Question

Assume that you have a firm that owns land that is currently valued at $100 million and that the firm has $80 million of zero-coupon debt outstanding. The debt will mature in 1 year. In 1 year, the value of the land will either be $200 million or $50 million. The probability of the value of land increasing is 36.67%. The risk-free rate is 5%. The firm has 2 million shares outstanding. What is the price of 1 share for this firm?

Answer 1

Let P1 denote the Probability that the value of land will increase and P2 denote the Probability that the value of land will decrease.
P1 = 36.67%
P2 = 100-36.67% = 63.33%

Value of Land in case of P1 = $200 m
Value of Land in case of P2 = $50 m

Expected value of Land after Year 1 will be = (P1 * Value of Land in case of P1 ) + (P2 * Value of Land in case of P2 )
                                                                             = (36.67% * 200m) + (63.33% * 50m)
                                                                 = 73.34 + 31.665
                                                                 = $105.005 million
Rf = 5%
Therefore, PV of Expected value of Land after Year 1 = $105.005 million/1.05
                                                                                 = $100 million

Value of ZCB = F/ (1+r)t
where
F = Face Value of Bond
r = rate of yield
t = No. of years to maturity

Value of ZCB = $80/(1.05)1
                      = $76.1905 million

Shareholders Equity = Value of Assets – Value of Liabilities
                                 = $100 - $76.1905
                                = $23.8095 million

Total No. of shares = 2 million
Price per share = Shareholders Equity / Total No. of Shares
                      = $23.8095 million / 2 million
                        = 11.9048