上限型股權連結保本票券之設計、評價和比較

Capped Equity-linked and Principal-protected Notes: Design, Valuation and Comparison

本文考量投資人保守的投資行為與設限型股權連結票券所存在的delta 跳躍(delta jump)現象，延伸Brennan and Schwartz (1976)模型，提出一個能在股價波動之際，使發行的避險部位delta 呈現平滑變動且兼具保本(protected principal)功用的一般化模型(general form)。相較於一般的設限型股權連結保本模型，本模型具有以下特色。第一，加入股價成長率的調整因子(adjustable factor)，當景氣低靡，股價不停下跌時，正的調整因子可減緩股價下滑之勢，進而增加投資人在票券到期日時獲取更多資本利得(capital gain)的機會。同時，調整因子縮小了當期股價成長率與股價上限成長率(capped stock growth rate)之間的差距，繼而減緩delta 跳躍的幅度，降低發行者的避險成本。並且在HJM 利率模型下，delta 隨股價與股價波動度的變化更顯平滑(smooth)。第二，在保本率(protection rate)和參與率 (participation rate)不變之下，本模型的期初合理價格(fair price)較低，投資人能以較低的成本取得同等的投資保障。第三，若將本票券的名目面額(notional principal)視作共同基金(mutual fund)的淨值(net value)，而該淨值與股價連動，則本模型即成為股權連結的保本型基金(principal-protected fund)。

The protected note with part or entire principal protected at maturity is considered one of the structured financial products. The percentage of the principal guaranteed on the maturity date is known as protected rate. The popularity of the protected notes started in 1980’s and leaped to the stage of the security market in the middle of 1990 as a result of global low interest rate. To protect investors’ interest, protected notes have become an important financial instrument in portfolio selections in late 1990’s. According to Hong Kong Investment Funds Association, the net sales of the protected notes accounted for only 2.94%, but rocketed to 84% in 2001~2002 period. The protected notes started in 2003 due to legal restrictions in Taiwan, but began to pick up the momentum most recently by Winbond Electronics. Despite the mushrooming of the equity-linked and principal-protected (hereafter ELPP) notes across the world, academic research both in theory and empirical testing has lagged far behind. The seminal work by Brennan and Schwartz (1976) ushered in the concept of convex product and protected put strategy in protecting the final payoff of the insured. This paper extends the work by Brennan and Schwartz (1976) to propose a new design of capped, equity-linked and protected-principal note and to investigate the delta jump phenomenon. We design the financial product which can reduce the amplitudes of the delta jump and at the same time protect the principal. It differs from other capped equity-linked, and principal-protected models in three respects. First, we have added an adjustable factor to growth rate of stock price in such a way that a positive factor slows down the downward momentum in a bear market. As such it can increase the probability of realizing a capital gain at maturity. In the meantime, the adjustable factor narrows the gap between the current stock growth rate and the capped stock growth rate and thus really reduces the magnitude of the delta jump and hence lowers the hedging cost for brokers. The numerical results show the stabilizing effect of the adjustable factor of the growth rate of stock price on the delta jump in both the constant interest rate model and stochastic interest rates model. Furthermore, the delta appears to be smoother in the presence of changing stock prices and its volatility within the framework of the HJM interest rate model. Second, given a constant protection rate and participation rate, our approach provides a lower option premium. We conduct a comparative analysis on ELPP notes with a limit growth rate of stock price such as Winbond Electronic ELPP note. The result shows that the fair price of the note based on our model is lower than that of Winbond Electronic ELPP note for a given protection rate and participation rate. That is, investors can obtain equivalent guarantee with lower cost. Moreover, the magnitude of delta jump of our model is less than that of Winbond Electronic ELPP note. In our model, the empirical results illustrate that the fair price based on the HJM interest rate model is less than that of the constant interest rate model owing to the higher risk premium. Third, if we equate the notional principal of the note to the net value of a mutual fund whose value is linked to stock price, this model becomes the equity-linked and principal-protected fund. As a whole, previous studies about ELPP notes with a restricted growth rate of stock price never explicitly discussed how the delta jump could be solved. The purpose of this paper is to introduce an adjustable factor to dampen the magnitude of delta jump and reduce the hedging cost. The adjustable factor reveals that the return at maturity of the capped, equity-linked and principal-protected note can rise during an economic recession with a positive adjustable factor. On the other hand, it can rapidly mark the expired return up in an economic boom. The approach for reducing delta jump could be applied to other barrier options. In addition, the fair price of the capped, equity-linked and principal-protected note is lower than that of the note such as Winbond Electronic ELPP note. A rational investor can obtain an identical amount of return with a lower price. These would be useful to the issuers and the investors.