9. Appendixes. Page 73 of 95

Transcription

1 9. Appendixes Appendix A: Construction cost Appendix B: Cost of capital Appendix B.1: Beta Appendix B.2: Cost of equity Appendix C: Geometric Brownian motion Appendix D: Static NPV valuation Appendix E: Simulation output Appendix F: Binomial lattice model Appendix G: Closed-form model Appendix H: Probability of exercise Appendix I: Development of sequential compound option Page 73 of 95

2 Appendix A: Construction cost Construction cost index data in Excel file (Sheet: Construction cost) Table A.A: Descriptive statistics construction cost index Quaterly Annually Mean 0,80% 3,19% Standard Error 0,08% 0,33% Median 0,73% 2,91% Standard Deviation 0,81% 1,61% Sample Variance 0, ,00013 Kurtosis 0, Skewness 0, Range 4,78% Minimum 1,51% Maximum 3,27% Table A.B: Historical distribution of average construction costs Site cost (% of total cost) 19% 19% 19% 19% 19% Hard construction costs (% of total cost) 66% 65% 65% 65% 65% Soft construction costs (% of total cost) 15% 16% 16% 16% 16% Total 100% 100% 100% 100% 100% Source: The Ministry of Social Affairs Table A.C: Newly-built non-profit housing by type of property, type of building, region, acquisition prices, and time. Family housing Social housing Copenhagen Jutland Jutland (est) Cost of land Land development Total cost of land Hard costs Other hard costs Total hard costs Cost of land Land development Total cost of land Hard costs Other hard costs Total hard costs Cost of land Land development Total cost of land Hard costs Other hard costs Total hard costs Source: Boligselskabernes Landsforening Average Hard construction cost (kr/ gross condo m2) Average Soft construction cost (kr/ gross condo m2) Average total construction cost (kr/ gross condo m2) Page 74 of 95

3 Appendix B: Cost of capital Stock data in Excel file (Sheet: Beta estimation) Appendix B.1: Beta Beta for TK Development: Dependent Variable: TKD Method: Least Squares Date: 10/29/11 Time: 17:31 Sample: Included observations: 129 Variable Coefficient Std. Error t-statistic Prob. C MSCI R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Beta for Sjælsø Gruppen: Dependent Variable: SJA Method: Least Squares Date: 10/29/11 Time: 17:34 Sample: Included observations: 129 Variable Coefficient Std. Error* t-statistic Prob. C MSCI R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression* Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) * Std. Error --> 2 standard errors is 95% confidence interval --> thereby shows range of error of beta estimate --> please note it is calculated for the levered beta Page 75 of 95

4 Unlevered Beta: 1 1 / where See Excel file for unlevered beta and rolling beta (Sheet name: Beta estimation) Industry Group # of firms Beta D/E Ratio Tax rate Beta Unlevere D/(E+D) Re R.E.I.T. 30 0,49 86,16% 4,63% 0,27 46,28% 9,73% Real Estate 49 0,75 291,07% 9,75% 0,21 74,43% 7,36% Real Estate (Dev) 35 0,71 170,79% 6,27% 0,27 63,07% 7,12% Average 0,65 182,68% 6,88% 0,25 61,26% 8,07% Source: Darmodoran Online 1. Industry beta is calculated by Damodaran unlevering a raw beta by the use of Hamada's formula for 5 years of monthly returns. 2. The betas could be adjusted/smoothed to one following e.g. Bloomberg's methodology: Raw-Beta*0.67+1*0.33 described in Koller et al This has not been done since we want to be able to compare to Damodaran (Are shown in Excel file Beta Estimate) Page 76 of 95

5 Appendix B.2: Cost of equity See Excel file (Sheet: Cost of capital) Risk free rate: The risk free rate is estimated based on a yield curve of spot rates from government bonds with 10 year maturity. The yield curve is based on the Svensson model. Risk free rate Mean 3,49% Standard Error 0,01% Median 3,58% Standard Deviation 0,46% Sample Variance 0,22% Kurtosis 0,25% Skewness 0,31% Range 2,31% Minimum 2,47% Maximum 4,78% Source: ECB % Yield curve spot rate, 10 year maturity Euro The risk free rate was estimated to 3,49% and the expected return on the market was estimated to 8,64%. Using CAPM the risk premium on the market is estimated, given as:, The market model is derived from CAPM and is given as:, 1 Table B.A: Cost of Equity Avg. Risk free rate 3,42% 2,21% 4,14% 4,12% 3,72% 2,40% 3,49% Beta 1,153 1,373 1,184 1,123 0,811 1,1 1,1 Risk premium 5,48% 5,94% 5,55% 4,13% 4,65% 5,15% 5,15% Re 4,92% 6,15% 9,11% 10,92% 9,70% 8,07% 9,15% Page 77 of 95

6 Appendix C: Geometric Brownian motion Descriptive statistics for sale prices/m 2 for total period and for Quaterly Annually Mean 1,64% 6,56% Standard Error 0,55% 2,20% Median 2,40% 9,58% Standard Deviation 4,83% 9,67% Sample Variance 0, ,0047 Kurtosis 0, Skewness 0, Range 23,64% Minimum 12,80% Maximum 10,84% Quaterly Annually Mean 1,00% 4,02% Standard Error 0,74% 2,95% Median 1,67% 6,68% Standard Deviation 4,83% 9,67% Sample Variance 0, ,0047 Kurtosis 0, Skewness 0, Range 20,04% Minimum 10,91% Maximum 9,13% To simulate the Geometric Brownian Motion equation above it is discretized by Itô s Lemma according to Marques et al. (2006):, To estimate α and σ in equation 6.6 the following equations are applied: 0,5 Where N is periods per year. Page 78 of 95

7 Dickey-Fuller test statistics In order to test for unit root a dickey fuller test is performed. Null Hypothesis: PRICE_M2 has a unit root Exogenous: Constant Lag Length: 1 (Automatic based on SIC, MAXLAG=11) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level 10% level *MacKinnon (1996) one-sided p-values. With trend and constant Null Hypothesis: PRICE_M2 has a unit root Exogenous: Constant, Linear Trend Lag Length: 4 (Automatic based on SIC, MAXLAG=11) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level 10% level *MacKinnon (1996) one-sided p-values. The Dickey-Fuller cannot be rejected and the data is assumed to be a random walk. In order to satisfy the above statement of the GBM following assumptions must be satisfied: 1. Normality of the log ratios 2. Independence from previous data (log ratios independent of their past values) Page 79 of 95

8 Normality: To test for normality goodness-of-fit test is used in this case a battery of test is used. The hypothesis states: H 0 : The distribution is normal H 1 : The distribution is not normal Before the tests are performed the plots below gives a good indication of how normal the data is. The p-value is compared to the specified significance level, it the p-value is greater than the level of significance the test statistic is not in the rejection region and the null hypothesis cannot be rejected. Empirical Distribution Test for PRICE Hypothesis: Normal Date: 10/30/11 Time: 11:19 Sample: 1 77 Included observations: 77 Method Value Adj. Value Probability Lilliefors (D) NA > 0.1 Cramer-von Mises (W2) Watson (U2) Anderson-Darling (A2) Method: Maximum Likelihood - d.f. corrected (Exact Solution) Parameter Value Std. Error z-statistic Prob. MU SIGMA Log likelihood Mean dependent var No. of Coefficients 2 S.D. dependent var Page 80 of 95

9 The null hypothesis is rejected meaning that there is a problem with normality in the data set. The dataset is divided into 2 groups in order to see is the data there is normality in the data before Period 1: Empirical Distribution Test for _92_03 Hypothesis: Normal Date: 10/30/11 Time: 11:25 Sample (adjusted): 1 43 Included observations: 43 after adjustments Method Value Adj. Value Probability Lilliefors (D) NA > 0.1 Cramer-von Mises (W2) Watson (U2) Anderson-Darling (A2) Method: Maximum Likelihood - d.f. corrected (Exact Solution) Parameter Value Std. Error z-statistic Prob. MU SIGMA Log likelihood Mean dependent var No. of Coefficients 2 S.D. dependent var The data in the first period ( ) is tested, the null cannot be rejected, meaning that the data is normal distributed. Page 81 of 95

10 Period 2: Empirical Distribution Test for _03_10 Hypothesis: Normal Date: 10/30/11 Time: 11:26 Sample (adjusted): 1 34 Included observations: 34 after adjustments Method Value Adj. Value Probability Lilliefors (D) NA > 0.1 Cramer-von Mises (W2) Watson (U2) Anderson-Darling (A2) Method: Maximum Likelihood - d.f. corrected (Exact Solution) Parameter Value Std. Error z-statistic Prob. MU SIGMA Log likelihood Mean dependent var No. of Coefficients 2 S.D. dependent var In the second period the null hypothesis is rejected and normality is not present. Heteroskedasticity test: The Geometric Brownian motion process assumes constant volatility. To test for this a heteroskedasticity test is applied, again a battery of test is used. The hypothesis states: H 0 : Equal variances (Homoskedasticity) H 1 : Unequal variances (Heteroskedasticity) Page 82 of 95

11 Before the test is performed a test for equal means is performed, in order to compare the results. Test for Equality of Means Between Series Date: 10/30/11 Time: 11:33 Sample: 1 43 Included observations: 43 Method df Value Probability t -test Satterthwaite-Welch t -test* Anova F -test (1, 75) Welch F -test* (1, ) *Test allows for unequal cell variances The hypothesis on equal means is rejected and the means in the two groups cannot be assumed to be equal. Thereby no constant drift rate. Test for Equality of Variances Between Series Date: 10/30/11 Time: 11:34 Sample: 1 43 Included observations: 43 Method df Value Probability F-test (42, 33) Siegel-Tukey Bartlett E Levene (1, 75) Brown-Forsythe (1, 75) The null hypothesis cannot be rejected and the variances are assumed to equal and thereby constant volatility. Page 83 of 95

12 Appendix D: Static NPV valuation See Excel file for calculations (Sheet: static DCF) Figure D.1: Schedule for simultaneous strategy Phase I Construction (1 year) Sale Phase II Construction (1 year) Sale Phase III Construction (2 year) Sale 2010 Q4: 2011 Q4: 2012 Q4: 2014 Q4: Decision for all Phase I completed Phase II completed Phase III completed all phases are taken Phase II constructions starts Phase III constructions star Total project completed Decision The estimated net present values are slightly different than the once inside the text, because they are simulated. Figure D.2: DCF model of the total project All Phases T Residentials for sale (net m2) Total: 2010 Q Q Q Q Q4 Phase I (net area) Phase II (net area) Phase III (net area) Phase III (Office) Vacancy (Office) 94% Total net area for sale Sales prices (m2) exp increse in pricesale price (m2) Static expected sales price Phase I (m2 price) 4,02% Phase II (m2 price) 4,02% Phase III (m2 price) 4,02% Yield (Office) 5,3% Sales revenue (total) Total 2010 Q Q Q Q Q4 Phase I Phase II Phase III Expected gross revenue ( ) Sales Commissions (% of price/m2) 2,5% Tax 25% Total net cash inflow Development costs 2010 Q4 prices 2010 Q Q Q Q Q4 Expetced Hard construction cost (kr/ gross condo area m2): Expected Soft construction cost (kr/ gross condo area m2): Total construction cost (kr/ gross condo area m2) excl. Cost of site: Phase I (Gross Condo) Phase II (Gross Condo) Phase III (Gross Condo) Phase III (Office area) Total construction cash outflow Re 9,15% Rf 3,49% PV expected total net cash inflow from residential sale PV of leaseable office area Sum PV expected total net cash inflow PV total construction cost outflow Sum PV total construction cost outflow Net cash flow NPV (Total) Page 84 of 95

13 Figure D.3: DCF model for Phase I & II Phase I T Sales (net m2) Total: 2010 Q Q Q Q Q4 Phase I (net area) Total (net area) Sales prices (m2) exp increse in pricesale price (m2) Phase I (m2 price) 4,02% Sales revenue (total) Total 2010 Q Q Q Q Q4 Phase I Expected gross revenue ( ) Sales Commissions (% of price/m2) 2,5% Tax 25% Total net cash inflow Development costs 2010 Q4 prices 2010 Q Q Q Q Q4 Expetced Hard construction cost (kr/ gross condo area m2): Expected Soft construction cost (kr/ gross condo area m2): Total construction cost (kr/ gross condo area m2) excl. Cost of site: Phase I construction cost Total construction cash outflow Re 9% Rf 3,49% PV expected total net cash inflow Sum PV expected total net cash inflow PV total construction cost outflow Sum PV total construction cost outflow Net cash flow NPV (Phase I) Phase II T Sales (net m2) Total: 2010 Q Q Q Q Q4 Phase II (net area) Total (net area) Sales prices (m2) exp increse in pricesale price (m2) Phase II (m2 price) 4,02% Sales revenue (total) Total 2010 Q Q Q Q Q4 Phase II Expected gross revenue ( ) Sales Commissions (% of price/m2) 2,5% Tax 25% Total net cash inflow Development costs 2010 Q4 prices 2010 Q Q Q Q Q4 Expetced Hard construction cost (kr/ gross condo area m2): Expected Soft construction cost (kr/ gross condo area m2): Total construction cost (kr/ gross condo area m2) excl. Cost of site: Phase II construction cost Total construction cash outflow Re 9% Rf 3,49% PV expected total net cash inflow Sum PV expected total net cash inflow PV total construction cost outflow Sum PV total construction cost outflow Net cash flow NPV (Phase II) Page 85 of 95

14 Figure D.4: DCF model for Phase III Phase III T Sales (net m2) Total: 2010 Q Q Q Q Q4 Phase III (net area) Leaseable office area Vacany (Office) 94% Total net area for sale Sales prices (m2) exp increse in pricesale price (m2) Phase III (m2 price) 4,0% Sales revenue (total) Total 2010 Q Q Q Q Q4 Phase III Expected gross revenue from sales ( ) Sales Commissions (% of price/m2) 2,5% Tax 25% Total net cash inflow Development costs 2010 Q4 prices 2010 Q Q Q Q Q4 Expetced Hard construction cost (kr/ gross condo area m2): Expected Soft construction cost (kr/ gross condo area m2): Total construction cost (kr/ gross condo area m2) excl. Cost of site: Site acquisition Phase III construction cost PhaseIII construction cost (Office area) Total construction cash outflow Re 9% Rf 3,49% PV expected total net cash inflow from residential sale PV leaseable office area Sum PV expected total net cash inflow PV total construction cost outflow Sum PV total construction cost outflow Net cash flow NPV (Phase III) NPV phase I NPV phase II NPV phase III Total Page 86 of 95

15 Appendix E: Simulation output Figure E.1: Output from random simulation of sale prices (dashed line = static drift rate) , ,00 Sale price/m , , , , ,00 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Time Figure E.2: Frequency chart and statistics for total project Page 87 of 95

16 Figure E.3: Frequency chart and statistics for Phase I Figure E.4: Frequency chart and statistics for Phase II Figure E.4: Frequency chart and statistics for Phase III Page 88 of 95

17 Figure E.5: Frequency chart and statistics for value of underlying asset Page 89 of 95

18 Appendix F:Binomial lattice model Inputs to model Annualized Standard deviation: 9,67% : 4,83% Annual riskless rate: 3,49% Risk free (quaterly): 0,87% Dividend yield: 6,00% Dividend yield Quaterly 1,50% Growth in constuction cost: 0,00% Growth in constuction cost (Quaterly): 0,00% Time to expiration (years): 5 n: 20 Binomial inputs Up movement in binomial: 1, Down movement in binomial : 0, p: 0, (in kr. 1000) V(0): K1+K2+K3: Underlying asset Years Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Period (j) Down moves (i)

19 Phase III (in kr. 1000) V K Years Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Period (j) Down moves (i) Phase II (in kr. 1000) V K Years Q Q Q Q Q Q Q Q Q Q Q Q Q Period (j) Down moves (i) Phase I (in kr. 1000) V K Years Q Q Q Q Q Period (j) Down moves (i)

20 Appendix G: Closed-form model MV of project: V total Construction cost PH3: K3 Construction cost PH2: K2 Construction cost PH1: K1 Time to Maturity of PH3: 5,00 T3 Time to Maturity of PH2: 3,00 T2 Time to Maturity of PH1: 1,00 T1 Volatility in sale price/m2: 9,67% Vol Risk free rate: 3,49% rf Continuous Dividend Yield (q): 6,00% q Outputs 3 fold compound: NPV mean: Value of flexibility: Interim Calculations Notation from n fold compound b1 1,8795 b2 1,6743 b3 1, a1 1, a2 1, a3 1, rho(t1,t2) 0,77 rho(t1,t3) 0, rho(t2,t3) 0, CBND1 0, CBND2 0, V*t V*t Solve VT2* ,31 to equel K2 Solve VT1* ,69 to equel K1 b1(vt2*) 0,92118 a1(vt2*) 0,78445 b1(vt3*) 1,41678 a1(vt3*) 1,22342 b2(vt3*) 1,08245 a2(vt3*) 0,94572

21 Appendix H: Probability of exercise Probability of exercise 100,00% 80,00% 60,00% 40,00% 20,00% 0,00% ,67% 11,67% 1,67% Volatility Sale price/m 2 The figure illustrates the probability of the exercise of the options. Threoy predicts that the probability of exercise decreaese when uncertainty increases. This is also the case I this model, and the it decreases faster when the underlying asset value is small. 120,0% 100,0% Exercise probability 80,0% 60,0% 40,0% 20,0% 0,0% 1% 15% 30% 45% 60% 75% 90% 105% 120% 135% 150% Volatility Trivariate cumulative distribution function Page 93 of 95

22 Appendix I: Development of sequential compound option By using the notation from Geske (1979) 34, the Lee & Paxson (2001) model is expressed in the following equations. The formula for the American exchange option is given as follow:,,,,,,,, (I.1) is the value of a perpetual American exchange option, and the value is given by: (I.2) is the optimal exercise price where the American exchange option should be exercised. The optimal exercise price is given by 35 : (I.3) S 1 is the European exchange option price at t 2 using Margrabe (1978) and S 2 is a twice exercisable exchange option. S 2 is valued using the formula from Geske & Johnson (1984) (Eq I.4): ; ; ; ; Knowing S 1, S 2 and the value of an American exchange option can be estimated. The values covers the flexibility in the investment opportunities, but the sequential element is not modeled. This is calculated using the formula of a European sequential exchange option developed by Carr (1988). To do this some simplifying assumptions are assumed, the process of K 1 is identical to the stochastic process of the K 2 development cost 36. The formula from Carr (1988) is given as: 34 S A = value of American exchange option σ K = Volatility of investment cost 35,, 36 W = value of European sequential exchange option W A = value of American sequential exchange option Page 94 of 95

23 ,,,,,,,,, (I.5) ; ; ; ; The above formula gives the value of a European sequential option, with fixed exercise times. Page 95 of 95