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必过1030_ · 2020年01月26日

问一道题:NO.PZ201902210100000104

* 问题详情,请 查看题干

问题如下:

Considering only the US, UK, and Euro markets, the most attractive duration-neutral, currency-neutral carry trade could be implemented as:

选项:

A.

Buy 3-year UK Gilts, Sell 3-year German notes, and enter a 6-month FX forward contract to pay EUR/receive GBP.

B.

Receive fixed/pay floating on a 3-year GBP interest rate swap and receive floating/pay fixed on a 3-year EUR interest rate swap.

C.

Buy the T-note futures contract and sell the German note futures contract for delivery in six months.

解释:

B is correct.

In order to be duration-neutral and currency-neutral, the trade must lend long/borrow short in one market and do the opposite (lend short/borrow long), with the same maturities, in another market. The best carry is obtained by lending long/borrowing short on the steepest curve and lending short/borrowing long on the flattest curve. The GBP curve is the steepest and the EUR curve is the flattest. The largest yield spread between these markets is 0.55% at the 3-year maturity, and the narrowest spread is 0.35% at the 6-month maturity. Hence, the best trade is to go long the GBP 3-year/short the EUR 3-year and long the EUR 6-month/short the GBP 6-month. This can be implemented in the swaps market by receiving 3-year fixed/paying 6-month floating in GBP and doing the opposite in EUR (receiving 6-month floating/paying 3-year fixed). The net carry is +0.10% = [(0.95% – 0.50%) + (0.15% – 0.40%)]/2 for six months.

A is incorrect. The FX forward position as stated (pay EUR/receive GBP) corresponds to implicitly borrowing EUR for six months and lending GBP for six months. Correct execution of the trade would require the opposite, receiving EUR and delivering GBP 6 months forward.

C is incorrect. This combination of futures positions does create a duration-neutral, currency neutral carry trade, but it is not the highest available carry. Since the T-note futures price reflects the pricing of the 5-year note as cheapest to deliver, the long position in this contract is equivalent to buying the 5-year Treasury and financing it for 6 months. This generates net carry of 0.275% = (1.95% – 1.40%)/2. Similarly, the short position in the German note futures is equivalent to being short the 5-year German note and lending the proceeds for 6 months, generating net carry of –0.225% = (0.15% – 0.60%)/2. The combined carry is 0.05%, half of what is available on the position in B.

A为什么不对?

1 个答案

发亮_品职助教 · 2020年02月03日

嗨,努力学习的PZer你好:


“A为什么不对?”


A选项后半句:Enter 6-month FX forward contract to pay EUR/receive GBP,这个Forward的方向反了。正确的方向应该是:To pay GBP/Receive EUR.

这道题让做的是特殊的Inter-market carry trade,也就是Duration-neutral, Currency-neutral,Most attractive的Carry trade。

做这种Carry trade,就是在一国内部(收益率最陡峭的国家)借低利率、投高利率,一般向上倾斜的收益率曲线就是借短期、投长期,来赚取息差。收益率最陡峭,代表着息差收益最大。因为是在一国内部赚取息差,所以他是没有汇率风险的,实现了Currency-neutral。但是借短期、投长期,会存在一个正Duration。

为了让策略的Duration为零,我们需要在另外一个国家,借长期、投短期,这样可以构建出一个负Duration。对于向上倾斜的收益率曲线,借长期、投短期势必会有负的收益,所以我们需要在收益率曲线最平坦的国家,借长期、投短期,这样即便有负的收益,收益拖累也是最小的。这样正负Duration相互抵消,实现Duration-neutral。

这样整个策略实现了Currency-neutral,Duration-neutral,又因为是在最陡峭的曲线上做息差收益,构建Duration-neutral时,也是最平坦的曲线,所以获取的Inter-market carry trade的收益一定是最Attractive的,所以实现了这种特殊Carry trade的3个目标。


这是这种策略的基本思路。因为Currency Forward,bond Futures,interest rate swap他们归根到底,最终是利率产品,例如Forward的定价公式里有两国利率,Bond futures的最终标的物是Underlying bond代表的利率,Interest rate swap也是利率产品,所以我们可以用这三种衍生品来构建这种特殊的Inter-market carry trade。

那我们看看A选项的利率头寸(先不看Forward):

Buy 3-year UK Gilts, Sell 3-year German notes,这是借EUR 3-Year,投资UK 3-year,如果要构成这种特殊的Carry trade,我们还差两个利率头寸。因为Currency forward定价公式里自带两个国家的短期利率,所以可以利用Forward来构建剩余的两个利率头寸。

剩下额外需要用Forward构建的两个利率头寸为:

我们需要在UK市场上构建借短期利率,这样和原来Buy 3-year UK利率,实现在UK市场上借短期、投长期,来赚取息差;同时需要在EUR市场上构建投资短期利率,这样和原来Sell 3-Year German一起来实现在EUR市场上借长期、投短期,来构建负Duration。

所以,利用Forward,需要构建的利率为:借UK短期、投资EUR短期

和原来的两个利率头寸结合起来,这四个利率总体为:

借UK 6-Month,投资UK3-year;赚取息差,但是有正Duration;

借EUR 3-year,投资EUR 6-month;构建负Duration,同时产生负收益的拖累。

至于是否是Most attractive carry trade,就要看UK 6-Month与 UK 3-year的息差收益是否最大,同时EUR 3-year与EUR 6-month的收益拖累是否最小。不过这道题因为Forward的方向反了,所以直接排除A,不需要在判断是否是收益最大的一个了。


所以,A选项需要让我们利用Currency forward,实现借UK 6-Month,投资EUR 6-month的利率头寸。

由于Interest rate parity里,Currency forward的定价公式涉及2个利率,所以我们签订一个6个月的Forward,就一定能引入2国的6个月利率。

利用Forward hedge掉Currency的收益为:

发现,Hedge的收益是两国短期利率之差,所以利用Forward hedge currency risk就相当于借B国短期利率,投资A国短期利率。


我们用Forward构建的利率头寸目标是:借UK 6-month,投资EUR 6-month,这样的利率头寸产生的收益为:EUR 6-month – UK 6-month

利用Currency forward将UK hedge成EUR可以产生上面的收益公式,所以我们需要签订一个Forward在期末支付UK收到EUR。所以是Pay UK/Receive EUR的Forward。A选项刚好写反了。

在Forward里,将B国货币Hedge成A国货币,就相当于投资了A国的短期利率,也不用记汇率的标价方式,在固收这里如果要算Hedged return,Hedge的目标是哪个国家,这个国家的利率就写减号前面。


例如,把RMB hedge成USD,我们最终的目标是拿到USD,所以利用Forward hedge产生的收益是:(USD的6个月利率 减去 RMB的6个月利率)。使用6个月利率,是因为Forward的期限是6个月,如果Forward是3个月的Forward,那就使用3个月的利率。

这道题,我们可以分析出,需要利用Forward构建的利率头寸是:

EUR 6-month – UK 6-month,前面提到过目标是哪个货币,哪个货币写减号前面,因为EUR在减号前面,所以我们的目标是EUR,于是我们需要把GBP hedge成EUR,所以签订的Forward合约是支付GBP,收到EUR。这样就可以判断出A选项的Forward方向反了。


答案对A的解释,就直接可以套用上面的结论:

他说pay EUR/receive GBP,也就是期末支付EUR,收到GBP,所以目标货币是GBP,目标是把EUR hedge成GBP,所以GBP写在减号前面,A选项里利用Forward构建的利率头寸收益为:(UK 6-month 减去 EUR 6-month),这就相当于借EUR短期、投资UK短期。真正我们需要用Forward构建一个借UK 6-month,投资EUR 6-month,表示的收益应该是:EUR 6-month – UK 6-month,所以A选项用Forward构建的利率头寸刚好反了。


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