I need help with all parts of question 3 and 4 in this problem set.

Problem Set 1 Name (Print): Fill out this copy of problem set. Write each answer in box provided. Correct Answer = True Value 10%. 1. You're thinking about buying a property which will be worth either $10M next year with probability 0.5562 or $6M with probability 0.4438. Discount rate R = 8%. Riskless rate Rf = 4%. (a) Draw the binomial tree representing this property's payouts. $10 62 0.55 P0 0.44 38 $6 (b) What is the property's value today? P = $7.6156 = 1 ( 0.5562 $10 + 0.4438 $6 ) 1 + 0.08 (c) What is the portfolio of up and down state assets that replicates the property's payouts? $7.6156 = 10 SU + 6 SD (d) What is the portfolio of up and down state assets that replicates the riskless asset's payouts? $1 = $0.9615 = SU + SD 1 + 0.04 (e) What are the prices of the up and down state assets implied by the property's sale price? \u0013 \u0012 $1 $1 SD = SU and $7.6156 = 6 SU + 10 SU 1 + 0.04 1 + 0.04 SU = $0.4615 = $7.6156 6 10 6 $1 1+0.04 and SD = $0.5000 = $1 SU 1 + 0.04 (f) Roger Cannaday tells you that he paid $12.5 for another property in the area that will be worth $15 next year if things go well and only $9 if things go poorly. Did he overpay? Yuuup. $12.5 > $11.4225 = 15 0.4615 + 9 0.5000 Problem Set 1 Page 2 of 4 Due Oct. 5th, 2012 2. You're buying a property which will be worth either $110 next year with probability 0.4421 or $72 with probability 0.5579. Discount rate R = 11%. Riskless rate Rf = 7%. (a) Draw the binomial tree representing this property's payouts. $110 21 0.44 $80 0.55 7 9 $72 1 P = $80 = ( 0.4421 $110 + 0.5579 $72 ) 1 + 0.11 (b) You take out a $70 mortgage to buy the property at a RMtg= 14.4293% rate. Draw the binomial tree representing the mortgage's payouts to the lender. 21 0.44 $70 (1 + RMtg ) = $80.1005 $70 0.55 7 9 $72 (c) Is the mortgage risky? Yes. (d) Draw the binomial tree representing the payouts to your equity position. 21 0.44 $110 $80.1005 = $29.8995 $10 0.55 79 $0 (e) What is your return on equity? $10 = 1 ( 0.4421 $29.8995 + 0.5579 $0 ) 1 + REquity so REquity = 32.19% Problem Set 1 Page 3 of 4 Due Oct. 5th, 2012 3. Roger Cannaday comes to you with a proposal to buy a property for $25M which will be worth either $40.545 or $21.250 next year. Discount rate R = 10%. Riskless rate Rf = 8%. (a) What is the portfolio of up and down state assets that replicates the property's payouts? $25 = 40.545 SU + 21.250 SD (b) What is the portfolio of up and down state assets that replicates the riskless asset's payout? $1 = $0.9259 = SU + SD 1 + 0.08 (c) What are the market implied prices of the up and down state assets? $1 SD = SU 1 + 0.08 \u0012 and $1 $25 21.250 1+0.08 SU = $0.2759 = 40.545 21.250 $25 = 21.250 and $1 SU 1 + 0.08 SD = $0.6500 = \u0013 + 40.545 SU $1 SU 1 + 0.08 (d) Suppose Roger asks you to lend him $21 to finance the purchase. Draw the binomial tree representing the payouts to this mortgage. $21 (1 + RMtg ) $21 $21.250 (e) What is the fair mortgage rate you should charge him? $21 = 0.2759 $21 (1 + RMtg ) + 0.6500 $21.250 so RMtg = 24.05% (f) Suppose that the fair mortgage rate is too high for Roger, and in response he offers to make a $1 personal guarantee. Draw the binomial tree representing the payouts to this new mortgage. $21 (1 + RMtg ) $21 $21.250 + $1 (1 + Rf ) = $22.33 (g) What is the fair mortgage rate you should charge him after taking into consideration the personal guarantee? $21 = 0.2759 $21 (1 + RMtg ) + 0.6500 $22.330 so RMtg = 11.94% Problem Set 1 Page 4 of 4 Due Oct. 5th, 2012 4. You're buying a house for $150k using both a $120k first mortgage as well as a $15k second mortgage. You know that next year the house will be worth either $200k with probability 0.50 or $130k with probability 0.50. Riskless rate Rf = 5%. (a) Draw the binomial tree representing the first mortgage's payouts. Is it risky? $120k (1 + Rf ) = $126k 0.50 $120k 0.50 $120k (1 + Rf ) = $126k (b) Draw the binomial tree representing the second mortgage's payouts. Is it risky? $15k (1 + RM2 ) 0.50 $120k 0.50 $4k (c) What is the portfolio of up and down state assets that replicates the property's payouts? $150k= 200k SU + 130k SD (d) What is the portfolio of up and down state assets that replicates the riskless asset's payout? $1 = $0.952381 = SU + SD 1 + 0.05 (e) What are the market implied prices of the up and down state assets? \u0012 \u0013 $1 $1 SD = SU and $150k = 130k SU + 200k SU 1 + 0.05 1 + 0.05 SU = $0.3741 = $1 $150k 130k 1+0.05 200k 130k and SD = $0.5782 = $1 SU 1 + 0.05 (f) What is the fair mortgage rate on the second mortgage? $15k = 0.3741 $15k (1 + RM2 ) + 0.5782 $4k so RM2 = 126.0928%