固定收益课题 5 ：利率的期限结构 / Term structure of interest rate收益曲线策略 / Yield curve strategies

陈国辉博士 ／ 南洋理工大学商学院

现货利率 Spot Rate or cash market• 现货利率 ＝零息利率 (zero)• N 期的现货利率＝ N 期利息债券的到期收益率（ Spot rate of n-year maturity = Yield to Maturity of a n-year zero ）

• 不同期限的现货利率可以从 US STRIPS 取得• 如果没有 US STRIPS ，我们应如何取得现货利率？见下几页

现货利率和到期收益率Spot Rate and Yield to Maturity (YTM)• YTM 总汇了购买一支债券的预期回报率。 • 除非 YTM 的期限曲线 (yield curve) 是平行的，用 YTM来折现不同期限的现金流是不准确的。• YTM 是现货利率的“平均值”

• 当现货利率曲线 (spot curve) 成正向时， YTM 曲线 (yield curve) 也成正向(When spot curve is upward sloping, yield curve is also upward sloping but lies below the spot curve)

隐含现货曲线 Implied Spot Curve

• 如何建立现货曲线？• 应用自助方法 (bootstrapping method)• 最准确来自平价债券 (par bond)• 假如

• 1 年平价债券的票息＝ 5% p.a.• 2 年平价债券的票息＝ 5.5% p.a.

• 假设每年只付一次票息• 请问

• 1 年的零息利率＝ 5%• 2 年的零息利率＝ ?

从平价 YTM 曲线到隐含现货曲线From Par Yield Curve to Implied Spot Curve

T Price Coupon Discount Factor Cumulative Factor (up to n-1) Zero Rate1 1 5.00% 0.952380952 0.952380952 0.052 1 5.50% 0.898217107 1.850598059 0.0551381823 1 5.80% 0.843729029 2.694327088 0.0582761364 1 5.95% 0.792531891 3.486858979 0.0598534395 1 6.10% 0.742037325 4.228896304 0.0614874156 1 6.20% 0.694734867 4.923631172 0.0625845187 1 6.25% 0.651551108 5.575182279 0.063111438 1 6.30% 0.61031375 6.185496029 0.0636674179 1 6.40% 0.567789713 6.753285742 0.064909001

10 1 6.50% 0.526794767 7.280080509 0.066193069

nn

n

n

tt

n

ZFactor

whereCouponPar

FactorCouponParFactor

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远期利率 Forward Rate• 远期利率是两交易方今天定下的一未来利率，其期限从一未来的日期到另一未来日期。

A forward rate is the interest rate for a loan between any two dates in the future, contracted today.

• 远期利率市场叫远期利率合同 (FRA, Forward Rate Agreement) 。 应用于投机和对冲。

• 给予一现货利率曲线，我们可以求导远期利率的期限结构A given term structure of spot rate gives rise to a specific term structure of forward rates.

现货利率和远期利率Spot and Forward Rate

1-period forward rate

Spot rate

计算远期利率 Calculate Forward Rate

• 第 1 期到 2 期的远期利率 forward rate ＝ (f1.2)• 应用 1- 年的现货利率 (s1) 和 2- 年 (s2) 的现货利率我们可寻找 :

)1()1()1(1

22

2,1 ssf

如何理解远期利率• 现货利率是一组远期利率的几何平均率

Spot rate is the geometric average of a series of forward rates.• 远期利率是所有同一信用评级债券的损亏率，即每一债券的收益率都相同

Forward rate is the break-even rate that make all bonds of same credit class earn the same return over a holding period.• 如果远期利率实现了，那么所有同信用评级的债券都得到同样回报。

If the forward is realized, all bonds of same credit class regardless of maturity earn the same return. Nobody makes more.

• 杠铃式－子弹式得零收益Barbell-bullet makes zero return.

利率曲线 Yield Curves• 利率曲线的种类

• 平价利率曲线 (Benchmark yield curve / Par yield curve)• 零息利率曲线 (Zero coupon curve)

• 利率曲线的用途• 为企业债券定价 ( 利差， credit spreads ）• 反映市场对未来利率的走向 (market expectation)• 利率曲线投资战略 (yield curve strategies)

平价利率曲线 Par Yield Curve• 由平价债券的到期收益率所组成的曲线• 市场一般观察此曲线因为它所表示的利率比较能准确的反映市场利率。这是由于折价或溢价债券受其它因素影响，所以其 YTM 不能反映真实利率。

(The par curve is important as the yields on bonds selling at par are likely to be more representative of the underlying term discounting rates implicit in the market. Bonds selling at a substantial discount or premium to par are often subject to special forces which distort their prices. )

收益率曲线 －正常曲线

• 一般上借贷期越长，利率就越高。这反映了期限越长的债券，其风险越大。这样的曲线我们叫正常曲线。

1M 3M 6M 1Y 2Y 5Y 10Y 30Y

收益率曲线 －正常移动

• 当联储局提高或降低利率（联邦基金利率）时，短期利率会变化，其它期限的利率也会跟着变，曲线保持正常状态。

旧线新线

1M 3M 6M 1Y 2Y 5Y 10Y 30Y

收益率曲线 －平型

• 曾经联储局都在提高联邦基金利率，但是长期利率却一直在下降，造成收益曲线成水平型。

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新线

1M 3M 6M 1Y 2Y 5Y 10Y 30Y

收益率曲线 －反向型

• 当短期利率不断的下降，但长期利率不上升时，收益率曲线就会成反向型。这是很不正常的。人们一般认为这种情形意味经济将会出现不景气。 1989 年和 2000 年时的反向曲线就是好例子。，但是 1998 年是的反向曲线却没给经济都带来萧条。

旧线

新线

1M 3M 6M 1Y 2Y 5Y 10Y 30Y

http://stockcharts.com/charts/YieldCurve.html

利率期限结构理论Theories of Term Structure

• 预期理论 Expectation theory• 纯预期理论 Pure expectation• 流动性理论 Liquidity theory• 偏好理论 Preferred habitat theory

• 市场分割理论 Market segmentation theory

纯预期理论 Pure Expectation• 假设你有 $1 投资 2 年时间选项 A:投资 1 年然后再投资（滚动） 1 年选项 B:一次性投资 2 年• 假设以下的现货利率曲线 :

1-year spot: 7%2-year spot: 8%

纯预期理论 Pure Expectation• 请问市场认可的 1 年后的 1 年利率是多少？

At what expected rate can the market agree?• 答案是 9.01% （为什么？）

• 预期理论主张远期利率是预期未来利率Rate expectation asserts that the forward rates represent the expected future rates.

• 因此今天的利率曲线完全反映市场对未来利率的预期Thus the entire term structure at a given time reflects the market’s current expectations of the family of future short-term rates.• 如果此理论成立的话，正向（反向）曲线意味市场预期利率将升高（下降） Under this view, upward (downward) slope curve indicates market

expects short-term rates to rise (fall).• 不管是投资短期债券还是长期债券，投资者都会得到同样回报。为什么？

(Ans: If long-term bonds that have higher yields than the short-term bond yield then long-term bonds are expected to suffer capital losses that offset their yield advantage.)

• 预期理论是不是真能决定利率曲线的形状？• 如果是对的话，那么利率曲线的形状应该是有时正向有时成反向。那为什么我们一般观察到的利率曲线 90% 的时候是正向的？

If pure expectation is true, then sometimes the yield curves will be upward sloping and sometimes downward sloping. But why 90% of the time the yield curve is upward sloping.

• 如果是对的话，那么远期利率能准确的预测未来利率。可能吗？If pure expectation is true, then forward rate is the perfect predictor of future interest rate. If this is the case, then holding any maturity of bonds is irrelevant. No risk!

纯预期理论 Pure Expectation

流动性理论 Liquidity Theory

• 投资偏好流动性，因此如果要他们持有长期债券，他们要求在回报上得到一些溢价 (premium)In order to induce these investors to hold longer term bonds, they must be compensated with a liquidity premium.• 同样假设一投资者面对两只债券 :

A: 1 年期B: 2 年期

• 再假设他的投资期限为 2 年。• 请问他会选择哪一只债券？

流动性理论 Liquidity Theory• 虽然他的期限为 2 年，但他还是会选择 1 年期债券因为他不喜欢被“锁住”

This is in case funds are needed in the middle of investment, ie., investors dislike being “lockup”.

• 因此如果要他买 B ， B 债券的回报率必须带有溢价。He will choose Bond A unless he is compensated enough to buy Bond B.

• 假设他想在一年后把 B 债券卖出，他可能面对价格风险。If the investor chooses Bond B and in case he wants to liquidate it in one year time, he will face price risk (the bond has another year to maturity).

• A 债券提供了再投资的灵活性Bond A gives flexibility (liquidity)

• 所以对他最有利的投资策略是先投 A ，再考虑是否要向前滚动， 除非 B 能给回报溢价。

流动性理论 Liquidity Theory• 应用同一例子：

1 年零息利率 = 7%2 年零息利率 = 8% (FV of $1 = (1.08)(1.08) =$1.1664).

• 我们也知道远期利率 = 9.01%.• 市场预期的未来利率应该高过、低过、或等于 9.01% ？

流动性理论 Liquidity Theory

• 远期利率 (f) = 预期利率 E(s) + 流动风险溢价 (L) Forward rate (f)= expected rate (E(s)) + Liquidity Risk premium (L)

• 因此推断：(1 + s2) (1 + s2) > (1 + s1) (1 + E(s))

• where s1 = 1-yr spot, s2 = 2-yr spot • 应用以上的例子：

• 假设预期利率 E(s) ＝ 8.6%, 那么 L ＝ ？• 滚动战略的未来值是多少？

• 流动性理论解释为什么曲线会成反向• 假设 s2 = 6%, s1=7%, L = 0.41%.• 远期利率是多少？• 预期未来利率 E(s) 由是多少？

• 结论：• 反向曲线是由于市场预期未来利率将下降。这符合纯预期理论的推测

Downward sloping is due to expectation of lower interest rate. Consistent with the pure expectation theory but much lower.

流动性理论 Liquidity Theory

• 流动性理论解释为什么曲线有时会成平行 Liquidity premium explains flat yield curve• 假设 s2 = 7%, s1=7%, L = 0.41%.• 远期利率是多少？• 预期未来利率 E(s) 由是多少？

• 结论：• 平行曲线是由于市场预期未来利率将下降。这和纯预期理论的推测相反

Flat yield curve is due to expectation of lower future interest rate (in contrast to pure expectation theory.)

流动性理论 Liquidity Theory

• 流动性理论解释为什么曲线会成正向• 假设 s1 = 7% and s2 = 7.1%.• 远期利率是多少？• 预期未来利率 E(s) 由是多少？

• 结论：• 轻微正向曲线可能意味市场预期未来利率将下降。这和纯预期理论的推测相反

Thus, slight upward sloping means the market expects a slight decline in the spot rate (contradict to pure expectation theory).

• 再假设 s1 = 7% and s2 = 7.3%.• 那么 f = 7.6% ， L = 0.41%, E(s) = 7.19%• 结论是什么？

流动性理论 Liquidity Theory

流动性理论 Liquidity Theory

• 汇总 :• 反向曲线 : 利率将下降• 平行曲线 : 利率将下降• 正向曲线 : 看斜率有多大，可以是利率将下降或上升• 假设有一半的时候市场预期未来利率会上升，一半的时候预期未来利率会下降，那么多数时候我们会观察到正向曲线• 流动性溢价也应该因期限的加长而增加

Since upward sloping yield curve happen most of the time, the liquidity premium must rise fast enough as maturity lengthens.

偏好理论 Preferred Habitat Theory

• 同意曲线受市场预期和流动风险导致• 但不同意流动溢价一定要和期限成正比。正或负流动溢价都可能• 当供应和需求不匹配时，借款人和贷款人会通过利率来调节平衡供需。因此任何的曲线都可能

市场分割理论 Market segmentation theory

• 借款人和贷款人各有其期限的局限Markets are segmented by maturities due to their constraint.

• 借款人和贷款人不会从一期限区移动到另一期限区Lenders and borrowers won’t shift from segment to segment.

凸性偏爱 Convexity Bias

• 凸性偏爱影响利率曲线Convexity bias measures the impact of convexity on the yield curve.• 凸性是有价值的• 因此投资者会愿意牺牲回报率来得到高凸性的债券 investors tend to

demand less yield for more convex positions because of the prospect of making higher returns.

Yield Curve Strategies

Parallel up

flattening

Positive butterfly

Yield Curve Strategies

Parallel down

steepening

Negative butterfly

杠铃对子弹Barbell vs Bullet

• 3 bonds:

Bond Coupon Maturity Dirty price

YTM $ Duration

$ Convexity

A 8.5 5 100 8.5 4.005 19.8164

B 9.5 20 100 9.5 8.882 124.1702

C 9.25 10 100 9.25 6.434 55.4506

杠铃对子弹Barbell vs Bullet

• Bullet: invest 100% in bond C.• Barbell: invest 50.2% in bond A and 49.8% in Bond B

(the weight is due to duration matching)

Strategy $ Duration $ Convexity Yield

Bullet 6.434 55.4506 9.25%

Barbell 0.502*4.005+ 0.498*8.882 = 6.434

0.502*19.8164+ 0.498*124.1702 = 71.7846

0.502*8.5+ 0.498*9.5 = 8.998%

杠铃对子弹Barbell vs Bullet• 不难发现 :

• 两个组合有同等的久期both portfolio has the same duration.

• 但是 Yield of Bullet > Yield of Barbell• 问题：

• 哪为什么人们要买杠铃呢？Why would anyone want to buy barbell in this case (with a lower yield)?

应用远期曲线交易Using forward curve for trading

• “如要在债券市场上有所收获，你不该认同远期利率是未来利率，当然你的预测必须是正确的”“To make money in fixed-income markets, you must disagree with the forwards – and be right.”• 因为如果远期利率真的实现的话，所有的投资战略都赚一样的回报

Because if forward rates are realized, all positions make the same return.

• 如果利率上升而且比远期高，那么• 看跌（利率上升）会有盈利

bearish positions (rates rise) make money• 看涨（利率下降）会亏钱

bullish positions lose money

正向曲线 Upward Sloping Curve• 假设 S3 > S1

• 买入 3 年，同时卖空 1 年 = 正持有 positive carry• 1 年后， 如果 2 年利率＝远期 (f1,3). 利润 = 0.• 但是如果 2 年利率低过远期 (f1,3) ，再融资的成本将会更低，有盈利• 但是如果 2 年利率高过远期 (f1,3) ，再融资的成本将会更高，但是由于正持有减轻了成本 (positive carry provides cushion), 损失可减少

T Spot Rate t to t+1One year forward 1 to t

Implied Spt Rate One Year Forward

1 6% 0,1 6% 1,2 8.01%2 7% 1,2 8.01% 1,3 8.64%3 7.75% 2,3 9.27% 1,4 9.09%4 8.31% 3,4 10.02% 1,5 9.43%5 8.73% 4,5 10.44% 1,6 9.67%6 9.05% 5,6 10.65% 1,7 9.85%7 9.29% 6,7 10.72% 1,8 9.97%8 9.47% 7,8 10.72% 1,9 10.06%9 9.60% 8,9 10.67% 1,10 10.12%

10 9.70% 9,10 10.60%

现货、远期和远期现货Spot, Forward Rate, and Forward Spot Rate

1-period forward rate

Spot rate

0 1

Spot rate 1 year forward

0%

2%

4%

6%

8%

10%

12%

0 2 4 6 8 10 12

Maturity

Spot Curve

Implied Spot Rates One Year Forward

Forward Curve

In this example, the forward curve implies flattening yield curve

杠铃式－子弹式 （蝴蝶式）Barbell-Bullet Trade (Butterfly)

• 举例 :• 卖空 3 年零息 （卖子弹） (Sell bullet)• 买入 1 年 和 5 年 （买杠铃） (buy barbell)• 预期 : 曲线会平行 （在 2 年到 4 年期限内） (curve fattening

between 2 to 4 years)• 中性久期 (Duration neutral) ：

• 达到免疫作用 (immune from parallel shift) 和• 自融基金 self financing (cash neutral).

• 可能是 正持有或负持有 (positive carry or negative carry)• 曲线平行的话此战略就会盈利 (For the trade to make money, curve

must flatten)

杠铃式－子弹式 （蝴蝶式）Barbell-Bullet Trade (Butterfly)

• 应用先前的例子• Sell 3-year zero at 7.75%• Buy 1-year zero at 6%• Buy 5-year zero at 8.73%• Negative carry ＝ 0.5*(6.00-7.75%) + 0.5*(8.73-7.75) = -0.88 +

0.49 = -0.39%.• 曲线必须平行来对冲负持有

For the trade to make money, capital gains caused by future flattening of the spot curve must offset the negative carry.

信贷利差Yield Spread Strategy – Credit Spread

Treasuries

Current non-treasury

Expected non-treasury

Spread between treasuries and non-treasuries tends to widen when economy is heading to recession. This is because non-treasury companies are likely to suffer credit downgrade in downturn.