Simulation Methods

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Learning Module 6: Simulation Methods

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Continuously Compounded Returns

\[ r_{0,T} = r_{T-1,T} + r_{T-2,T-1} + \cdots + r_{0,1} \tag{1} \]

Where:

• \(r_{0,T}\): continuously compounded return to time \(T\)
• \(r_{T-1,T}\): continuously compounded return from time \(T-1\) to time \(T\)
• \(T\): total number of periods

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## Continuously Compounded Returns

$$
r_{0,T} = r_{T-1,T} + r_{T-2,T-1} + \cdots + r_{0,1} \tag{1}
$$

Where:

* $r_{0,T}$: continuously compounded return to time $T$
* $r_{T-1,T}$: continuously compounded return from time $T-1$ to time $T$
* $T$: total number of periods

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confirm name?? Expected Continuously Compounded Returns

\[ E(r_{0,T}) = E(r_{T-1,T}) + E(r_{T-2,T-1}) + \cdots + E(r_{0,1}) = \mu T \tag{2} \]

Where:

• \(E(r_{0,T})\): expected continuously compounded return from time \(0\) to time \(T\)
• \(E(r_{t-1,t})\): expected continuously compounded return from time \(t-1\) to time \(t\)
• \(\mu\): mean of the one-period continuously compounded return
• \(T\): total number of periods

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## confirm name?? Expected Continuously Compounded Returns

$$
E(r_{0,T}) = E(r_{T-1,T}) + E(r_{T-2,T-1}) + \cdots + E(r_{0,1}) = \mu T \tag{2}
$$

Where:

* $E(r_{0,T})$: expected continuously compounded return from time $0$ to time $T$
* $E(r_{t-1,t})$: expected continuously compounded return from time $t-1$ to time $t$
* $\mu$: mean of the one-period continuously compounded return
* $T$: total number of periods

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name?? Variance of Expected Continuously Compounded Returns

\[ \sigma^2(r_{0,T}) = \sigma^2 T \tag{3} \]

Where:

• \(\sigma^2(r_{0,T})\): variance of the continuously compounded return from time \(0\) to time \(T\)
• \(\sigma^2\): variance of the one-period continuously compounded return
• \(T\): total number of periods

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## name?? Variance of Expected Continuously Compounded Returns

$$
\sigma^2(r_{0,T}) = \sigma^2 T \tag{3}
$$

Where:

* $\sigma^2(r_{0,T})$: variance of the continuously compounded return from time $0$ to time $T$
* $\sigma^2$: variance of the one-period continuously compounded return
* $T$: total number of periods

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