Question

On January 1, 1997, an investment account is worth 100,000. On April 1, 1997, the value has increased to 103,000 and 8,000 is withdrawn. On January 1, 1999, the account is worth 103,992. Assuming a dollar weighted method for 1997 and a time weighted method for 1998, the annual effective interest rate was equal to x for both 1997 and 1998.Calculate x.(A) 6.0% (B) 6.25% (C) 6.50% (D) 6.75% (E) 7.0%

Answer 1

(B) 6.25%

Step-by-step explanation:

January 1, 1997 = $100,000

April 1. 1997 = $103,000 - $8,000 = $95,000

January 1, 1999 = $103,992

annual interest rate for 1997 = i = (x - 100,000 + 8,000) / [100,000 - 8,000(1 - ³/₁₂) = (x - 100,000 + 8,000) / [100,000 - 8,000(1 - 0.25) = (x - 92,000) / 94,000

x = 92,000 + 94,000i

annual interest rate for 1998 = 1 + i = 103,992/x

x = 103,992/(1 + i)

0 = x(1 + i) - 103,992

now we replace x by 92,000 + 94,000i

0 = (92,000 + 94,000i)(1 + i) - 103,992

0 = (94,000 (1 + i) - 2,000)(1 + i) - 103,992

we now replace 1 + i by Y

0 = (94,000Y - 2,000)Y - 103,992

0 = 94,000Y² - 2,000Y - 103,992

using a calculator, Y = 6.25%