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Alex · 2021年04月03日

请问第二步hedge的价格是怎么确定的。

NO.PZ2019103001000063

问题如下:

Susan Winslow manages bond funds denominated in US Dollars, Euros, and British Pounds. Each fund invests in sovereign bonds and related derivatives. Each fund can invest a portion of its assets outside its base currency market with or without hedging the currency exposure, but to date Winslow has not utilized this capacity. She believes she can also hedge bonds into currencies other than a portfolio’s base currency when she expects doing so will add value. However, the legal department has not yet confirmed this interpretation. If the lawyers disagree, Winslow will be limited to either unhedged positions or hedging into each portfolio’s base currency.

Winslow thinks the Mexican and Greek markets may offer attractive opportunities to enhance returns. Yields in these markets are given in Exhibit 1, along with those for the base currencies of her portfolios. The Greek rates are for euro-denominated government bonds priced at par. In the other markets, the yields apply to par sovereign bonds as well as to the fixed side of swaps versus six-month Libor (i.e., swap spreads are zero in each market). The six-month Libor rates also represent the rates at which investors can borrow or lend in each currency. Winslow observes that the five-year Treasury-note and the five-year German government note are the cheapest to deliver against their respective futures contracts expiring in six months.

Winslow expects yields in the US, Euro, UK, and Greek markets to remain stable over the next six months. She expects Mexican yields to decline to 7.0% at all maturities. Meanwhile, she projects that the Mexican Peso will depreciate by 2% against the Euro, the US Dollar will depreciate by 1% against the Euro, and the British Pound will remain stable versus the Euro. Winslow believes bonds of the same maturity may be viewed as having the same duration for purposes of identifying the most attractive positions.

Based on these views, Winslow is considering three types of trades. First, she is looking at carry trades, with or without taking currency exposure, among her three base currency markets. Each such trade will involve extending duration (e.g., lend long/borrow short) in no more than one market. Second, assuming the legal department confirms her interpretation of permissible currency hedging, she wants to identify the most attractive five-year bond and currency exposure for each of her three portfolios from among the five markets shown in Exhibit 1. Third, she wants to identify the most attractive five-year bond and hedging decision for each portfolio if she is only allowed to hedge into the portfolio’s base currency.

If Winslow is limited to unhedged positions or hedging into each portfolio’s base currency, she can obtain the highest expected returns by

选项:

A.

buying the Mexican 5-year in each of the portfolios and hedging it into the base currency of the portfolio.

B.

buying the Greek 5-year in each of the portfolios, hedging the currency in the GBP-based portfolio, and leaving the currency unhedged in the dollar-based portfolio.

C.

buying the Greek 5-year in the Euro-denominated portfolio, buying the Mexican 5-year in the GBP and USD-denominated portfolios, and leaving the currency unhedged in each case.

解释:

B is correct.

Winston should buy the Greek 5-year bond for each portfolio. In the US dollar portfolio, she should leave the currency unhedged, accepting the exposure to the Euro, which is projected to appreciate by 1% against the USD. In the UK portfolio, she should hedge the bond’s EUR exposure into GBP. In the Euro-based portfolio there is no hedging decision to be made because the Greek bond is denominated in EUR.

Because yields are projected to remain unchanged in the US, UK, Euro, and Greek markets, the 5-year Greek bonds will still be priced at par in six months and the US, UK, and Euro bonds will realize a negligible price appreciation when they have 4.5 years to maturity. Hence, the local market return for each of these bonds will equal half of the coupon: 0.975%, 0.55%, 0.30%, and 2.85%, respectively. The Mexican 5-year will be priced to yield 7.0% at the end of the period. Its price will be


Its local market return is therefore 4.576% = (100.9501 + 7.25/2)/100. By covered interest parity, the cost of hedging a bond into a particular currency is the short-term (six months here) rate for the currency into which the bond is hedged minus the short-term rate for the currency in which the bond is denominated. For hedging US, UK, and Mexican bonds into Euros for six months the calculation is:

USD into EUR: (0.15% – 1.40%)/2 = –0.625%

GBP into EUR: (0.15% –0.50%)/2 = –0.175%

MXN into EUR: (0.15% – 7.10%)/2 = –3.475%

(Note that a negative number is a cost while a positive number would be a benefit.)

Combining these hedging costs with each bond’s local market return, the returns hedged into EUR, which can now be validly compared, are:

US: 0.975% + (–0.625%) = 0.350%

UK: 0.550% + (–0.175%) = 0.375%

MX: 4.576% + (–3.475%) = 1.101%

GR: 2.850% + 0 = 2.850%

EU: 0.300% + 0 = 0.300%

The Greek bond is by far the most attractive investment. This would still be true if returns were hedged into USD or GBP. So, the Greek 5-year should be purchased for each portfolio. Whether or not to actually hedge the currency exposure depends on if the cost/benefit of hedging is greater than the projected change in the spot exchange rate. For the dollar-denominated portfolio, hedging the Greek bond into USD would “pick up” 0.625% (the opposite of hedging USD into EUR). But EUR is expected to appreciate by 1.0% against the dollar, so it is better to leave the bond unhedged in the USD-denominated portfolio. Hedging EUR into GBP picks up 0.175% of return. Since EUR is projected to remain unchanged against GBP, it is better (from an expected return perspective) to hedge the Greek bond into GBP.

A is incorrect because it can be seen from the explanation for B above that the Greek 5-year bond is by far the most attractive investment, returning 2.85% compared to the Mexican 5-year bond’s return of 1.101%. If the returns for these bonds were hedged into USD or GBP (instead of EUR), in each case the return on the Mexican 5-year bond would still be inferior to that of the Greek 5-year bond.

C is incorrect because it can be seen from the explanation for B above that the Greek 5-year bond is by far the most attractive investment, returning 2.85% compared to the Mexican 5-year bond’s return of 1.101%. If the returns for these bonds were hedged into USD or GBP (instead of EUR), in each case the return on the Mexican 5-year bond would still be inferior to that of the Greek 5-year bond. Moreover, over the 6-month investment horizon the Mexican Peso is expected to depreciate against both the GBP and USD, further impairing the unhedged returns on the Mexican 5-year bond in GBP and USD terms.

第二步需要计算hedge的成本;题目用的都是六个月利率相减。请问题干什么地方说远期利率价格F就是六个月利率了呢。 倒是不hedge计算时,那些基金经理的expected观点写的很明白。
2 个答案
已采纳答案

发亮_品职助教 · 2021年04月06日

嗨,爱思考的PZer你好:


第二步需要计算hedge的成本;题目用的都是六个月利率相减。请问题干什么地方说远期利率价格F就是六个月利率了呢。


这是汇率远期合约(Currency forward)定价公式里所决定的。6个月的利率来自6个月的Forward合约里的利率。


注意,在Currency forward里面定未来的汇率时,使用利率和Forward的期限一致。


由于这道题里面的利率预期是未来6个月的,所以本题的所有策略都是基于未来这6个月的利率预期,所以本题的投资期就是6-month。


那么,为了对冲这6个月的投资,我们使用的Forward应该是6个月的Forward,那这样的话,Forward合约里面定价的利率自然就是6个月的利率了。所以我们算Forward hedge之后的收益时,就约等于两个国6个月的利率相减。


下面大概具体推一下如何计算Forward hedge的收益,我们固定收益里面最终只用记结论即可(Hedge收益等于两国利率相减):


Forward约定的汇率为:


两步同时减去S,经过变形之后为:



而用Foward hedge的收益为:

(F-S)/ S


刚好再变形一下就等于:

最终发现,由于rb非常小,就认为(1+rb)等于1,所以Hedge的收益,就约定于两国的利率差。

这两个利率是来自Forward合约里的定价利率,那本题之中,由于Forward合约是6个月的合约,因此Hedged收益是两国6个月的利率之差。


如果Forward是3个月的合约,那Hedged的收益就是两国3个月的利率相减。


固收里面碰到这种题目,在算Hedged收益时,直接用对应期限的利率相减就好了,但是注意表格里面的利率都是年化利率,如果Hedged的期限小于1年,例如是Hedge半年,那么在算Hedged收益时,是两国利率相减再除以2。

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加油吧,让我们一起遇见更好的自己!

Alex · 2021年04月06日

我再次听了何老师的课件,发现这个公式的前提条件是covered irp成立,此时f就等于1+r外国/1-r本国,约等于r外国-r本国。所以就像您说的,直接利率相减就行了。

发亮_品职助教 · 2021年04月07日

嗨,从没放弃的小努力你好:


我再次听了何老师的课件,发现这个公式的前提条件是covered irp成立,此时f就等于1+r外国/1-r本国,约等于r外国-r本国。所以就像您说的,直接利率相减就行了。


正确。Forward合约就是建立在Covered IRP成立的基础上的。


或者换句话说,Forward合约的这个定价公式,就完全是基于Covered IRP成立的,所以,只要用到Forward算合约的Hedged return,近似收益就是直接利率相减就好了。


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虽然现在很辛苦,但努力过的感觉真的很好,加油!

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