Question

With the probabilities of recession, stagnation and boom equal to \(1 / 2\), \(1 / 4,1 / 4\) and the predicted annual returns in the first two of these scenarios at \(-5 \%\) and \(6 \%\), respectively, find the annual return in the remaining scenario if the expected annual return is known to be \(6 \%\).

Short answer

The annual return in the boom scenario is 28%.

Step-by-step solution

- Write down given probabilities

Given probabilities are:Probability of recession, P(Recession) = \(\frac{1}{2}\)Probability of stagnation, P(Stagnation) = \(\frac{1}{4}\)Probability of boom, P(Boom) = \(\frac{1}{4}\)

- Write down given returns

Given annual returns for the scenarios are:Return during recession, R(Recession) = -5% = -0.05Return during stagnation, R(Stagnation) = 6% = 0.06

- Write down the expected annual return

The expected annual return is given as 6%, or E(R) = 6% = 0.06

- Set up the equation for the expected return

The expected return can be written as the weighted sum of the returns in each scenario with their probabilities:E(R) = P(Recession) * R(Recession) + P(Stagnation) * R(Stagnation) + P(Boom) * R(Boom)

- Plug in the known values

Substitute the known values into the equation:0.06 = \(\frac{1}{2}\) * (-0.05) + \(\frac{1}{4}\) * 0.06 + \(\frac{1}{4}\) * R(Boom)

- Calculate the known parts

Calculate the contributions of the known returns:\(\frac{1}{2}\) * (-0.05) = -0.025\(\frac{1}{4}\) * 0.06 = 0.015

- Solve for the unknown return in the boom scenario

Substitute back into the equation and solve for R(Boom):0.06 = -0.025 + 0.015 + \(\frac{1}{4}\) * R(Boom)0.06 = -0.01 + \(\frac{1}{4}\) * R(Boom)0.07 = \(\frac{1}{4}\) * R(Boom)R(Boom) = 0.07 * 4R(Boom) = 0.28 or 28%

Key concepts

Probability in Finance

Probability is a key concept in finance, especially when forecasting returns under different economic conditions. It helps investors gauge the likelihood of certain outcomes and make informed decisions. Here, we're working with three scenarios—recession, stagnation, and boom—with respective probabilities of \(\frac{1}{2}\), \(\frac{1}{4}\), and \(\frac{1}{4}\).
In essence, probability allows us to quantify uncertainty. By understanding probabilities, investors can prepare for different market conditions and potentially mitigate risks.

Scenario Analysis

Scenario analysis involves evaluating the impact of different economic conditions on your investment returns. It helps in understanding how various scenarios can affect the overall return.
In our exercise, we look at three scenarios—recession, stagnation, and boom. Each scenario has a particular predicted return, such as \(-5\text{\textbackslash%}\) in a recession and \(6\text{\textbackslash%}\) in stagnation.
This helps investors prepare for different outcomes and make more robust financial plans. By analyzing each scenario, one can anticipate potential gains or losses and adjust their investment strategy accordingly.

Weighted Average Return

The weighted average return represents the expected annual return, considering the probabilities of different scenarios. It's a critical metric for investors as it combines multiple possible outcomes into a single number.
To calculate this, we multiply the return of each scenario by its probability and sum these products. For our exercise, this means:
Expected return \(E(R) = P_{\text{Recession}} \times R_{\text{Recession}} + P_{\text{Stagnation}} \times R_{\text{Stagnation}} + P_{\text{Boom}} \times R_{\text{Boom}}\).
Substituting the given values, we solve for the unknown return in the boom scenario and find it to be \(28\text{\textbackslash%}\).
Calculating the weighted average return provides a comprehensive view of what to expect under various conditions, helping to make informed investment decisions.