Implementation of Real Options
CHAPTER 9. CONCLUSION
Orkla. Through the case study, we discovered that there was a clear issue of justifying the complexity of our model, especially for the valuation purposes of a company like Orkla, which has a relatively low and symmetric cash flow exposure towards the market. However, we found that the case participants could see the value of our model when used in specific areas, such as when modeling real options, pricing assets with cash flows that vary non-symmetrically with the state of the market, and when combining our proposed model with other models, such as the CAPM. Hence, we believe that the proposed model can be an alternative to conventional models in applied capital budgeting.
Furthermore, the case study with Orkla’s M&A department revealed that even though our proposed model builds on relatively complicated and less known subjects, practitioners seemed to understand the economic intuition behind the model. In our opinion, this is probably a result of a strong familiarity with the concept of systematic risk, established through a business school education. Concerning the concept of systematic risk, the case study also revealed that some of the assumptions in our proposed model are probably quite unrealistic. More specifically, it revealed that the assumption of absence of systematic risk in cash flows within states, and the assumption of non-increasing systematic risk with time, are quite inconsistent with reality and will lead to the overvaluation of many assets/projects. To counter these issues, we introduced two potential extensions to our model, namely the certainty equivalent approach to valuation and a non-recombining trinomial tree. The former deals with the absence of systematic risk within states and non-increasing systematic risk with time, while the latter only deals with non-increasing systematic risk with time. However, while the non-recombining trinomial tree does not deal with the absence of systematic risk within states, it allows for the modelling of systematically dependent cash flows, which through our case study appeared to be highly natural. In addition, the trinomial tree is implementable in practice, while the certainty equivalent approach to valuation, for now, only is a theoretical solution. A drawback is that the trinomial tree probably requires the use of a computer algorithm for the valuation of projects of more than three periods. However, we believe that in our modern world, such a requirement is probably a smaller issue. Hence, we believe that our proposed model, although perhaps with some extensions and when used for specific purposes, increases the applicability of a state-contingent asset pricing model à la Banz & Miller. We also believe that this can be done with relatively little compromise of theoretical consistency and empirical validity. However, whether or not practitioners will use our proposed model remains to be seen.
List of Figures
2.1 Illustration of a state price . . . 27
2.2 Single-period path system . . . 28
2.3 Multi-period path system . . . 28
3.1 CAPM’s sensitivity to the risk-free rate in perpetuity . . . 35
4.1 European stock market returns vs. government bond yields . . . 40
4.2 STOXX cumulative real net return and rolling 1-year volatility . . . 41
4.3 Hexagonal bin plot . . . 42
4.4 Liquidity spirals . . . 47
4.5 Economic indicator and rolling 1-year volatility for STOXX . . . 49
5.1 A 4-state normal distribution . . . 54
5.2 Non-overlapping return categorization . . . 57
5.3 Plot of rolling 1-year returns and corresponding states . . . 58
6.1 Distributional mechanics behind changes in the risk-free rate . . . 72
6.2 State price under risk-neutral measure and changed volatility . . . 73
6.3 The sensitivity of the Low state price towards volatility and the risk-free rate . . 74
6.4 Selection of Low state prices . . . 74
6.5 The sensitivity of the Mid state price towards volatility and the risk-free rate . . 75
6.6 Selection of Mid state prices . . . 75
6.7 The sensitivity of the High state price towards volatility and the risk-free rate . 76 6.8 Selection of High state prices . . . 76
6.9 Sampling from a mixture distribution . . . 78
6.10 A comparison between a sampled mixture distribution and a normal distribution 82 7.1 Orkla’s 6-month trailing market beta towards STOXX . . . 85
8.1 Real option expansion after one year . . . 95
8.2 Real option expansion after two years . . . 96
8.3 Real option no expansion . . . 96
8.5 A Present value approach to real options . . . 97
8.6 Present value comparison (DCF and state-contingent pricing model) . . . 101
8.7 Regime shifting geometric Brownian motion 10,000 simulations . . . 104
8.8 Valuation in a non-recombining trinomial tree . . . 106
List of Tables
3.1 Orkla’s shareholders . . . 33
4.1 Average of STOXX 600 rolling 1-year volatility for respective economic states . . 49
5.1 Estimates of unconditional parameters . . . 53
5.2 Probability bounds . . . 56
5.3 Return bounds . . . 56
5.4 Number of observations . . . 57
5.5 State-contingent annualized volatility and returns . . . 58
5.6 Levene’s test for equality of variances . . . 59
5.7 Strike prices (absolute states) . . . 60
5.8 1-Year state price matrix . . . 60
5.9 2-Year state price matrix . . . 60
5.10 5-Year state price matrix . . . 61
5.11 Perpetuity state price matrix . . . 63
5.12 Valuation example from Banz and Miller (1978) . . . 64
6.1 Summary statistics for the S&P 500 index and the STOXX 600 index . . . 67
6.2 Unconditional parameter estimates for the S&P 500 and the STOXX 600 . . . . 68
6.3 S&P and STOXX return intervals . . . 68
6.4 S&P and STOXX one year state price matrices . . . 68
6.5 State-contingent risk-free rates from Banz and Miller (1978) . . . 69
6.6 Return bounds . . . 70
6.7 Summary statistics S&P 500 . . . 70
6.8 State price matrix comparison with Banz & Miller . . . 71
6.9 Central moments of the sampled mixture distribution and the normal distribution 83 8.1 Comparison of present values calculated using a non-recombining trinomial tree 107 8.2 Discount factors for extreme scenarios . . . 108
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Appendices
.1 Simulation Algorithm 1
Algorithm 1 1-Year Simulation of GBM
Input: drift parameter: µ, volatility parameter: σ, initial index value: S0, number of trading days per year: ntd
1: procedure Calculate Time Step(ntd)
2: ∆t= 1/ntd
3: procedure Simulate Brownian Motion(µ, σ, S0, ntd, ∆t)
4: simulations ←−[...] . Empty column vector, number of elements: ntd
5: simulations[1] ←− S0 .Populate the first element as S0
6: for i=2 tontd do
7: W←− generate X ∼N(0,1)
8: simulations[i] ←−simulations[i−1]∗exp
(µ−0.5σ2)∆t+σ√
∆tW return simulations
.2 Simulation Algorithm 2
Algorithm 2 Simulation of Regime Switching Stochastic Process
Input: Number of Paths: NP, number of years: NY, number of trading days per year: ntd, vector of state contingent drift parameters: µ : µT = [µLow, µM id, µHigh], vector of state contingent volatility σ : σT = [σLow, σM id, σHigh], initial index value: S0, vector of upper state boundaries (in returns): R:RT = [RLow, RM id]
1: for i=j to NP do
2: procedure First Year Simulation(µM id, σM id, S0, ntd, ∆t) . Start in Mid
3: simvec ←−run Algorithm 1(µM id, σM id, S0, ntd, ∆t)
4: hpr ←−ln
simvec[ntd]/simvec[1]
5: procedure Simulation of Remaining Years(µ, σ, S0, ntd)
6: for i=1 to NY−1do
7: if hpr ≤R[1] then . Determine the new state
8: state ←− 1
9: else if (hpr> R[1]) and (hpr ≤R[2])then
10: state ←− 2
11: else if hpr > R[2] then
12: state ←− 3
13: procedure Simulation of Next Year(µ[state], σ[state], simvec[ntd×j], ntd)
14: nextsim ←− run Algorithm 1(µ[state], σ[state], simvec[ntd×j], ntd)
15: nextsim ←− nextsim\nextsim[1] . Drop first element
16: hpr ←− ln
nextsim[ntd−1]/simvec[ntd∗j]
. Update holding period return
17: procedure Merging of Simulations(simvec,nextsim)
18: simvec ←− [simvec,nextsim] . Concatenate vectors along column axis
19: procedure Storage of Finished Simulations(simvec)
20: if j=1 then
21: allsims ←−simvec
22: else
23: allsims ←−[allsims,simvec] . Concatenate vectors/matrix along row axis return allsims
.3 State Price Matrices (STOXX)
Our State Price Matrices (STOXX)
Row sum
Implied Annual Real Riskless
v1 0 1 2
0 0.479757 0.241803 0.284056042 1.005616 -0.0056 1 0.427568 0.367918 0.210129758 1.005616 -0.0056 2 0.422755 0.380221 0.202639444 1.005616 -0.0056
v2 0 1 2
0 0.45364 0.312974 0.244648735 1.011263 -0.0056 1 0.451272 0.318647 0.241344369 1.011263 -0.0056 2 0.451057 0.319161 0.241044663 1.011263 -0.0056
v3 0 1 2
0 0.454881 0.317861 0.244199846 1.016942 -0.0056 1 0.454773 0.318119 0.244049553 1.016942 -0.0056 2 0.454764 0.318142 0.244035894 1.016942 -0.0056
v4 0 1 2
0 0.457376 0.319788 0.245488288 1.022653 -0.0056 1 0.457371 0.3198 0.245481455 1.022653 -0.0056 2 0.457371 0.319801 0.245480834 1.022653 -0.0056
v5 0 1 2
0 0.459942 0.321591 0.246863108 1.028396 -0.0056 1 0.459942 0.321591 0.246862798 1.028396 -0.0056 2 0.459942 0.321591 0.24686277 1.028396 -0.0056
v6 0 1 2
0 0.462525 0.323397 0.248249248 1.034171 -0.0056 1 0.462525 0.323397 0.248249235 1.034171 -0.0056 2 0.462525 0.323397 0.248249233 1.034171 -0.0056
v7 0 1 2
0 0.465122 0.325213 0.249643335 1.039978 -0.0056 1 0.465122 0.325213 0.249643335 1.039978 -0.0056 2 0.465122 0.325213 0.249643335 1.039978 -0.0056
v8 0 1 2
0 0.467734 0.327039 0.251045259 1.045819 -0.0056 1 0.467734 0.327039 0.25104526 1.045819 -0.0056 2 0.467734 0.327039 0.25104526 1.045819 -0.0056
v9 0 1 2
0 0.470361 0.328876 0.252455055 1.051692 -0.0056 1 0.470361 0.328876 0.252455056 1.051692 -0.0056 2 0.470361 0.328876 0.252455056 1.051692 -0.0056
v10 0 1 2
0 0.473002 0.330723 0.253872769 1.057598 -0.0056 1 0.473002 0.330723 0.25387277 1.057598 -0.0056 2 0.473002 0.330723 0.25387277 1.057598 -0.0056
vperp 0 1 2
0 40.3551 32.3719 26.7738 99.5008 0.0100 1 40.2835 32.5107 26.7066 99.5008 0.0100 2 40.2768 32.5245 26.6995 99.5008 0.0100
.4 Banz & Miller’s State Price Matrices (S&P)
Banz & Miller State Price Matrices (S&P) Our State Price Matrices (S&P)
Row sum
Implied Annual Real Riskless rate
1 2 v1 0 1 2 v1
0.226355 0.271049 0 0.5251 0.2935 0.1735 0.9921 0.0079 0
0.389234 0.164177 1 0.5398 0.2912 0.1672 0.9982 0.0018 1
0.369153 0.177747 2 0.5544 0.2888 0.1612 1.0044 -0.0044 2
1 2 v2 0 1 2 v2
0.301025 0.220487 0 0.5304 0.2897 0.1681 0.9882 0.0059 0
0.312293 0.213051 1 0.5333 0.2915 0.1693 0.9941 0.0030 1
0.310961 0.213931 2 0.5364 0.2934 0.1705 1.0003 -0.0001 2
1 2 v3 0 1 2 v3
0.305067 0.216146 0 0.5281 0.2886 0.1676 0.9843 0.0053 0
0.305841 0.215636 1 0.5313 0.2903 0.1686 0.9902 0.0033 1
0.305749 0.215696 2 0.5345 0.2921 0.1696 0.9962 0.0013 2
1 2 v4 0 1 2 v4
0.304209 0.215046 0 0.526 0.2874 0.1669 0.9803 0.0050 0
0.304262 0.215011 1 0.5291 0.2892 0.1679 0.9862 0.0035 1
0.304256 0.215015 2 0.5324 0.2909 0.1689 0.9922 0.0020 2
1 2 v5 0 1 2 v5
0.303019 0.214171 0 0.5239 0.2863 0.1662 0.9764 0.0048 0
0.303023 0.214169 1 0.527 0.288 0.1672 0.9822 0.0036 1
0.303023 0.214169 2 0.5302 0.2897 0.1682 0.9881 0.0024 2
1 2 v6 0 1 2 v6
0.301811 0.213315 0 0.5217 0.2851 0.1655 0.9723 0.0047 0
0.301812 0.213315 1 0.5249 0.2868 0.1665 0.9782 0.0037 1
0.301812 0.213315 2 0.5281 0.2886 0.1676 0.9843 0.0026 2
1 2 v7 0 1 2 v7
0.300607 0.212463 0 0.5197 0.284 0.1649 0.9686 0.0046 0
0.300607 0.212463 1 0.5228 0.2857 0.1659 0.9744 0.0037 1
0.300607 0.212463 2 0.526 0.2874 0.1669 0.9803 0.0028 2
1 2 v8 0 1 2 v8
0.299407 0.211615 0 0.5176 0.2828 0.1642 0.9646 0.0045 0
0.299407 0.211615 1 0.5207 0.2845 0.1652 0.9704 0.0038 1
0.299407 0.211615 2 0.5239 0.2863 0.1662 0.9764 0.0030 2
1 2 v9 0 1 2 v9
0.298211 0.210771 0 0.5155 0.2817 0.1636 0.9608 0.0044 0
0.298211 0.210771 1 0.5186 0.2834 0.1645 0.9665 0.0038 1
0.298211 0.210771 2 0.5218 0.2851 0.1655 0.9724 0.0031 2
1 2 v10 0 1 2 v10
0.297021 0.209929 0 0.5134 0.2806 0.1629 0.9569 0.0044 0
0.297021 0.209929 1 0.5165 0.2823 0.1639 0.9627 0.0038 1
0.297021 0.209929 2 0.5197 0.284 0.1649 0.9686 0.0032 2
1 2 vperp 0 1 2 vperp
77.04337 54.57223 0 132.5 72.41 42.05 246.96 0.0040 0
77.21835 54.45738 1 133.31 72.85 42.3 248.46 0.0040 1
77.19684 54.47189 2 134.14 73.29 42.55 249.98 0.0040 2
.5 Our State Price Matrices (S&P)
Our State Price Matrices (S&P)
Row sum
Implied Annual Real Riskless
v1 0 1 2
0 0.4986 0.2264 0.2710 0.996008 0.0040 1 0.4426 0.3892 0.1642 0.996008 0.0040 2 0.4491 0.3692 0.1777 0.996008 0.0040
v2 0 1 2
0 0.47052 0.301024942 0.22048671 0.992032 0.0040 1 0.466688 0.312293484 0.21305082 0.992032 0.0040 2 0.467141 0.310960545 0.21393076 0.992032 0.0040
v3 0 1 2
0 0.466858 0.305066962 0.21614639 0.988072 0.0040 1 0.466595 0.305840546 0.21563588 0.988072 0.0040 2 0.466626 0.305749093 0.21569624 0.988072 0.0040
v4 0 1 2
0 0.464872 0.30420912 0.21504596 0.984127 0.0040 1 0.464854 0.304262227 0.21501092 0.984127 0.0040 2 0.464856 0.30425595 0.21501506 0.984127 0.0040
v5 0 1 2
0 0.46301 0.30302 0.21417 0.980199 0.0040 1 0.46301 0.30302 0.21417 0.980199 0.0040 2 0.46301 0.30302 0.21417 0.980199 0.0040
v6 0 1 2
0 0.461159 0.301811457 0.21331508 0.976286 0.0040 1 0.461159 0.301811715 0.21331492 0.976286 0.0040 2 0.461159 0.301811685 0.21331494 0.976286 0.0040
v7 0 1 2
0 0.459318 0.300606736 0.21246345 0.972388 0.0040 1 0.459318 0.30060676 0.21246344 0.972388 0.0040 2 0.459318 0.300606758 0.21246344 0.972388 0.0040
v8 0 1 2
0 0.457484 0.299406715 0.21161529 0.968506 0.0040 1 0.457484 0.299406724 0.21161529 0.968507 0.0040 2 0.457484 0.299406724 0.21161529 0.968507 0.0040
v9 0 1 2
0 0.455658 0.298211478 0.21077051 0.96464 0.0040 1 0.455658 0.298211485 0.21077052 0.96464 0.0040 2 0.455658 0.298211485 0.21077052 0.96464 0.0040
v10 0 1 2
0 0.453839 0.297021011 0.20992911 0.960789 0.0040 1 0.453839 0.297021018 0.20992912 0.960789 0.0040 2 0.453839 0.297021018 0.20992912 0.960789 0.0040
vperp 0 1 2
0 117.884 77.04336974 54.5722318 249.4996 0.0040 1 117.8239 77.2183501 54.4573773 249.4996 0.0040 2 117.8309 77.19683818 54.4718923 249.4996 0.0040
.6 Sensitivity Analysis State Prices
LOW -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.08 0.377 0.327 0.280 0.238 0.199 0.165 0.135 0.109 0.087 0.068 0.053 0.040 0.1 0.409 0.367 0.326 0.289 0.253 0.220 0.190 0.163 0.139 0.117 0.098 0.081 0.12 0.432 0.395 0.360 0.326 0.294 0.264 0.235 0.209 0.185 0.162 0.142 0.123 0.14 0.450 0.417 0.386 0.355 0.326 0.298 0.272 0.247 0.223 0.201 0.180 0.161 0.16 0.464 0.435 0.406 0.379 0.352 0.326 0.301 0.278 0.255 0.234 0.214 0.195 0.18 0.476 0.449 0.423 0.398 0.373 0.349 0.326 0.304 0.282 0.262 0.242 0.224 0.2 0.487 0.462 0.438 0.414 0.391 0.369 0.347 0.326 0.306 0.286 0.268 0.250 0.22 0.496 0.473 0.450 0.428 0.407 0.386 0.365 0.345 0.326 0.308 0.290 0.272 0.24 0.505 0.483 0.462 0.441 0.421 0.401 0.381 0.362 0.344 0.326 0.309 0.292 0.26 0.512 0.492 0.472 0.452 0.433 0.414 0.396 0.378 0.360 0.343 0.326 0.310 0.28 0.520 0.500 0.481 0.462 0.444 0.426 0.408 0.391 0.374 0.358 0.342 0.326 0.3 0.526 0.508 0.490 0.472 0.454 0.437 0.420 0.404 0.387 0.372 0.356 0.341 MID -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.08 0.5386 0.5568 0.5686 0.5734 0.5711 0.5618 0.5458 0.5237 0.4963 0.4644 0.4292 0.3917 0.1 0.4581 0.4681 0.4742 0.4763 0.4743 0.4682 0.4582 0.4446 0.4276 0.4078 0.3855 0.3613 0.12 0.3955 0.4013 0.4045 0.4053 0.4035 0.3991 0.3924 0.3834 0.3722 0.3591 0.3443 0.3280 0.14 0.3466 0.3499 0.3516 0.3517 0.3501 0.3468 0.3420 0.3356 0.3278 0.3187 0.3083 0.2969 0.16 0.3076 0.3096 0.3105 0.3101 0.3087 0.3061 0.3025 0.2977 0.2920 0.2853 0.2778 0.2694 0.18 0.2761 0.2773 0.2776 0.2771 0.2758 0.2737 0.2708 0.2671 0.2627 0.2577 0.2519 0.2456 0.2 0.2502 0.2508 0.2508 0.2502 0.2490 0.2472 0.2449 0.2419 0.2385 0.2345 0.2300 0.2251 0.22 0.2285 0.2288 0.2286 0.2279 0.2268 0.2253 0.2233 0.2209 0.2181 0.2149 0.2113 0.2073 0.24 0.2101 0.2102 0.2099 0.2092 0.2082 0.2068 0.2051 0.2031 0.2008 0.1981 0.1952 0.1919 0.26 0.1944 0.1943 0.1939 0.1932 0.1923 0.1911 0.1896 0.1879 0.1859 0.1836 0.1812 0.1785 0.28 0.1807 0.1805 0.1801 0.1794 0.1786 0.1775 0.1762 0.1747 0.1729 0.1710 0.1689 0.1666 0.3 0.1688 0.1685 0.1681 0.1674 0.1666 0.1656 0.1645 0.1631 0.1616 0.1599 0.1581 0.1561 HIGH -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.08 0.0945 0.1163 0.1411 0.1692 0.2003 0.2343 0.2708 0.3094 0.3495 0.3906 0.4319 0.4727 0.1 0.1430 0.1652 0.1894 0.2154 0.2430 0.2722 0.3025 0.3339 0.3658 0.3981 0.4303 0.4621 0.12 0.1825 0.2035 0.2256 0.2488 0.2729 0.2978 0.3234 0.3494 0.3756 0.4019 0.4281 0.4539 0.14 0.2137 0.2329 0.2527 0.2732 0.2943 0.3158 0.3376 0.3595 0.3816 0.4036 0.4255 0.4470 0.16 0.2383 0.2556 0.2734 0.2915 0.3099 0.3286 0.3474 0.3663 0.3852 0.4040 0.4226 0.4409 0.18 0.2577 0.2733 0.2892 0.3053 0.3216 0.3379 0.3544 0.3708 0.3872 0.4035 0.4195 0.4354 0.2 0.2731 0.2872 0.3014 0.3158 0.3303 0.3448 0.3593 0.3737 0.3881 0.4023 0.4164 0.4302 0.22 0.2855 0.2982 0.3110 0.3239 0.3368 0.3497 0.3626 0.3754 0.3881 0.4007 0.4131 0.4253 0.24 0.2954 0.3069 0.3185 0.3301 0.3417 0.3533 0.3648 0.3762 0.3876 0.3988 0.4098 0.4207 0.26 0.3033 0.3138 0.3243 0.3348 0.3453 0.3557 0.3661 0.3764 0.3865 0.3966 0.4065 0.4162 0.28 0.3097 0.3193 0.3288 0.3384 0.3479 0.3573 0.3667 0.3760 0.3851 0.3942 0.4031 0.4119 0.3 0.3148 0.3236 0.3323 0.3410 0.3496 0.3582 0.3667 0.3751 0.3834 0.3916 0.3997 0.4076
.7 1-Year State Price Matrix for Perpetual Matrix
v1 0 1 2
0 0.4428 0.2419 0.3053
1 0.3756 0.3723 0.2422
2 0.3692 0.3853 0.2355