finance

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Today, Mark invested $5,000 into an account that guarantees 7.50% p.a., compounded monthly and Madonna invested $5,000 into account guaranteeing 8.125% p.a., compounded quarterly.

How long will it take (in years) for the value of Madonna's investment to be three times as much as Mark's?

  • finance -

    For mark:
    interest rate
    = 7.5% p.a.
    = 7.5%/12 per period of one month
    = .625%
    Future value of $5000 in n years, FVK(n)
    = 5000*(1.00625)12*n

    For Madonna,
    interest rate
    = 8.125% p.a.
    = 8.125%/4 per quarter
    = 2.03125% per quarter
    Future value of $5000 in n years, FVD(n)
    = 5000*(1.0203125)4*n

    We look for n where
    FVD(n) = 3FVK(n)
    5000*(1.0203125)4*n
    = 3*5000*(1.00625)12*n
    (1.0203125)4*n / (1.00625)12*n = 3

    Take log on both sides
    4*n*log(1.0203125)-12*n*log(1.00625) = 3
    n = log(3)/(4log(1.0203125-12log(1.00625))
    = 193.785 years

    Check:
    in 193.785 years,
    Mark will have $9,801,861,882.04
    Madonna will have $29,405,482,100.99

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