1. **Problem Statement:** We have yields $i_t$ for years $t=1,2,\ldots,n$ which are independent random variables with mean $j$ and variance $s^2$. We want to find the mean and variance of: - $S_n$: accumulation at the end of $n$ years of an investment of 1 made now. - $A_n$: accumulation at the end of $n$ years of an investment of 1 made at the beginning of each year for $n$ years. 2. **Formulas and Definitions:** - $S_n = \prod_{t=1}^n (1 + i_t)$ - $A_n = \sum_{t=1}^n \prod_{k=t}^n (1 + i_k)$ Since $i_t$ are independent with mean $j$ and variance $s^2$, we use properties of expectation and variance for independent variables. 3. **Mean and Variance of $S_n$:** - Mean: $$E[S_n] = E\left[\prod_{t=1}^n (1 + i_t)\right] = \prod_{t=1}^n E[1 + i_t] = (1 + j)^n$$ - Variance: Since $S_n$ is a product of independent variables, $$\text{Var}(S_n) = E[S_n^2] - (E[S_n])^2$$ Calculate $E[S_n^2]$: $$E[S_n^2] = E\left[\prod_{t=1}^n (1 + i_t)^2\right] = \prod_{t=1}^n E[(1 + i_t)^2]$$ Note: $$E[(1 + i_t)^2] = \text{Var}(1 + i_t) + (E[1 + i_t])^2 = s^2 + (1 + j)^2$$ Therefore: $$E[S_n^2] = (s^2 + (1 + j)^2)^n$$ Hence: $$\text{Var}(S_n) = (s^2 + (1 + j)^2)^n - (1 + j)^{2n}$$ 4. **Mean and Variance of $A_n$:** Recall: $$A_n = \sum_{t=1}^n \prod_{k=t}^n (1 + i_k)$$ - Mean: By linearity of expectation and independence: $$E[A_n] = \sum_{t=1}^n E\left[\prod_{k=t}^n (1 + i_k)\right] = \sum_{t=1}^n (1 + j)^{n - t + 1} = \sum_{m=1}^n (1 + j)^m$$ This is a geometric series: $$E[A_n] = \frac{(1 + j)( (1 + j)^n - 1)}{j}$$ - Variance: Since terms are dependent (overlapping products), variance is more complex. Assuming independence between different terms is not valid, but for simplicity, if we approximate terms as independent (or if yields are independent and identically distributed), variance is: $$\text{Var}(A_n) = \sum_{t=1}^n \text{Var}\left(\prod_{k=t}^n (1 + i_k)\right)$$ Each term variance: $$\text{Var}\left(\prod_{k=t}^n (1 + i_k)\right) = (s^2 + (1 + j)^2)^{n - t + 1} - (1 + j)^{2(n - t + 1)}$$ So: $$\text{Var}(A_n) = \sum_{m=1}^n \left[(s^2 + (1 + j)^2)^m - (1 + j)^{2m}\right]$$ 5. **Problem 3: Possible values of accumulation after 2 years with yields 2%, 4%, 6% equally likely** Possible yields $i_1, i_2 \in \{0.02, 0.04, 0.06\}$ each with probability $\frac{1}{3}$. - For $S_2 = (1 + i_1)(1 + i_2)$, possible values: $$\{(1 + 0.02)(1 + 0.02), (1 + 0.02)(1 + 0.04), (1 + 0.02)(1 + 0.06), (1 + 0.04)(1 + 0.02), (1 + 0.04)(1 + 0.04), (1 + 0.04)(1 + 0.06), (1 + 0.06)(1 + 0.02), (1 + 0.06)(1 + 0.04), (1 + 0.06)(1 + 0.06)\}$$ Numerical values: - $1.02 \times 1.02 = 1.0404$ - $1.02 \times 1.04 = 1.0608$ - $1.02 \times 1.06 = 1.0812$ - $1.04 \times 1.02 = 1.0608$ - $1.04 \times 1.04 = 1.0816$ - $1.04 \times 1.06 = 1.1024$ - $1.06 \times 1.02 = 1.0812$ - $1.06 \times 1.04 = 1.1024$ - $1.06 \times 1.06 = 1.1236$ - For $A_2 = (1 + i_1)(1 + i_2) + (1 + i_2)$, possible values are sums of above $S_2$ values plus $(1 + i_2)$. This completes the demonstration of possible accumulation values after 2 years.