2024 Level 1 CFA® Exam:
Returns - Other Concepts

Last updated: July 23, 2023

Continuously Compounded Returns for Level 1 CFA Candidates

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Here are 3 formulas for continuously compounded returns that you can use in your level 1 CFA exam:

Continuously Compounded Return ('1' Holding Period)
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\(r_{t,t+1} = ln(\frac{S_{t+1}}{S_{t}}) = ln(1+R_{t,t+1})\)

  • \(S_{t}\) - stock price at time "t"
  • \(S_{t+1}\) - stock price at time "t+1"
  • \(R_{t,t+1}\) - holding period return for "t" to "t+1" period
  • \(r_{t,t+1}\) - continuously compounded return for "t" to "t+1" period
Continuously Compounded Return ('T' Holding Periods)
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\(r_{0,T} = r_{T-1,T} + r_{T-2,T-1} + \ldots r_{0,1}\)

  • \(r\) - continuously compounded return
  • \(T\) - investment horizon
Variance ('T' Holding Periods; Continuously Compounded Return)
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\(\sigma^{2}(r_{0,T}) = \sigma^{2}\times T\)

  • \(\sigma^{2}(r_{0,T})\) - variance for T periods
  • \(\sigma^{2}\) - variance for one period
  • assumption: returns are independently and identically distributed (IID)
Example 2 (continuously compounded return)

The price of a stock at the beginning of a year is equal to USD 56 and at the end of the year equals USD 70. What is the holding period return and continuously compounded return for this year?

Very often securities valuation models assume that continuously compounded returns of securities are normally distributed. Based on this assumption, it can be proved that prices of securities are lognormally distributed. Thanks to it, we can use the lognormal distribution to model the probability distribution of security prices, which is very useful for modeling prices of securities.

You should also remember that even if continuously compounded returns of securities are not exactly normally distributed, then according to the central limit theorem, security prices can be well described by the lognormal distribution.

Anualized Return & Real Return & Leveraged Return

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Annualized Return

As far as the computation of returns and the comparison of returns on different investments are concerned, it is important to be able to convert rates of return to a single period. The most commonly used measure is the annualized rate of return, which you can compute using the following formula in your level 1 CFA exam:

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

  • \(r_{annual}\) - annualized return
  • \(r_{period}\) - the return for the period
  • \(c\) - number of periods in a year

For calculation purposes assume that 1 year = 52 weeks and 1 week = 5 days.

Leveraged Return

Leveraged return involves using borrowed money to amplify the potential gains of an investment.

Two main methods to achieve leverage:

  1. Futures Contracts: By investing a small margin (like 10%), you control a much larger position. Profits and losses are then typically magnified by a factor related to this margin.
  2. Borrowing: You can borrow money to invest more than your original capital. If you borrow half your investment amount, the potential profit doubles, but you need to deduct interest costs.

Formula to remember:

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  • \(R_l\) - leveraged return
  • \(R_p\) - portfolio return
  • \(R_d\) - cost of debt
  • \(V_e\) - proportion of investment financed using equity
  • \(V_d\) - proportion of investment financed using debt

In essence, if the return on your investment (RP) exceeds the cost of borrowing (rD), the return with leverage surpasses that without leverage. Conversely, if it doesn't, losses amplify. Remember, while leverage boosts potential profits, it also increases the risk.

Nominal Return vs Real Return

An investor must realize that return on investment should be adjusted for inflation and must be able to distinguish between nominal return and real return. Real returns are nominal returns adjusted for inflation. Have a look at the equation: