Method Documentation
Expected value

4min
Method
This method implements the calculation of the expected value for a savings plan. The objective of this calculation is to project the future value of a savings plan, which comprises:
an initial deposit,
regular monthly savings.
The calculation makes use of the following formula to determine the expected value (x) of the savings plan at a given month (t):
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x_t = (x_{t-1} + y_{t-1}) \times (1+r)
x_t = (x_{t-1} + y_{t-1}) \times (1+r)
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In this formula: 
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\begin{align*}
x_t & = \text{expected value at time t (month)}, \\
x_{t-1} & = \text{expected value at time t-1 (previous month)}, \\
y_{t-1} & = \text{cashflow at time t-1 (previous month)}, \\
r & = \text{monthly return}. \\
\end{align*}

\begin{align*}
x_t & = \text{expected value at time t (month)}, \\
x_{t-1} & = \text{expected value at time t-1 (previous month)}, \\
y_{t-1} & = \text{cashflow at time t-1 (previous month)}, \\
r & = \text{monthly return}. \\
\end{align*}
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The cashflow variable (y) encapsulates both the initial deposit and the monthly savings. The initial deposit is factored in at the beginning of the savings period. The monthly savings (cashflow) are introduced at the start of each subsequent month and initial monthly saving should be set to zero, i.e.
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y_0=0.
 
y_0=0.
 
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The expected value (x) is calculated at the end of each month.
Additionally, the method determines the monthly return (r) from the annual return (R) using the formula:
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r = \left(1 + R\right)^{\frac{1}{12}} - 1
r = \left(1 + R\right)^{\frac{1}{12}} - 1
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This formula consists of:
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\begin{align*}
r & = \text{monthly return}, \\
(1 + R) & = \text{one plus the annual return}, \\
\left(1 + R\right)^{\frac{1}{12}} & = \text{the (1 + annual return) raised to the power of 1/12 to convert to a monthly equivalent}, \\
\left(1 + R\right)^{\frac{1}{12}} - 1 & = \text{subtract 1 to bring it back to a return-like format}. \\
\end{align*}

\begin{align*}
r & = \text{monthly return}, \\
(1 + R) & = \text{one plus the annual return}, \\
\left(1 + R\right)^{\frac{1}{12}} & = \text{the (1 + annual return) raised to the power of 1/12 to convert to a monthly equivalent}, \\
\left(1 + R\right)^{\frac{1}{12}} - 1 & = \text{subtract 1 to bring it back to a return-like format}. \\
\end{align*}
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The monthly return (r) is therefore a transformation of the annual return (R) that factors in the monthly nature of the savings plan.

Through the use of these formulas and variables, the method provides a projected expected value for a given savings plan.
Updated 05 Jun 2023
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