Question

n°5" Jon PC (3) Franklin's investment fund had a balance of $290,000 on January 1, 1995 and a balance of $448,000 two years later. The amount of interest earned during the two years was $34,000, and the annual yield rate on the fund was 5.4%. Estimate the (dollar-weighted) average date of contributions to the account. LA T . 1 1. cc.teco0o0o1 1 .. .

Answer 1

The whole investment can be divided into summation of 2 investments to solve the question -

1. Initial balance of $290,000 invested for 2 years
2. Some intermediate contribution happens at a later date, which remains invested in the fund for duration less than 2 years.

Let's focus on the 1st investment -
Initial amount invested P1 = $290,000
Annual yield on this investment r = 5.4%
Time duration of investment T1 = 2 years.

Final Value of this investment F1 = P1*(1 + r)^T1 = 290,000*(1 + 0.054)^2 = 290000*1.110916 = 322,165.64
Hence Interest earned on initial investment I1 = F1 - P1 = 322,165.64 - 290,000 = 32,165.64

Let's check the 2nd investment now -
Total value of fund after 2 years F= 448,000
Total Interest earned after 2 years I = 34,000

Total Value of investment after 2 years F = F1 + F2
where,
F1 = total value of investment-1 after 2 years
F2 = total value of investment-2 after 2 years,

so,
448,000 = 322,165.64 - F2
F2 = 125,834.36

Similarly,
Total Value of interest I = I1 + I2
where,
I1 = total interest from investment-1
I2 = total interest from investment-2

So,
34000 = 32165.64 - I2
I2 = 1834.36

So, Principal invested in 2nd investment P2 = F2 - I2
P2 = 125,834.36 - 1834.36 = 124,000

So, intermediate investment of $124,000 was invested for time period T2 to become 125,834.36

Using,
F2 = P2*(1 + r)^T2
125,834.36 = 124,000*(1 + 0.054)^T2
125,834.36/124,000 = 1.054^T2
1.01479 = 1.054^T2
T2 = LOG(1.01479)/LOG(1.054)
T2 = 0.279 years.

So, intermediate contributed remained invested for Time
T2 = 0.279 years
T2 in days = 0.279*No. of days in the year of maturity(year 1996)
T2 in days = 0.279*366 = 102.114

Hence T2 in days = 102 days approximately.

Date of maturity = 1st January 1995 + 2 years = 31st December 1996
Date of Contribution = Date of Maturity - No. of days of investment
Date of Contribution = 31st December 1996 - 102 days = 20th September 1996

Dates can be calculated in Excel as given below -

1 Date of Maturity 2 No. of Days of Investment 3 Day of Investment =DATE(1996,12,31) =102 =B1-B2

Final Answer : Dollar-weighted date of contributions is September 20, 1996

1 Date of Maturity 2 No. of Days of Investment 3 Day of Investment December 31, 1996 102 September 20, 1996 4