Question

How long does the capital of 4765,65 will amount to 5716,93 euro if we use the compound nominal convention with capitilization period equal to two-months and we know that the interest is null in the firs year, 1 month and 21 days and then it is nominal three monthly convertible 3%?

Answer 1

nominal three monthly convertible = r = 3%

Let nominal 2 month rate be r₂

Hence, (1 + r / 4)⁴ = (1 + 3% / 4)⁴ =1.030339191 = (1 + r₂)⁶

Hence, r₂ = 1.030339191^1/6 - 1 = 0.4994%

Let, the maturity time period is "m" periods of two months + p days after the first year, 1 month and 21 days. Please note that "m" will be a positive integer.

Hence, FV = 5716.93 = PV x (1+ r₂)^m x (1 + r₂ / 60)^p = 4765.65 x (1 + 0.4994%)^m x (1 + 0.4994% / 60)^p

Hence, (1 + 0.4994% / 60)^p x (1 + 0.4994%)^m = 5716.93 / 4765.65 =  1.1996

Please note that,

(1 + 0.4994%)^m < 1.1996 < (1 + 0.4994%)^{m +1}

Hence, m x ln (1 + 0.4994%) < ln (1.1996) < (m + 1) x ln (1 + 0.4994%)

Hence, m < ln (1.1996) / ln (1 + 0.4994%) = 36.54

and m > n (1.1996) / ln (1 + 0.4994%) - 1 = 35.54

Hence, the only positive integer to satisfy the two criteria is m = 36

When m = 36, we get: (1 + 0.4994% / 60)^p x (1 + 0.4994%)³⁶ = 1.1996

Hence, (1 + 0.4994% / 60)^p =  1.1996 / (1 + 0.4994%)³⁶ = 1.0027

Hence, p x ln (1 + 0.4994% / 60) = ln (1.0027)

Hence, p = ln (1.0027) / (1 + 0.4994% / 60) = 32

Hence, total time =  "m" periods of two months + p days after the first year, 1 month and 21 days = m x 2 months + p days + 1 year + 1 month + 21 days = 36 x 2 months + 32 days + 1 year + 1 month + 21 days = (72 months + 1 year) + 1 month + 53 days

36 x 2 months = 6 years

53 days = 1 month + 23 days

Hence, total time = (6 + 1) years + 1 month + (1 month + 23 days) = 7 years, 2 months, 23 days