Consider an apartment property development project A$.$ Developer D will invest $$30$M now at t$=0,$ spend one year on construction, hold the completed property for one year between year t$=1$ and t$=2,$ and sell the property at year t$=2.$

The total development cost is $$80$M$,$ but D will make two payments to the general contractor: $$40$M at year t$=0$ and the remaining $$40$M at year t$=0\mathrm{.}5.$ D will put $$30$M equity at t$=0.$ The remaining cost is funded with a construction loan at $6\%$ per annum $($one$-$year loan with two draws at year t$=0$ and year t$=0\mathrm{.}5($month $6)).$ Interests are accrued monthly until the maturity at t$=1.$

At the completion of property A at t$=1,$ D will arrange a one$-$year, zero$-$coupon, non$-$recourse, bridge loan $($Loan B$)$ collateralized by property A until t$=2.$ The face value is $$95$M $($The loan contract states that D should repay $$95$M at t$=2).$ Part of this bridge loan will be used to pay off the construction loan. Interest accrual is annual.

At t$=1,$ the value of property A will be $$100$M$.$ At t$=2,$ the property value, which includes operating cash flows between t$=1$ and t$=2,$ will be worth either $$117$M with a probability of $0\mathrm{.}4$ or $$93$M with a probability of $0\mathrm{.}6.$

Debt

$1.$ What is the construction loan due at t$=1?$

$ M

$2.$ Consider a one$-$year, zero$-$coupon, riskless loan, with a face value of $$95$M$,$ originated at t$=1($The only payment that the lender will receive is $$95$M at t $=2).$ The riskless rate of return is $1\%$ per year, which accrues annually. What is the value of this riskless loan at t$=1?$

$ M

$3.$ Consider the bridge loan B that was explained above. What is the bridge loan lender$$s payoff in each state of nature at t$=2?($Hint: D promises to pay $$95$M at t$=2$ but may strategically default on the loan.$)$

$ M if the property value is $117$;

$ M if the property value is $93$

$4.$ What is the amount that Loan B$$s lender will forgive $($i$.$e$.,$ the payoff to the borrower$$s default option$)$ in each state of nature at t$=2?($Hint: The lender has to forgive the difference between the promised payment and the collateral value.$)$

$ M if the property value is $117$;

$ M if the property value is $93$

$5.$ The value $($i$.$e$.,$ premium$)$ of the borrower$$s default option for Loan B is $$1\mathrm{.}32$M at t$=1.$

What is the value of Loan B at t$=1?$ This amount will be the loan amount that Developer D will receive at t$=1$ from the Bridge loan lender.

$ M

$6.$ What is the YTM and credit spread for Loan B$?$

YTM is $\%$ and the credit spread is $\%$

$($Hint: The credit spread is the difference in YTM between a credit$-$risky loan and a riskless loan. The YTM is the IRR on the basis of the promised payment.$)$

$7.$ What is the expected rate of return and the risk premium to Loan B$?$

The expected rate of return is $\%$ and the risk premium is $\%.$

$($Hint: The expected rate of return is based on the expected payoff in the future. The risk premium is the difference in the expected return between a credit$-$risky loan and a riskless loan.$)$

Equity

$8.$ How much can Developer D extract cash out of the project A at t$=1$ after the refinancing? $($This amount will be transferred to D$$s personal account and not remain in Project A$.)$

$ M

$9.$ What is Property A$$s equity payoff in for each state of nature at t$=2?$ At t$=2,$ the property will be sold at the market price, and Loan B will be paid off. The equity payoff equals sales proceeds less loan repayment.

$ M if the property value is $117$;

$ M if the property value is $93$

$10.$ What is the value of the equity at t$=1?$

$ M $($Hint: There are two methods to calculate the equity value.$)$

Method $1$: Recall the simple method based on the Modigliani$-$Miller Theorem:

$($Property value at t$=1)=($Debt value at t$=1)+($Equity value at t$=1).$

Method $2$: The equity DCF based on the expected rate of return to the equity that will be calculated in Q$10.$

$11.$ What is the expected rate of return to equity between t$=1$ and t$=2?$

$\%$

$($Hint: If you already calculated the equity value in Q$9,$ you can simply calculate the expected rate of return. Alternatively, if you want to first calculate the expected rate of return to equity, you can use the WACC formula.$)$

$12.$ Compare the present value of equity with the present value of the following asset portfolio: a long $(+)$ property, a short $(-)$ riskless debt, and a long $(+)$ default option. Should they be equal or different?

The PV of equity is $ M $($copied from Q$10).$

The PV Portfolio: Property value $($$$100$M$)$ Riskless debt $($$ M copied from Q$2)+$ Default Option $($$$1\mathrm{.}32$Mcopied from Q$5)=$ $ M$.$

These two values should be $\{$equal $/$ different$\}.$ Pls show work