Understanding Absolute, Discrete, And Continuous Proportional Returns

Understanding the computation of asset returns is crucial in financial analysis and risk management because returns, or "shocks", are not a one-size-fits-all; they come in various forms and serve different purposes across different financial/risk models.

Whether it’s calculating the daily returns on stocks or assessing how these returns impact a portfolio under stress, understanding the different types of returns is essential for capturing the realistic behavior of many financial variables.

It essentially represents the money that has been made or lost on an investment over a certain period of time, capturing the profit or loss generated due to changes in asset prices.

In risk management, these returns are often referred to as "shocks", representing the sudden changes in asset prices that can affect the overall portfolio. And by calculating and analyzing these shocks, we can better understand potential risks and prepare strategies to mitigate them.

01: Absolute Returns (Dollar Returns)

Absolute return is the total return an asset achieves over a specific period. It represents the actual gain or loss of an investment without comparing it to any benchmark or other investments.

Where:

𝑟Absolute = Absolute return

S𝑡 = Price of the equity at time 𝑡

S𝑡-1 = Price of the equity at time 𝑡-1

for example, if an investor bought a stock at $100 and sold it at $120, the absolute return is $20. This measure does not consider the scale of the investment (relative percentage), just the absolute increase or decrease.

02: Discrete Proportional Returns (Simple Returns)

Discrete proportional return (also known as simple return) expresses the relative percentage change in the price of an equity from one period to the next.

Where:

𝑟discrete = Discrete proportional return

S𝑡 = Price of the equity at time 𝑡

S𝑡-1 = Price of the equity at time 𝑡-1

for example, if the price of a stock increased from $100 to $120, the discrete proportional return would be (120 − 100) / 100 = 0.20 or 20%. This measure is useful for comparing returns over different periods or investments, making it easier to assess performance.

03: Continuous Proportional Returns (Logarithmic Returns)

Continuous proportional returns (or log returns) are calculated using the natural logarithm of the price ratio. This measure is useful when analyzing compounding effects over multiple periods.

Where:

𝑟continuous or 𝑟log = Continuous proportional (log) return

S𝑡 = Price of the equity at time 𝑡

S𝑡-1 = Price of the equity at time 𝑡-1

for example, if the price of a stock increased from $100 to $120, the log return would be ln(120/100) ≈ 0.1823 or 18.23%. These returns are additive over time (unlike discrete returns) and are often used in financial modeling, particularly in Geometric Brownian Motion (GBM) models for stock prices.

Important Considerations:

• Discrete returns are more intuitive for short-term analysis. Continuous returns are favored in models that require continuous compounding.

  
  

• Continuous returns are often preferred in risk modeling since they exhibit more stable statistical properties, especially over longer periods.

  
  

• Many pricing models, such as the Black-Scholes option pricing model, rely on continuous returns because they reflect the assumption of continuously compounded returns.

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Which shock type should be used—'Discrete Proportional Shock' or 'Continuous Proportional Shock' for equities, and in which scenarios would each be most appropriate?

Discrete Proportional Returns/Shocks are typically used when we are dealing with data that naturally follows discrete intervals, such as daily, weekly, or monthly returns. Discrete shock type is appropriate when we assume that returns are compounded periodically, for example, at the end of the day (EOD) or month (EOM).

• When managing a portfolio and calculating daily or monthly returns, discrete proportional shocks are ideal because the returns are realized at discrete intervals.

  
  

• Discrete returns are often used for financial reporting, where performance metrics are calculated based on specific periods, such as quarterly or annual returns.

Continuous Proportional Returns/Shocks are used when we need to assume continuous compounding, which is more common in theoretical models and certain financial contexts like option pricing (BSM). Continuous shock type is useful when we are modeling scenarios where prices or returns change continuously over time, without distinct intervals.

• The BSM model uses continuous compounding to price options. In such models, returns are treated as being continuously compounded because the underlying asset’s price is assumed to move continuously over time.

  
  

• Continuous shocks are often applied in asset price modeling, where the assumption is that prices change continuously rather than at fixed intervals.

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