Consider an integrated real$-$options$-$based model of corporate investment and financing. This model synthesizes the capital structure decisions of Leland $(1994)$ and investment as a real option exercising problem of McDonald and Siegel $(1986).$

$-$The firm starts with no cash flows and simply holds a perpetual American$-$style investment option, which can be exercised at any time ${}_i$ by paying a one$-$time fixed investment cost I.

$-$After exercising its investment option, the firm starts generating a stochastic perpetual cash flow process, $Y_t,$ which is given by

$d{Y}_t=Y_{t}dt+Y_{t}d{Z}_t,$ and $Z_t$ is a standard Brownian motion.

$-$The risk$-$free interest rate $r$ is constant. For convergence, let the expected growth rate $$ be lower than the interest rate, in that $r>.$ Assume no production cost after the asset is in place.

$-$At the endogenously chosen $($stochastic$)$ investment time ${}_i,$ the firm issues a mixture of debt and equity. As in standard trade$-$off models of capital structure, debt has a tax advantage. The firm faces a constant tax rate $>0$ on its income after servicing interest payments on debt. To balance the tax benefits, debt induces deadweight losses when the firm does poorly. The firm dynamically trades off the benefits and costs of issuing debt. For analytical convenience, assume that debt is perpetual and is issued at par as in Leland $(1994).$ Let $C$ denote the firm's coupon payment for its debt outstanding. The assumption of perpetual debt simplifies the analysis substantially and has been widely adopted

in the literature. For your technical convenience, we assume that the firm can only issue debt at investment times ${}_i.$

$-$That is$,$ once debt is issued, the equity holders receive $(1-)(Y-C)$ and debt investors collect $C$ until the firm defaults at ${}_d.$

$-$Upon default at ${}_d,$ the firm's recovery value is

$L(Y)=(1-)(\frac{1-}{r-})Y.$

Equity holders are protected by limited liability. Absolute priority rule $($APR$)$ holds in that equity investors will only get paid upon liquidation$/$default if creditors are fully repaid. Here debt holders will seize all the recovery upon default. All agents have rational expectations.

Answer the following questions:

$($a$)$ Let $Y_B$ be the bankruptcy boundary which is determined endogenously. Please write down the HJB equation of debt value $D(Y)$ for $Y>Y_B.$ Provide appropriate value matching condition at $Y_B$ and the asymptotic behavior of $D(Y)$ as $Y.$

$($b$)$ Solve the boundary value problem in $($a$)$ by postulating that the

solution is of the form $D(Y)=A_0+A_{1}Y+A_2{Y}^{}.$

$($c$)$ Write down the HJB equation of equity value $E(Y)$ and solve it$.$

$($d$)$ What is the firm's leverage choice $($e$.$g$.,$ the amount of debt issuance and coupon payments$)$ at the moment of exercising the growth option? What is the firm's default strategy after debt is issued at ${}_i?($Hint: you may pin down the default boundary $Y_B$ by using the smooth pasting condition.$)$

$($e$)$ What is the firm's optimal growth option exercising strategy?

$($f$)$ Let $r=5\%,=20\%,=1\%,=20\%,I=1.$ Use these parameter

values to plot the model solution, e$.$g$.,$ equity value, debt value, firm value, after the firm exercises the investment and financing options at ${}_i$; and the firm value $($which is the same as the option value$)$ before ${}_i.$ Use $E(Y),D(Y),$ and $F(Y)$ to denote the firm's levered equity, debt, and total firm value, respectively. Graph these value functions and their sensitivities$,E^{\text{'}}(Y),D^{\text{'}}(Y)$ and $F^{\text{'}}(Y).$