rF}) \times (1 + π) \times (1 + RP)

r_{rF} = Real risk-free rate of return

π = Inflation

RP = Risk premium

**Real rate of return**

(1 + r_{real}) = (1 + r_{rF}) \times (1 + RP) = \frac {(1+r)}{(1+π)}

r_{rF} = Real risk-free rate of return

π = Inflation

RP = Risk premium

**Population variance**

σ^2=\frac {\displaystyle\sum_{i=1…n}^N (x_i-μ)^2}{N}

x_{i} = Return for period i

N = Total number of periods

μ = Mean

**Population standard deviation**

\sqrt {σ=\frac {\displaystyle\sum_{i=1…n}^N (x_i-μ)^2}{N}}

x_{i} = Return for period i

N = Total number of periods

μ = Mean

**Sample variance**

S^2=\frac {\displaystyle\sum_{i=1…n}^n(x_i-\overline x)^2}{n-1}

X_i = Return for period i

N = Total number of periods

\overline x = Mean of n returns

**Sample standard deviation**

\sqrt {s=\frac {\displaystyle\sum_{i=1…n}^n(x_i-\overline x)^2}{n-1}}

X_i = Return for period i

N = Total number of periods

\overline x = Mean of n returns

**Covariance**

COV_{1,2} =\frac {\textstyle\sum_{t=1}^n \big\{[R_{t,1}-\overline {R_1}][R_{t,2}-\overline {R_2}] \big\}}{n-1}=ρ_{1,2}σ_{1}σ_{2}

R_{T1} = Return on Asset 1 in period t

R_{T2} = Return on Asset 2 in period t

ρ = Correlation

\overline R = Mean of respective assets

**Correlation**

ρ_{1,2}=\frac {COV_{1,2}}{σ_{1}σ_{2}}

σ = Standard deviation

**Utility function**

U = E(r) - \frac {1}{2}Aσ^2

U = Utility of an investment

E(r) = Expected return

σ^2 = Variance of the investment

A = Risk aversion level

**Portfolio return** (Many risky assets)

R_{P} =\displaystyle\sum_{i=1}^Nw_iR_i~,~\displaystyle\sum_{i=1}^Nw_i=1

R_i = Return of asset i

w_i = Weight within the portfolio

**Portfolio variance**

σ^2_P = \displaystyle\sum_{i, j = 1}^N w_iw_jCOV (R_i, R_j)

w = Weights

R = Returns

**Portfolio variance** (Two-asset portfolio)

σ^2_P = w^2_1σ^2_1+w^2_2σ^2_2+2w_1w_2 COV(R_1, R_2)

**Portfolio standard deviation** (Two-asset portfolio)

σ_P = \sqrt {w^2_1σ^2_1+w^2_2σ^2_2+2w_1w_2 COV(R_1, R_2)}

**Portfolio return of two assets** (when one asset is the risk-free asset)

E(R_p) = w_1R_f + (1 - w_1)E(R_i)

R_f = Returns of respective asset

R_i = Returns of respective asset

w_1 = Weight in asset 1

1 - w_1 = w_2

**Portfolio standard deviation of two assets** (when one asset is the risk-free asset)

σ_P = \sqrt {w^2_1σ^2_f+(1-w_1)^2σ^2_i+2w_1(1-w_1)ρ_{1,2}σ_fσ_i}=(1 - w_1)σ_i

f = Risk-free asset

i = Asset

σ = Standard deviation

w = Weight

**Portfolio Risk And Return: Part II**

**Capital Asset Pricing Model (CAPM)**

E(Ri) = R_F + β_i \big[E (R_M) - R_F\big]

β_i = Return sensitivity of stock i to changes in the market return

E(R_M) = Expected return on the market

E(R_M)-R_F = Expected market risk premium

R_F = Risk-free rate of interest

*Capital allocation line*

E(R_p) = R_f+ \Big (\frac {E (R_M) - R_f}{σ_m}\Big) \times σ_p

E(R_m) = Expected return of the market portfolio

R_f = Risk-free rate of return

σ_m = Standard deviation of the market portfolio

σ_p = Standard deviation of the portfolio P

**Expected return** (multifactor model)

E(Ri) - R_f = β_{i1} \times E(Factor 1) + β_{i2} x E(Factor 2) + … + β_{ik} x E(Factor k)

β_{ik} = Stock i’s sensitivity to changes in the k^{th} factor

(Factor k) = Expected risk premium for the k^{th} factor

**Beta of an asset**

β_i = \frac {Cov(R_i , R_m)}{σ_m^2}=\frac {ρ_{i,m}~σ_iσ_m}{σ_m^2}=\frac {ρ_{i,m}~σ_i}{σ_m}

σ = Standard deviation

m = Market portfolio

i = Asset portfolio

**Portfolio beta**

β_P =\displaystyle\sum_{i=1}^n w_iβ_i \qquad \displaystyle\sum_{i=1}^n w_i=1

**Sharpe ratio**

Sharpe~ratio = \frac {R_p - R_f}{σ_p}

R_p = Portfolio return

R_f = Risk-free rate of return

σ_p = Standard deviation (volatility) of portfolio return

*M ^{2} ratio*

M^2~ratio = (R_p - R_f)\frac {σ_m}{σ_p} - (R_m - R_f)

m = Market portfolio

*Treynor ratio*

Treynor~ratio = \frac {E(R_p) - R_f}{β_p}

β_p = Portfolio beta

R_p = Portfolio return

R_f = Risk-free rate of return

*Jensen’s alpha*

α_P = R_p - \big[R_f + β_p (R_m - R_f)\big]

**Follow the links to find more formulas on Quantitative Methods, Economics, Corporate Finance, Alternative Investments, Financial Reporting and Analysis, Equity Investments, Fixed-Income Investments, and Derivatives, included in the CFA® Level 1 Exam.**