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中二 · 2020年02月15日

问一道题:NO.PZ2016070201000065 [ FRM II ]

问题如下:

Suppose an investor expects that the 1-year rate will remain at 6% for the first year for a 2-year zero-coupon bond. The investor also projects a 50% probability that the 1-year spot rate will be 8% in one year and a 50% probability that the 1-year spot rate will be 4% in one year. Which of the following inequalities most accurately reflects the convexity effect for this 2-year bond using Jensen’s inequality formula?

选项:

A.

$0.89031 > $0.89000.

B.

$0.89000 > $0.80000.

C.

$0.94340 > $0.89031.

D.

$0.94373 > $0.94340

解释:

The left-hand side of Jensen’s inequality is the expected price in one year using the 1-year spot rates of 8% and 4%

E[$1(1+r)]=0.5×$1(1.08)+0.5×$1(1.04)=0.5×0.92593+0.5×0.96154=$0.94373E{\lbrack\frac{\$1}{(1+r)}\rbrack}=0.5\times\frac{\$1}{(1.08)}+0.5\times\frac{\$1}{(1.04)}=0.5\times0.92593+0.5\times0.96154=\$0.94373

The expected price in one year using an expected rate of 6% computes the right-hand side of the inequality as:

$10.5×1.08+0.5×1.04=$11.06=0.94340\frac{\$1}{0.5\times1.08+0.5\times1.04}=\frac{\$1}{1.06}=0.94340

Next, divide each side of the equation by 1.06 to discount the expected 1-year zero-coupon bond price for one more year at 6%. The price of the 2-year zero-coupon bond equals $0.89031 (calculated as $0.94373 / 1.06), which is greater than $0.89000 (the price of a 2-year zero-coupon bond discounted for two years at the expected rate of 6%). Thus, Jensen’s inequality reveals that $0.89031 > $0.89000

题目能做正确,但是我是算除了0.89031这个书,再联想convexy涨多跌少的性质会使得实际债权价格更大才选正确A的,讲课的时候有提到Jensen’s ineuality这个知识点么?似乎没有印象了,如果讲义上有,麻烦告知一下在那一页,谢谢!

1 个答案

orange品职答疑助手 · 2020年02月15日

同学你好,詹森不等式在基础班讲义这里: