Mr. Jones was considering a new grapefruit venture that would generate a random sequence of yearly cash flows. He asked his son. Gavin, "People tell me I should use a cost of capital figure to discount the stream. They say it's based on the CAPM. Have you given up on that? I haven't heard you talk about it for awhile."

Gavin replied, "Special conditions are required to justify it for more than one period. We had a complicated final exam question on it."

Consider a two-year model. The risk-free rate for each is \(r\). The (random) rates of return for the Markowitz portfolio in the two years are \(r_{1}\) and \(r_{2}\), respectively, and they are independent. There is a single random cash flow \(x_{2}\) at the end of the second year. Denote by \(x_{210}\) and \(x_{2 \mid 1}\) the random variable \(x_{2}\) given the information at times zero and one, respectively, and let \(\mathrm{E}_{0}\) and \(\mathrm{E}_{1}\) denote expectation at times zero and one. Likewise let \(V_{0}\) and \(V_{1}\) denote the value at time zero and one, respectively, of receiving \(x_{2}\) at time 2 . Assume that \(\mathrm{E}_{0}\left\{\mathrm{E}_{1}\left[x_{2 \mid 1}\right]\right\}=\mathrm{E}_{0}\left[x_{2 \mid 0}\right]\) and that \(\operatorname{cov}\left[x_{2 \mid 1} / V_{1}, r_{2}\right]\) is independent of the information received at time one. Show that the value at time zero of receiving \(x_{2}\) at time 2 is \[V_{0}=\frac{E_{0}\left|x_{2}\right| 0 \mid}{\left[1+r+\beta_{1}\left(\bar{r}_{1}-r\right)\right]\left[1+r+\beta_{2}\left(\bar{r}_{2}-r\right)\right]},\]
where \[\beta_{1}=\operatorname{cov}\left[V_{1} / V_{0}, r_{1}\right] / \sigma_{r_{1}}^{2} .\]
Find \(V_{1}\) and \(\beta_{2}\).