Portfolio insurance Directory UMM :Data Elmu:jurnal:I:Insurance Mathematics And Economics:Vol27.Issue1.2000:

86 J.M. Chamorro, J.M- a . P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 The paper is organized as follows: in Section 2, we explain the key theoretical elements of “portfolio insurance”; we then show how to work out indicators of risk and return, without and with homogenous transactions costs, that allow us to compare the different funds. In Section 3, we apply these measures to a heterogeneous group of 35 Spanish funds and build some classifications. We dedicate Section 4 to summarize the main results, make a wary comment, and suggest some further extensions. Finally, Appendix A reports some statistics concerning the goodness of the dynamic strategy as it impinges on our reliance on the results.

2. Portfolio insurance

There are n mutual funds in the market, the composition of which can be fixed income, variable income or mixed. Let F tj be the value in t of a share in fund j, where 0 ≤ t ≤ 1 and j = 1, 2, . . . , n. We assume that the instantaneous return on a share in a fund can be represented by the stochastic differential equation: dF j F j = µ j · dt + σ j · dZ, 1 where µ j is the instantaneous expected return on the share in fund j, σ 2 j the instantaneous variance of the return, and dZ is an increment to a standard Gauss–Wiener process. Initially we also assume “frictionless” markets: there are no transactions costs or differential taxes. Trading takes place continuously and borrowing and short-selling are allowed without restriction; investors can borrow and lend at the same rate. 2.1. Insurance premia as indicators of risk Let us consider in t = 0 the possibility of investing A dollars in any one of these funds. We theoretically could aim at obtaining in t = 1 either a predetermined fraction of our initial wealth z · A if the fund goes wrong or the whole value of our investment in the fund if this performs nicely. In other words, we set a floor below which we do not want our wealth to fall while, at the same time, we do want to benefit from the upside return should there be any gain in the value of the share. In this formulation z is a parameter that limits the chances of losing z 0. The existence of a “free lunch” is not possible due to the z e r restriction, i.e., there is no way of insuring with certainty a greater return than a riskless asset, like a Treasury Bill. If z = 1, this structure of results would allow us to either benefit from any possible revaluation of the fund, or at least, to rescue the initial capital. We would be aiming at complementing our share in a fund with an insurance against losses. In the cases of values of z different from 1, this characteristic remains, though somehow qualified. The complementary insurance would obviously imply a cost. It is well known that protecting oneself in this way is the same as buying put options on shares in the fund with maturity at t = 1 in a number w j and with an exercise price K j such that w j K j = z · A. In this context, an “insured share” can be characterized as a compound asset involving a basic share and a put, the value of which would be the sum F 0j + P 0j F 0j , K j , where P 0j denotes the initial price of the derivative asset, which depends, among other factors, on the value of the underlying asset and on the exercise price to be fixed. Following Black and Scholes’ B–S option pricing model, the value of this compound asset would be F 0j + P 0j F 0j , K j = F 0j · Nh j + K j · e −r · Nσ j − h j , 2 where N. is the cumulative normal distribution function, h j ≡ lnF 0j · e r K j σ j + σ j 2, r the interest rate of a riskless asset Treasury Bills, and σ j is standard deviation of the changes in the logarithm of the share price F tj . The number of insured shares that we could initially t = 0 buy with a sum of money A would be w j = A F 0j + P 0j F 0j , K j = A F 0j · Nh j + K j · e −r · Nσ j − h j . 3 J.M. Chamorro, J.M- a . P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 87 The value w j determines the size of the global insured share, what we call “insured portfolio”. As stated previously, in order to enjoy a perfect hedge, the minimum return insured by this portfolio must satisfy the equality w j K j = z · A, 4 from where the following expression is obtained K j z = F 0j · Nh j + K j · e −r · Nσ j − h j . 5 This is a nonlinear equation with only one unknown, K j . Although it cannot be solved analytically, it is possible to compute its solution by means of numeric procedures. Once K j has been calculated, w j , P 0j are determined, and obviously, F 0j + P 0j , which is the initial value of the synthetic asset or unitary insured share. It should be noted that, from the start, we have been using hypothetical terms. The reason is that there are not any markets, at least for the time being, where call and put options on shares in funds are negotiated. Thus, the insurance price P 0j is merely notional, and therefore cannot be observed. Nevertheless, it can be computed, and we deem this exercise to be useful from an informative point of view. The information embedded in them would allow us to give an answer to questions such as how much the fund managers ought to pay an insurance company should they want to insure shareholders’ contributions. The fact that this kind of insurance is not traded does not invalidate the informative function of P 0j . These prices, though merely notional, are still a monetary evaluation of the risk of losing in a fund. For the same reason, they are useful to us as a criterion for establishing a ranking based on risk. Thus, in t = 0, fund j is perceived as involving a higher risk than fund h if P 0j F 0j P 0h F 0h . The idea is very clear: notionally more expensive insurance reveals greater latent risks. 2.2. Risk-adjusted returns as indicators of performance Needless to argue that investors are concerned with both risk and return. Financial theory focuses on risk measuring and pricing, while risk-adjusted return RAR is assumed to be the ultimate objective investors pursue to maximize. Whatever the criterion considered to deal with risk, the basic idea remains the same, namely that raw return must be adjusted for or discounted by risk. In the CAPM framework, for instance, RAR is formulated as the mean return minus β times the excess market return, β being the risk parameter. In our case of no-lose investors, a naive way to introduce the notion of downside-risk-adjusted return DRAR would be as follows: DRAR j = R j − P 0j F 0j · e r if R j 0, − P 0j F 0j · e r if R j ≤ 0, where R j is the raw return on fund j at the end of the investment period t = 1, and P 0j F 0j e r appraises the risk of losing. It can be seen that DRAR j has a floor, but no ceiling. In fact, this is the return on an insured portfolio, i.e., a portfolio made from basic shares and put options at t = 0 so that investors could take no risk of losing. Clearly, DRAR j is an adjusted return since it is net of the payment for protection P 0j F 0j e r . Note, though, that DRAR j has a serious drawback, as it is deeply related to the last day of the period; this suggests that some sort of average value ought to be used in order to evaluate funds comparatively. A more accurate way to grasp the same idea is the following. We know that, in the B–S framework, the synthetic portfolio of basic shares and hypothetical puts can be replicated by means of another made from shares in the fund in percentage α and riskless assets in percentage 1 −α and managed dynamically. As in this case assets are real, this dynamic management can be carried out. That is, an investor who, in principle, had no puts and would nevertheless like to become insured, could protect himself in the same way or others could do it for him dynamically adjusting this replicating portfolio. 88 J.M. Chamorro, J.M- a . P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 Initially, the percentages would be α 0j = F 0j · Nh j F 0j + P 0j = F 0j · Nh j F 0j · Nh j + K j · e −r · Nσ j − h j , 1 − α 0j = K j · e −r · Nh j − σ j F 0j + P 0j = K j · e −r · Nh j − σ j F 0j · Nh j + K j · e −r · Nσ j − h j . 6 With the passage of time and as the involved variables change in value, these weights must be readjusted. Having reached t = 1, this portfolio, with its final composition, will have a certain value CV 1j which will define, in relation to the initial wealth A, a rate of return R cj = CV 1j − AA which we shall call covered or hedged return. Analogously to DRAR j , R cj is the rate of return on an insured share in fund j i.e., insured against the risk of losing. We are therefore also dealing with a net-of-downside-risk return. However we think R cj is a more precise indicator than DRAR j , since it evolves continuously through the whole investment period. Although it is more complex to calculate, we opt to rely on R cj for the sake of accuracy. As we are thinking about “no-lose” investors, R cj is supposed to be the key return in order to compare different funds. So we claim that fund j is better ranked than fund h if R cj R ch in a given period. To be sure: we place the different mutual funds in the same starting point at least to guarantee the initial capital A and we rank them from best to worst according to the net-of-downside-risk returns they afford at the end of the investment period in t = 1. Concerning its empirical implementation below, since we analyze 16 quarterly periods, it seems convenient to look at some average measure of these quarterly values. As we argue later on, the geometric mean fits reasonably well in this loss aversion context. 2.3. Introduction of transactions costs Up to now, we have not taken into account the transactions costs in the P 0j and R cj formulations. It is nevertheless clear that these do exist and that they affect, to a greater or less extent, the dynamic management required to replicate the “insured portfolio”. Following Leland 1985 we can distinguish two effects derived from these costs: 1 A “transactional frequency” effect takes place. Logically, if there are costs, the trade in assets diminishes. Because of this, the dynamism of portfolio revision slows down and the hedge or replication of the synthetic portfolio turns out more imperfect. 2 The “volatility” effect is the other and can be put in the following way: ˆσ 2 = σ 2 1 + √ 2π 2c σ √ 1t σ 2 , 7 where 2c is the cost of a round trip transaction purchase and sale, or the other way round, ˆσ 2 the variance with costs, and σ 2 is the variance without costs. The idea underlying this formula is that transactions costs drive a wedge between the effective closing prices of mutual funds. The introduction of transactions costs leads to a rising revision of the P 0j insurance costs due to greater volatility, and in general, it tends to reduce the return on the replicating portfolio R cj . We shall now proceed to look over the arithmetics of the model when transactions costs are incorporated. Let α t j be the percentage of A to be invested in fund j at date t. Obviously, the existence of costs implies that a different percentage β tj ought to be purchased. The revision of the model basically consists in defining the relation between α tj and β tj . For this, let us once again place ourselves in t = 0. If an order to acquire a percentage β 0j of shares is placed, a cost will have to be paid on the difference between the acquired percentage and that of the start which is zero, that is, c times δ 0j , where δ 0j = β 0j − 0. The initial net wealth after having placed the order will be A · 1 − c · δ 0j = A · 1 − c · δ 0j + β 0j − β 0j = A · [β 0j − c · δ 0j + 1 − β 0j ], 8 J.M. Chamorro, J.M- a . P´erez de Villarreal Insurance: Mathematics and Economics 27 2000 83–104 89 that is, the sum of the quantities invested in the fund and in the riskless asset. Note that β 0j − c · δ 0j is a fraction, as δ 0j = β 0j , and for the same reason, β 0j − c · δ 0j = β 0j · 1 − c. If α 0j is the desired investment, then the following ought to take place α 0j = A · β 0j − c · δ 0j A · 1 − c · δ 0j , 9 from where we obtain β 0j = α 0j 1 − c + c · α 0j . 10 Up to now we have referred to the initial moment. In order to consider dates within the investment period which we have normalized to be 0 ≤ t ≤ 1, let us agree on dividing this in n days and on using the subindex τ = 1, 2, . . . , n to date the variables on a daily basis. Let us also assume that after one more day τ = 1 the fund has grown by a factor of x 1 , and the riskless asset according to e r ·1t . The wealth at the beginning of the following day would be A · [α 0j · x 1 + 1 − α 0j · e r ·1t ] · 1 − c · δ 0j ≡ A 1 . 11 With this new initial wealth and the current price of the share, the corresponding proportion α 1j is determined. Be β 1j the new purchase order under transaction costs; we then have δ 1j ≡ A · α 0j · x 1 · 1 − c · δ 0j A 1 − β 1j ≡ |D| . 12 Thus c ·δ 1j will be paid and the net wealth will be A 1 1 − c · δ 1j . As in the previous stage, the question is to calculate β 1j from the expression α 1j = A 1 · β 1j − c · δ 1j A 1 · 1 − c · δ 1j . 13 This equation has two solutions: if D 0, β 1j = α 1j 1 + c · 1 − α 1j + 1 − α 1j 1 + c · 1 − α 1j · c · α 0j · x 1 · 1 − c · δ 0j · A A 1 , 14 if D 0, β 1j = α 1j 1 − c · 1 − α 1j − 1 − α 1j 1 − c · 1 − α 1j · c · α 0j · x 1 · 1 − c · δ 0j · A A 1 . 15 Similarly for τ = 2, 3, . . . , n.

3. Empirical results