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## Mathematical Theory

This section presents the theory from four prominent papers on the subject of adjusted return series. There is a subdivision between these papers where the works of Dimson (1979) and Scholes and Williams (1977) sort as Nonsynchronous trading while the works of Getmansky et al. (2004) and Okunev and White (2003) sort as Smoothed returns. The first subsection will state the basic market model which is common to all of the methods discussed subsequently. After that an exposé of the diﬀerent methods is presented, each with its own assumptions and suggested formulas based on the common market model.

**Market model**

There are diﬀerent ways of modelling a financial market and the returns of assets traded on the market. One common way to model the return Rt of an asset is to assume a simple linear model Rt = + Mt + t: (1).

Here, Rt is the true return, is the intercept, represents the correlation with the market index Mt and t is an error term. The return Rt is referred to as true since it represents the actual return of an asset during a specified time span, as opposed to the observed return which for various reasons might diﬀer from the true return. The details will be laid out in the upcoming section. The market index Mt represents the return of an index relevant to the market in question. This model composes the core of the theory in each of the methods discussed below. Assumptions regarding distributions and correlations between the constituents of (1) diﬀer slightly between the methods and will therefore be presented in the corresponding subsections.

The real issue is not associated with this particular choice of model, which is prevailing through much of the literature and is not very controversial. In fact, Stapleton and Subrah-manyam (1983) showed that a linear one-factor market model such as (1) is suﬃcient for deriving the widely accepted relationship between returns and market beta in the Capital Asset Pricing Model introduced in its original form by Sharpe (1964). Instead, the problem is that in reality one does not observe the true returns Rt. Rather, what we measure are the observed returns Rto. It is with respect to the observed returns diﬀerent authors have devised various assumptions and models. In particular, it is an open question concerning how to best link the observed returns to the true returns.

The following sections include methods which suggest diﬀerent solutions to the problem of adjusting the risk and performance measures with respect to illiquidity by assuming certain re-lations between observed and true returns. These methods have been divided into the subgroups of nonsynchronous trading and smoothed returns.

**Non synchronous trading**

As mentioned briefly in Section 1, nonsynchronous trading is related to the case where assets are valued at diﬀerent times but treated as if they were all valued at the same time. Based on this there will arise a deviance between true and observed returns. Adopting the notation of Section 2 in Scholes and Williams (1977) the setting is the following. Suppose that there are n = 1; :::; N securities and that true returns are to be calculated over time intervals [t 1; t] for t = 1; :::; T . However, a security typically trades at stochastic discrete times and has a price quoted only at the time of the actual trade. Such discontinuous trading patterns introduce errors in the market model (1).

To be more specific, consider any time interval [t 1; t]. During any such time window there is either a trade and a corresponding price is set, or there is no trade. If a trade takes place it is assumed that it is conducted at a random time t snt where snt is the remainder of time period t in which no trade occurs for security n and 0 snt 1 (Figure 1). If there is, for example, a trade reported in two consecutive periods for security n, then the observed rate of return rnts can be calculated over the time period [t 1 snt 1; t snt]. The crucial observation is that rnts in general diﬀers from the true return series rnt which is taken over [t 1; t]: The issue of nonsynchronous trading is still relevant, and maybe the issue is more pressing than ever with a market where highly illiquid alternatives are gaining ground. The more illiquid the security, the longer the possible space between actual trade and ascribed time of trade, hence increasing the risk for a large deviance between observed and true returns. There has been research on the subject of nonsynchronous trading and two of the most cited works are presented here.

### Method of Scholes and Williams (1977)

Scholes and Williams stress the assumption that the prices of securities are following a lognormal distribution in the market model (1). Assuming n = 1; :::; N securities and time periods t = 1; :::; T , this means that the true returns Rnt are joint normally distributed with constant means n, constant variances n2 and constant covariances nm. (Note that we abandon the notation given in Figure 1 in order to conform with the rest of the paper.) This yields the market model Rnt = n + nMt + nt (2). which is simply the market model (1) considered for each of the n securities. The coeﬃcients are as usual defined as n = n n M and n = nM = M2 , where subscript M denotes the market index.

Due to nonsynchronous trading, observed returns will in general diﬀer from true returns (Figure 1). The observed returns are generated by another process Rnto = no + noMto + nto (3).

where the coeﬃcients are given by o = E[Ro ] oE[Mo] and o = Cov(Ro ; Mo)=Var(Mo). n nt nt n nt t t

These expressions do in general not coincide with the corresponding expressions appearing in (2). The diﬀerence between equations (2) and (3), i.e. the diﬀerence between true and observed returns, will now be investigated more thoroughly. If we impose the further assumption that the non-trading periods St (s1t; s2t; :::; sNt) are independent and identically distributed (IID) random variables it can be shown that the expectation of observed returns is unchanged with respect to true returns, whereas the variance of the observed returns in general diﬀer from the variance of true returns. Beginning with the expected value, the derivation makes use of the law of total expectation by conditioning on the periods of non-trading St. After realizing that the period of trading corresponding to Rnto is 1 snt + snt 1 (see Figure 1) the derivation goes E[Rnto ] = E[E[RntojSt; St 1]] = (4).

E[(1 snt + snt 1) n] = (1 E[snt snt 1]) n = n.

This proves that the expected value of observed returns is indeed equal to the expected value of true returns. The calculation of the variance is a bit more involved. First we notice that the period of overlap between two observed returns Rnto and Rmto, conditional on St and St 1, has a length given by min(1 snt; 1 smt) + min(snt 1; smt 1).

**Method of Getmansky et al. (2004)**

The authors assume that in the market model (1) we have that E[Mt] = E[ t] = 0 and that both Mt and t are IID. In addition, the variance of the true return is Var(Rt) 2 and it has expected value E[Rt] : The model (1) describes the true returns which are unobservable. In practice we measure the observed returns Rto which are assumed to be given by a weighted average of true returns as Rto = 0Rt + 1Rt 1 + ::: + kRt k (26) where j 2 [0; 1] for j = 0; 1; :::; k (27) and k X j=0j = 1: (28). In other words, the observed returns are assumed to be a weighted average of the last k + 1 time periods. There is no theoretical practice regarding the choice of k here, it is found empirically. A rule of thumb is that the more illiquid the asset, the higher the k. The intuition behind (28) is that eventually there will always be a trade and at that time all the relevant information will be included in the price.

This smoothing process alters the performance statistics. It is easy to see that the expected value of observed returns coincides with that of true returns E[Rto] = E[ 0Rt + ::: + kRt k] = 0E[Rt] + ::: + kE[Rt k] = k (29) X 0 + ::: + k = j = = E[Rt].

#### Method of Okunev and White (2003)

This method is based on the theory introduced by Geltner (1993) to find the underlying market values from appraised values on the real estate market. Since then, variations of the method have been applied to other markets. Besides Okunev and White, there has been implementations on hedge funds by e.g. Brooks and Kat (2001). The specific method applied by Okunev and White oﬀers means to remove any order of serial correlation in the return series. However, considering the scoop of this paper, it will suﬃce to present the method for removing first order serial correlation. The assumption is that observed returns can be represented as a weighted average of the true return and lagged instances of observed returns. Note the distinction between this view and the one presented in Getmansky et al. (2004) which assume the weighted average to contain only lagged and current values of the true returns. Thus, the process can be written as Rto = (1 c1)Rt + c1Rto 1: (38).

**Table of contents :**

**1 Introduction **

1.1 Background

1.2 Purpose and Aim

**2 Mathematical Theory **

2.1 Market model

2.2 Nonsynchronous trading

2.2.1 Method of Scholes and Williams (1977)

2.2.2 Method of Dimson (1979)

2.3 Smoothed returns

2.3.1 Method of Getmansky et al. (2004)

2.3.2 Method of Okunev and White (2003)

**3 Data **

3.1 Private equity

3.2 Real estate

3.3 Infrastructure

3.4 Market Index

3.5 Risk-free asset

**4 Results **

4.1 Adjusted performance statistics

4.2 Portfolio optimization

**5 Discussion **

**6 Conclusion **

**7 Acknowledgements**