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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)		 \(\ln \left \vert \dfrac{y_t}{y_h} \right \vert = \ln \vert y_t \vert - \ln \vert y_{t_h} \vert\) âś… 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

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

Risk

Chance of actual return differing from expected return

Statistically quantified through variance of expected returns

Types of Risk

Systematic risk Unsystematic risk Uncertainty
Meaning Sensitivity to market fluctuations Personal factors Unknown effects
Type External Internal External
Minimization Cannot be reduced Can be reduced through portfolio optimization Cannot be reduced
Applicable to all corporations ✅ ❌ ✅
Risk Compensation expected ✅ ❌ ❌

Risk Measures
Standard Deviation \(\sigma (R_p)\)
Beta (systematic risk) \(\dfrac{\text{cov} (R_p, R_m)}{\sigma^2(R_m)}\)
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

\[ R_p - R_f = \alpha + \sigma_p \underbrace{\left ( \dfrac{R_m - R_f}{\sigma_m} \right )}_\text{Market Price of Risk} \]

Efficient Frontier

image-20240309173852371

Jensen’s Inequality

Using Jensen’s Inequality $$ E[u(R)] > u(E[R]) $$ 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

Yield

Yield to Maturity = IRR of security if held until maturity

Last Updated: 2024-05-12 ; Contributors: AhmedThahir

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