Risk and Returns¶

Note: Horizon need not always be $h=1$

Return¶

“Return is backward-looking” $$ r(t, h) = y_t - y_{t-h} $$

ROI¶

% change in series

Return on investment is in percentage relative to original investment

ROI	$R_t$	Time Additive?	Multi-Period Return is __ sum of individual returns
Simple	$\dfrac{y_t - y_h}{y_h}$	❌	Geometric
Continuous (Preferred)	$\ln \left \vert \dfrac{y_t}{y_h} \right \vert = \ln \vert y_t \vert - \ln \vert y_{t_h} \vert$	✅	Arithmetic

\[ \text{CR} = \ln \vert 1 + \text{SR} \vert \]

Re-Investment Benefit¶

\[ \text{Re-Investment Benefit} = \text{IRR} - \text{ROI} \]

Benefit that could be obtained by investing all intermediate inflows at the same ROI

Yield¶

“Yield is forward-looking” $$ Y_t = \dfrac{y_t - y_h}{y_t} $$

Dividends¶

Dividend rate are relative to face value, not your investment

Dates¶


Dividend Declaration Date
Ex-Dividend Date
Record Date
Payment Date

Return Series¶

Assumed to be a random walk

Expected Returns¶

\[ E(R) = \sum_i r_i \cdot P(r_i) \]

Risk¶

Chance of actual return differing from expected return

Statistically quantified through variance/standard deviation of returns’ PDF

Types of Unknowns¶

	Systematic risk	Unsystematic risk	Uncertainty
Meaning	Sensitivity to market fluctuations	Personal factors	Unknown effects
Type	External Macro	Internal Micro	External
Minimizable	❌	✅ through diversification (portfolio)	❌
Risk Compensation expected	✅	✅	❌

\[ \begin{aligned} \text{Risk: } \sigma^2 &= \text{SR} + \text{UR} \\ \text{SR} &= \beta^2 \cdot \sigma^2 (R_m) \end{aligned} \]

Risk Measures¶


Standard Deviation	$\sigma (R_p)$
Beta (Market sensitivity)	$\dfrac{\text{cov} (R_p, R_m)}{\sigma^2_{m}}$
Semi Deviation	$\sigma (\text{Loss}_p)$ $\text{Loss}_t = \arg \max(R_t, 0)$

where $p=$ portfolio and $m=$ market

Risk-Return Tradeoff¶

Investors are rational and risk-averse: prefer less risk investments
Investors expect risk premium: Investors are ready to take risk only with the expectation of higher return

securities_risk_premium

\[ R_\min = R_f + \underbrace{\left ( \dfrac{R_m - R_f}{\sigma_m} \right )}_\text{Market Price of Risk} \sigma \]

Jensen’s Inequality¶

Using Jensen’s Inequality $$ E[f(x)] \ne f(E[x]) \ \implies E[u(R)] > u(E[R]) $$ where

$R$ is the return obtained
$u(R)$ is the utility obtained from the return

Effect of Frequency on Volatility¶

\[ V \propto \nu \]

Trading Days¶

	Trading Days
Fixed-Income	365.25
Variable-Income	252

Annualization¶

\[ \begin{aligned} \text{Annual } E(R) &= 252 \times E(R) \\ \text{Annual } \sigma(R) &= \sqrt{252} \times \sigma(R) \end{aligned} \]

There are 252 trading days in a year

IDK¶

Fixed-income securities are also very volatile

YTM¶

Yield to Maturity = IRR of security if held until maturity

Last Updated: 2024-05-14 ; Contributors: AhmedThahir

ROI	\(R_t\)	Time Additive?	Multi-Period Return is __ sum of individual returns
Simple	\(\dfrac{y_t - y_h}{y_h}\)	❌	Geometric
Continuous (Preferred)	\(\ln \left \vert \dfrac{y_t}{y_h} \right \vert = \ln \vert y_t \vert - \ln \vert y_{t_h} \vert\)	✅	Arithmetic


Standard Deviation	\(\sigma (R_p)\)
Beta (Market sensitivity)	\(\dfrac{\text{cov} (R_p, R_m)}{\sigma^2_{m}}\)
Semi Deviation	\(\sigma (\text{Loss}_p)\) \(\text{Loss}_t = \arg \max(R_t, 0)\)