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债券价格对利率的敏感性

Macaulayand修改时间measure the sensitivity of a bond's price to changes in the level of interest rates.凸性衡量收益曲线中较小偏移的持续时间变化，从而测量键的二阶价格敏感性。两项措施都可以衡量债券投资组合对利率水平变化的价值的脆弱性。

另外，分析师可以使用持续时间和凸度来构建键组合，该键组合部分应对术语结构中的小移位。如果您将持续时间为零的投资组合中的债券组合在一起，则该投资组合在某种程度上是与利率变化相结合的。如果投资组合凸度也为零，则该绝缘材料甚至更好。但是，由于对冲成本或减少了预期收益，因此您必须知道与套餐的持续时间和凸度相比，仅套期保值持续时间产生了多少保护。

This example demonstrates a way to analyze the relative importance of duration and convexity for a bond portfolio using some of the SIA-compliant bond functions in Financial Toolbox™ software. Using duration, it constructs a first-order approximation of the change in portfolio price to a level shift in interest rates. Then, using convexity, it calculates a second-order approximation. Finally, it compares the two approximations with the true price change resulting from a change in the yield curve.

Step 1

Define three bonds using values for the settlement date, maturity date, face value, and coupon rate. For simplicity, accept default values for the coupon payment periodicity (semiannual), end-of-month payment rule (rule in effect), and day-count basis (actual/actual). Also, synchronize the coupon payment structure to the maturity date (no odd first or last coupon dates). Any inputs for which defaults are accepted are set to empty matrices ([]）作为适当的占位符。

Settle ='19-1999';Maturity = ['17 -Jun-2010';'09 -Jun-2015';'14 -May-2025'];face = [100;100;1000];优惠率= [0.07;0.06;0.045];

另外，指定收益曲线信息。

产量= [0.05;0.06;0.065];

Step 2

使用财务工具箱功能来计算价格，几年的修改持续时间以及每年债券的年份。

The true price is quoted (clean) price plus accrued interest.

[CleanPrice, AccruedInterest] = bndprice(Yields, CouponRate,...Settle, Maturity, 2, 0, [], [], [], [], [], Face); Durations = bnddury(Yields, CouponRate, Settle, Maturity, 2, 0,...[]， []， []， []， []， 脸）;凸度= bndconvy（收益率，优惠率，定居点，成熟度，2，0，0，...[]， []， []， []， []， 脸）;价格= CleanPrice + AregrueDestest
Prices = 117.7622 101.1534 763.3932

Step 3

Choose a hypothetical amount by which to shift the yield curve (here, 0.2 percentage point or 20 basis points).

dy = 0.002;

将三个债券平等加权，并计算投资组合中每个债券的实际数量，总价值为$ 100,000。

PortfolioPrice = 100000; PortfolioWeights = ones(3,1)/3; PortfolioAmounts = PortfolioPrice * PortfolioWeights ./ Prices
投资组合= 283.0562 329.5324 43.6647

第4步

Calculate the modified duration and convexity of the portfolio. The portfolio duration or convexity is a weighted average of the durations or convexities of the individual bonds. Calculate the first- and second-order approximations of the percent price change as a function of the change in the level of interest rates.

PortfolioDuration = PortfolioWeights' * Durations; PortfolioConvexity = PortfolioWeights' * Convexities; PercentApprox1 = -PortfolioDuration * dY * 100 PercentApprox2 = PercentApprox1 +...portfolioconvexity*dy^2*100/2.0
PercentApprox1 = -2.0636 PercentApprox2 = -2.0321

步骤5

使用两个估算值的价格变化百分比来估算新的投资组合价格。

PriceApprox1= PortfolioPrice +...PercentApprox1 * PortfolioPrice/100 PriceApprox2 = PortfolioPrice +...PERASEAPPROX2 *投资组合/100
PriceApprox1 = 9.7936E+04 PriceApprox2 = 9.7968e+04

步骤6

通过移动收益曲线来计算真正的新投资组合价格。

[CleanPrice, AccruedInterest] = bndprice(Yields + dY,...优惠率，定居，成熟度，2，0，[]，[]，[]，[]，[]，[]，，，...脸）;newPrice = Portfolioamounts' *（CleanPrice + AccrueDemptest）
新的价格= 9.7968e+04

步骤7

比较结果。分析结果如下：

  • 原始投资组合价格为100,000美元。

  • 产量曲线上升了0.2个百分点或20个基点。

  • The portfolio duration and convexity are 10.3181 and 157.6346, respectively. These are needed forBond Portfolio for Hedging Duration and Convexity。

  • The first-order approximation, based on modified duration, predicts the new portfolio price (PriceApprox1), which is $97,936.37.

  • 二阶近似基于持续时间和凸度，预测了新的投资组合价格（PriceApprox2），为$ 97,968.90。

  • The true new portfolio price (新的价格这个收益率曲线(s)hift is $97,968.51.

  • The estimate using duration and convexity is good (at least for this fairly small shift in the yield curve), but only slightly better than the estimate using duration alone. The importance of convexity increases as the magnitude of the yield curve shift increases. Try a larger shift (dy）查看这种效果。

此示例中的近似公式仅考虑术语结构中的平行移位，因为两个公式都是dy，产量的变化。除非每个收益量都变化相同，否则该公式的定义不明确。在实际的金融市场中，产量曲线水平的变化通常解释了债券价格变动的很大一部分。但是，产量曲线（例如斜率）的其他变化也可能很重要，并且在这里没有捕获。同样，这两个公式都给出了局部近似值，其准确性恶化为dy大小增加。您可以通过运行较大值的程序来证明这一点dy。

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