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罗密欧C · 2020年02月09日

问一道题：NO.PZ2020012005000020

  1. FRM I
  2. Financial Markets And Products

问题如下：

If a stock index, interest rate, and dividend yield remain constant, derive a formula for the futures price at time t in terms of the futures price at time zero. Suppose that the risk-free rate is 5% per year and the dividend yield on an index is 3% per year. If the stock index stays constant, at what rate does the futures price grow? (All rates are expressed with annual compounding.)

选项：

解释：

The relationship between the futures price, Ft, at time t and the spot price is:

Ft=S(1+R1+Q)T−t=S(1+R1+Q)T(1+Q1+R)t=F0(1+Q1+R)tF_t=S(\frac{1+R}{1+Q})^{T-t}=S(\frac{1+R}{1+Q})^T(\frac{1+Q}{1+R})^t=F_0(\frac{1+Q}{1+R})^t

where S, R, and Q are the index level, risk-free rate, and dividend yield, respectively and T is the initial time to maturity. This shows that the futures price grows at:

(1 + Q)/( 1 + R)- 1

When R = 5% and Q = 3%, the growth rate of the futures price per year is

(1 + Q)/( 1 + R)- 1=1.03/1.05-1=-0.019

or -1.9%.

请问一下这道题答案下面的公式为什么要减１　？

(1 + Q)/( 1 + R)- 1


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1 个答案

orange品职答疑助手 · 2020年02月09日

同学你好，这题稍微有点难，它问的其实是每年期货价格变动的多少。当我们求出 F0*((1+Q)/(1+R))^t后，这个是它价格的变达式，我们可以看到他是以每年(1+Q)/(1+R)的倍数去变动的。换个形式看就清楚了：S*(1+r)^T，它每年都是前一年的(1+r)倍，所以每年增加的是 r 这么多。回到本题中，那它每年的变动就是(1+Q)/(1+R)-1

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