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Kathy苏苏 · 2023年02月06日

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

  1. FRM I
  2. Financial Markets And Products

NO.PZ2020012005000020

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%.

(1 + Q)/( 1 + R)- 1是怎么得出的？谢谢

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2 个答案

pzqa27 · 2023年02月08日

嗨，爱思考的PZer你好：


比如现在设定T=1.我们就只看1年的情况。

S1=S0*(1+r)，可以推出S1/S0=1+r。

继续变形（S1/S0）-1=r

S1/S0-S0/S0=r

所以通分后可以得到(S1-S0)/S0=r

(S1-S0)/S0不就是S0到S1的增长率嘛

那么同理

F=S(1+Q)/(1+R)，把S除过去

F/S=(1+Q)/(1+R)，然后等式左右两边同时减1

F/S-1=(1+Q)/(1+R)-1

F/S-1通分后是(F-S/S)，因此有

(F-S/S)=(1+Q)/(1+R)-1

左边就是变化率，右边当然也是变化率

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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

pzqa27 · 2023年02月07日

嗨，从没放弃的小努力你好：


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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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

Kathy苏苏 · 2023年02月07日

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 老师，能解释下这句话吗？没看懂，谢谢。

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