$3.$ Valuing Rio Corporation

Sangria is tempted to acquire the Rio Corporation, which is also in the business of promoting relaxed, happy lifestyles. Rio has developed a special weight$-$loss program called the Brazil Diet, based on barbecues, red wine, and sunshine. The firm guarantees that within three months you will have a figure that will allow you to fit right in at Ipanema or Copacabana beach in Rio de Janeiro. But before you head for the beach, you've got the job of working out how much Sangria should pay for Rio.

Rio is a U$.$S$.$ company. It is privately held, so Sangria has no stock market price to rely on$.$ Rio has $1\mathrm{.}5$ million shares outstanding and debt with a market and book value of $$36$ million.

Rio is in the same line of business as Sangria, so we will assume that it has the same business risk as Sangria and can support the same proportion of debt. Therefore, we can use Sangria's WACC as the discount rate for Rio$$s cash flows.

Your first task is to forecast Rio's free cash flow $($FCF$).$ Free cash flow is the amount of cash that the firm can pay out to investors after making all investments necessary for growth. Free cash flow is calculated assuming the firm is all$-$equity$-$financed. Discounting the free cash flows at the after$-$tax WACC gives the total value of Rio $($debt plus equity$).$ To find the value of its equity, you will need to subtract the $$36$ million of debt.

We will forecast each year's free cash flow out to a valuation horizon $($H$)$ and predict the business's value at that horizon $($PVH$).$ The cash flows and horizon value are then discounted back to the present:

Of course, the business will continue after the horizon, but it's not practical to forecast free cash flow year by year to infinity. PVH stands in for the value in year H of free cash flow in periods H $+1,$ H $+2,$ etc.

Table $1$ below sets out the information that you need to forecast Rio's free cash flows. We will follow common practice and start with a projection of sales. In the year just ended Rio had sales of $$83\mathrm{.}6$ million. In recent years, sales have grown by between $5\%$ and $8\%$ a year. You forecast that sales will grow by about $7\%$ a year for the next three years. Growth will then slow to $4\%$ for years $4$ to $6$ and to $3\%$ starting in year $7.$

The other components of cash flow in Table $1$ are driven by these sales forecasts. For example, you can see that costs are forecasted at $74\%$ of sales in the first year with a gradual increase to $76\%$ of sales in later years, reflecting increased marketing costs as Rio's competitors gradually catch up$.$

Increasing sales are likely to require further investment in fixed assets and working capital. The projected investments in fixed assets and working capital in each year are given in the table.

Finally, Rio's free cash flow is calculated in the table as profit after tax, plus depreciation, minus investment. Investment is the change in the stock of $($gross$)$ fixed assets and working capital from the previous year.

Table $1$: Free cash flow projections and company value for Rio Corporation $($$ millions$)$

Question $3$: What is total firm value of Rio? What is the equity value of Rio? What is the maximum price Sangria would pay for each share Rio$$s equity?