Question

Six months ago you won $1,000,000 on a scratch-off lottery ticket and invested your winnings with a financial advisor at the investment firm Dewey, Lie, and Howe. The earnings on the investment are compounded monthly.You complain to the financial advisor that your returns after 6 months are inadequate and do not even cover the advisor's fees. The 6 monthly interest rates (in decimal form) have been R1 = -0.4, R2= 0.67, R3 = 1.0, R4 = -0.5, R5 = 0.2, R6 = -0.165.1. What is the total worth of your investment after 6 months?$The financial advisor responds that you shouldn't complain because the average return over the 6 months has been an impressive 13.4% (0.134 in decimal form). You angrily respond that the average return is not representative of the 6-month investment performance, and that the geometric mean should be used instead.2. What is the geometric mean of the above monthly returns? (express as a decimal; round off to 3 decimal places)geometric mean

Answer 1

Answer:

The total worth of the investment after 6 months is T =  $ 1004004

The geometric mean of the above monthly returns is   $\= G = 0.001$

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  $A = \ 1,000,000$

   The number period of the investment is  6 months

Generally the worth of the investment after each month is

              $G_i = G_p * (1 + R_i)$

Here $G_p$ is the worth of the investment the previous year

          $R_i$ is the growth for that month

So considering the first month

          $G_1 = G_p (1 + R_1)$

Here $G_p = A$

So

             $G_1 = 1000000 (1 -0.4)$

              $G_1 = 600000$

Considering the second month

  Here $G_p = 600000$

So

    $G_2 = 600000 (1 + 0.67)$

=>  $G_2 = 1002000$

Considering the third month

    Here $G_p = 1002000$

So    

    $G_3 = 1002000 (1 + 1)$

    $G_3 = 2004000$

Considering the fourth month

    Here $G_p = 2004000$

So  

    $G_4= 2004000 (1 + -0.5)$

    $G_4= 1002000$

Considering the fifth  month

    Here $G_p = 1002000$

So  

    $G_5= 1002000 (1 + 0.2)$

    $G_5= 1202400$

Considering the six month

       Here $G_p = 1202400$

So  

    $G_6= 1202400 (1 -0.165)$

    $G_6= 1004004$

Generally the total  worth of the investment after 6 months is T =  $ 1004004

Generally the geometric mean of the monthly returns is  

           $\= G = \sqrt[n]{ [(1 + R_1 ) * \cdots (1 + R_n)} ]-1$

Here  n represents the number of months which has a value  n =  6

So

  $\= G = \sqrt[6]{[(1+ (-0.4 )) * (1 + 0.67) * \cdots * (1 + (-0.165))]} - 1$

     

    $\= G = 0.001$