CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
3.2 The Market Portfolio
Due to the issues outlined in the previous section, it is natural to proceed in a similar manner as in Breeden and Litzenberger (1978) and Banz and Miller (1978), i.e. to use aggregate wealth as a proxy for aggregate consumption. As previously mentioned, Breeden and Litzenberger (1978) assume there exists a one-to-one correspondence between aggregate real consumption and ag-gregate real wealth. A nations real wealth is ultimately all the assets owned by its individuals and families, and includes more than just the financial assets (Hakansson (1976)). However, the financial assets are tradeable and easily observed. To most people the financial market is the market for stocks and bonds, but more broadly, it includes loans, insurances, option contracts, saving accounts and more. The value of written claims, such as stocks, are easily observed from their stock prices. This makes the stock market an attractive potential proxy for real wealth, as stocks are very well documented and the prices of these assets represent an equilibrium belief of their true underlying value. Thus, if we use a stock market index as the underlying asset, we measure the total value of most stocks within an economy and use this as a proxy for aggregate real wealth. Even though the stock market does not include all wealth, the wide diversification in today’s stock market make it exposed to the economy as a whole, and thus make it an acceptable proxy for the total wealth. In fact, stock prices are also strongly positively linked to consumption, as shown by Campbell and Cochrane (1995). However, it should be emphasized that it is an imperfect proxy, as it does not include all assets, which is the essence of Roll’s critique. Furthermore, even if the CAPM can be empirically verified in many different stock markets, this does not mean that the stock market is a good proxy for the true market portfolio. After all, reliability does not imply validity, and different stock markets might just be equally bad proxies for the true market portfolio.
An examination of the sharp declines of the stock market during recessions, gives an indication as to why the stock market is not a perfect proxy for real wealth. The demand for safer assets tends to rise significantly in recessions, a phenomenon known as ’flight to quality’, which increases the value of less risky assets. Such ’safe haven’ assets include government bonds, gold and cash. Hence, the true decline in aggregate wealth is probably not as sharp in recessions as the stock market indicates. It is reasonable to assume that a proportion of the decline in the stock market is due to shifted asset preferences that in themselves do not decrease aggregate wealth, but merely redistribute it. Nevertheless, the stock market is widely considered the best proxy we have. One could of course propose to create a composite portfolio with other assets that are not traded on the public stock markets. However, this would drastically increase the complexity of any modeling task. Hence, we emphasize that in a real world setting, the choice of underlying boils down to finding an asset that offers reasonable accuracy with a low degree of complexity. Therefore, our choice is to use a stock market index as done by Banz and Miller (1978), and argued for by Breeden and Litzenberger (1978).
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
3.2.1 Choosing the Proxy for the Market Portfolio
Even though the model we will propose is general, the exact choice of stock portfolio should reflect the systematic risk that the investor using it is exposed towards. In an ideal global capi-tal market, i.e. without frictions and with identical risk preferences across geographical regions, one could argue that a world stock market index would always be a reasonable choice, since investors would be exposed towards, and have the same notion of systematic risk. However, the real world is more complex. In the real world, frictions such as restrictions on capital flow across borders and tax discrepancies exist, just to mention a few imperfections. Since frictions are not equal for all market participants and may be specific for certain regions, the amount of risk market participants have to bear may differ. Not only may it differ for individuals, but it may also differ for larger groups. Said differently, since investors do not have the same investment-and diversification opportunities across regions, the notion of, investment-and exposure towards systematic (i.e. non-diversifiable) risk can be different across regions. Moreover, so could risk preferences be. Hence, for a company with an international or multinational shareholder base, there is probably no universally correct choice of the underlying asset. Practitioners should therefore be concerned with making the least worst choice, that is to choose an underlying asset that reflects the systematic risk the shareholders of the company are exposed towards as best as possible. However, as our proposed model requires the user to estimate the variation in cash flows between states, practitioners should also consider how difficult state-contingent cash flow estimations are for any given underlying asset. The reason is that there may be examples of cases where there is a systematic connection between cash flows and the underlying asset in the model, but that it is hard to quantify. In such cases, it is reasonable to compromise on the theoretical risk-based justification for choosing the underlying asset, and perhaps choose an underlying asset that reflects investor preferences more poorly, but ensures possibility of actual cash flow estimation.
To give a practical and intuitive example of the above, we begin by relating our proposed model to the case study with Orkla, which we present in chapter 7. Let us consider Orkla and its shareholder base, as shown in table 3.1
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
Country Shares % of Shares Shareholders % of Shareholders
Norway 472,986,956 46.42 46.42 89.42
United States 219,917,428 21.58 21.58 1.1
United Kingdom 107,730,926 10.57 10.57 0.81
Luxembourg 61,903,045 6.08 6.08 0.18
Netherlands 19,163,306 1.88 1.88 0.26
Japan 18,282,258 1.79 1.79 0.3
France 17,210,087 1.69 1.69 0.14
Sweden 14,209,212 1.39 1.39 2.49
Switzerland 11,331,482 1.11 1.11 0.21
Others 76,196,270 7.48 7.48 5.08
Total 1,018,930,970 100 38.06 100
Table 3.1: Orkla shareholders. Updated 26/02/2018. Source: Orkla
As can be seen in the table, Orkla’s investors are from multiple regions, with most investors being from Norway, the US and the UK, and the rest from Europe or other regions. Since there is a clear majority of European investors, it seems appropriate to choose some sort of European stock market index as the underlying asset, to represent the systematic risk investors in Orkla are exposed towards. Furthermore, choosing a European stock market index is advantageous to choosing a Norwegian one (even though Norwegians are the largest group of stockholders), since this increases the likelihood of choosing an underlying asset that also captures systematic risk that the relatively large investor base from the US are exposed towards. In addition, the Nordic and Baltic regions and selected countries in central Europe are Orkla’s main markets (see Orkla(a)). Combined with the risk-based reasoning, we argue that a value-weighted Eu-ropean stock index is the most sensible choice of the underlying asset. Hence, we have chosen the underlying in a way that balances risk exposure and ease of cash flow estimation.
The final choice is the value-weighted European stock index, STOXX Europe 600. The STOXX Europe 600 index represents large, mid and small capitalization companies across 17 countries of the European region: Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Ireland, Italy, Luxembourg, the Netherlands, Norway, Portugal, Spain, Swe-den, Switzerland and the United Kingdom. It is derived from the STOXX Europe Total Market Index and is a subset of the STOXX Global 1800 index (see STOXX(a)). We use the net return index from Thomson Reuters Datastream. The net return represents the theoretical aggregate growth in value of the constituents of the index. The index constituents are deemed to return an aggregate daily dividend which is included as an incremental amount to the daily change in the price index. More specifically, dividends are reinvested after the deduction of taxes imposed on a non-residential institutional investor. The final dataset on the net return index ranges from the 31st of December 1986 to the 28th of February 2018, with a total of 8005 return observations. Non-trading days are excluded. The exact calculation of the net return index supplied by Thomson Reuters is
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
N Rt =N Rt−1× P It P It−1
×
1 + DY 100×n
(3.1) WhereN Rtis the return index for dayt,P Itis the price index on dayt,DY is the net dividend yield of the price index and n is the number of days in a financial year (normally 260).