$30.$ Constant$-$growth DCF formula $($S$4\mathrm{.}4)$ The constant$-$growth DCF formula:

DIV $1$ Po $-$g

is sometimes written as:

ROE$(1$ b$)$BVPS

Po$=$

r$-$bROE

where BVPS is book equity value per share, b is the plowback ratio, and ROE is the ratio of earnings per share to BVPS$.$ Use this equation to show how the price$-$to$-$book ratio varies as ROE changes. What is price$-$to$-$book when ROE $=$ r$?$