Answer:
The total worth of the investment after 6 months is T = $ 1004004
The geometric mean of the above monthly returns is ![\= G = 0.001](https://tex.z-dn.net/?f=%5C%3D%20G%20%20%3D%20%200.001)
Step-by-step explanation:
From the question we are told that
The growth for each month are
R1 = -0.4, R2= 0.67, R3 = 1.0, R4 = -0.5, R5 = 0.2, R6 = -0.165
The amount invested is ![A = \$ 1,000,000](https://tex.z-dn.net/?f=A%20%3D%20%5C%24%201%2C000%2C000)
The number period of the investment is 6 months
Generally the worth of the investment after each month is
![G_i = G_p * (1 + R_i)](https://tex.z-dn.net/?f=G_i%20%3D%20%20G_p%20%2A%20%281%20%2B%20R_i%29)
Here
is the worth of the investment the previous year
is the growth for that month
So considering the first month
![G_1 = G_p (1 + R_1)](https://tex.z-dn.net/?f=G_1%20%3D%20%20G_p%20%281%20%2B%20R_1%29)
Here ![G_p = A](https://tex.z-dn.net/?f=G_p%20%3D%20%20A)
So
![G_1 = 1000000 (1 -0.4)](https://tex.z-dn.net/?f=G_1%20%3D%201000000%20%281%20-0.4%29)
![G_1 = 600000](https://tex.z-dn.net/?f=G_1%20%3D%20600000)
Considering the second month
Here ![G_p = 600000](https://tex.z-dn.net/?f=G_p%20%3D%20%20600000)
So
![G_2 = 600000 (1 + 0.67)](https://tex.z-dn.net/?f=G_2%20%3D%20%20600000%20%281%20%2B%200.67%29)
=> ![G_2 = 1002000](https://tex.z-dn.net/?f=G_2%20%3D%20%201002000)
Considering the third month
Here
So
![G_3 = 1002000 (1 + 1)](https://tex.z-dn.net/?f=G_3%20%3D%20%201002000%20%281%20%2B%201%29)
![G_3 = 2004000](https://tex.z-dn.net/?f=G_3%20%3D%20%202004000%20)
Considering the fourth month
Here ![G_p = 2004000](https://tex.z-dn.net/?f=G_p%20%3D%20%202004000%20)
So
![G_4= 2004000 (1 + -0.5)](https://tex.z-dn.net/?f=G_4%3D%202004000%20%281%20%2B%20-0.5%29)
![G_4= 1002000](https://tex.z-dn.net/?f=G_4%3D%201002000%20)
Considering the fifth month
Here ![G_p = 1002000](https://tex.z-dn.net/?f=G_p%20%3D%20%20%201002000%20)
So
![G_5= 1002000 (1 + 0.2)](https://tex.z-dn.net/?f=G_5%3D%201002000%20%281%20%2B%200.2%29)
![G_5= 1202400](https://tex.z-dn.net/?f=G_5%3D%201202400%20)
Considering the six month
Here ![G_p = 1202400](https://tex.z-dn.net/?f=G_p%20%3D%20%201202400%20)
So
![G_6= 1202400 (1 -0.165)](https://tex.z-dn.net/?f=G_6%3D%201202400%20%281%20-0.165%29)
![G_6= 1004004](https://tex.z-dn.net/?f=G_6%3D%201004004%20)
Generally the total worth of the investment after 6 months is T = $ 1004004
Generally the geometric mean of the monthly returns is
![\= G = \sqrt[n]{ [(1 + R_1 ) * \cdots (1 + R_n)} ]-1](https://tex.z-dn.net/?f=%5C%3D%20G%20%20%3D%20%20%5Csqrt%5Bn%5D%7B%20%5B%281%20%2B%20R_1%20%29%20%2A%20%20%5Ccdots%20%281%20%2B%20R_n%29%7D%20%5D-1)
Here n represents the number of months which has a value n = 6
So
![\= G = \sqrt[6]{[(1+ (-0.4 )) * (1 + 0.67) * \cdots * (1 + (-0.165))]} - 1](https://tex.z-dn.net/?f=%5C%3D%20G%20%3D%20%5Csqrt%5B6%5D%7B%5B%281%2B%20%28-0.4%20%29%29%20%2A%20%281%20%2B%200.67%29%20%2A%20%5Ccdots%20%2A%20%281%20%2B%20%28-0.165%29%29%5D%7D%20-%201)
![\= G = 0.001](https://tex.z-dn.net/?f=%5C%3D%20G%20%20%3D%20%200.001)