債券組合凸性極大化交易策略之比較

Abstract

摘 要 自1990年開始，即有學者指出，在現值相同、金額式存續期間相同的情況下，金額式凸性大的債券組合在殖利率變動的情況下，價值會高於金額式凸性小的債券組合。故1994年Christensen和Sorensen在假設時間不變的情況下，提出了賣出原本一個凸性較小的債券組合，同時買進一個現值相同、金額式存續期間，但是金額式凸性較大的債券組合凸性極大化的交易策略，以提升債券投資組合的績效。但是Christensen and Sorensen(1994)指出，只要利率走勢符合隨機過程，則債券的金額式凸性和時間價值呈反向關係，意即在加入時間變動的考量之下，投資人若要提昇金額式凸性，則要犧牲利息的時間價值，反之亦然，所以上述的交易策略在時間會變動的真實狀況中是不可行的。但是只要我們可以在市場上找到可以使得買進與賣出的債券組合的金額式時間價值相等的債券投資組合，則存在一個即使在時間會變動的真實情況之下，仍然可行的債券組合凸性極大化的交易策略。 2002年鄭千成以線性規劃的方式，在假設殖利率曲線只會水平移動的狀況下，提出可以在時間變動的情況下，可行的動態債券組合凸性極大化。2003年吳欣彰進一步放寬殖利率曲線只能水平移動的假設，提出了可以適用於殖利率曲線水平移動與斜率改變的動態債券組合凸性極大化的交易策略。但是真實的殖利率曲線，不但截距與斜率會改變，其形狀也可能變動，故鄭千成與吳欣彰的交易策略可能無法完全規避利率風險，而使得凸性極大化的交易策略的績效表現反而不及原先持有的債券組合。 本文提出一個新的模型，採用Nelson與Siegel在1987年的殖利率曲線配適方法，以及採用Willner在1996年根據Nelson和Siegel的配適方法，推導出的新的金額式存續期間定義，進而發展出一個可適用於殖利率曲線水平移動、斜率改變以及曲度改變的動態債券組合凸性極大化的交易策略。並且以台灣的公債實際歷史交易資料來測試鄭千成、吳欣彰與本文的交易策略的操作績效，探討三種交易策略於實際應用的可行性與限制。

Summary Since 1990s, scholars and financial analysts started to put emphasis on the convexity of bond portfolio. Given other conditions equal, the bond portfolio with greater dollar convexity will perform better than the bond portfolio with less dollar convexity no matter how yield to maturity changes. So under the assumption that the yield curve only moves parallel, Christensen and Sorensen proposed a trading strategy --- sell a bond portfolio with smaller dollar convexity while invest another bond portfolio with same price, same dollar duration and greater dollar convexity. Then whether the yield curve moves upward or downward, the trading strategy will bring better performance than return of original bond portfolio. But Christensen and Sorensen (1994) pointed out, so long as the change of interest rate obeys any stochastic process, the convexity and time value of the bond will be negative correlated, that means if we take the effect of time value into consideration, bond investors have to face the dilemma of gaining convexity while sacrificing the time value of the coupon payments and the opposite is also true,. Therefore, there does not exist the opportunity to enhance the performance without risk in the real world. But so long as we can find a bond portfolios (not a single bond) having same price, same dollar duration, same dollar theta and greater dollar convexity than the original bond portfolio we have, there are still a possibility to enhance the performance with risk. Cheng, in 2002, used linear programming to build up a dynamic model finding possibilities to maximize convexity in government bond market under the assumption of yield curve moving horizontally. Wu relaxed the assumption about yield curve in 2003 and improved Cheng’s model by allowing the slope of yield curve changeable. This paper proposes a new model adopting Nelson and Siegel’s method to construct yield curve and adopting Willner ‘s new definition of duration to derive a convexity maximization trading strategy suitable in most changes of yield curve—the changes in level, slope and cur vature of the shape of yield curve . Then I compare the performance of traditional models and the new model by scenario tests and historical data tests. And show that the new model indeed can be applied under the real changes of yield curve.