Bill realised that moeny compounded annually, motnly, daily etc. earsn difefrent amounts of interest, even if th interest rates remain unchnaged. He even thought to himself, 'What if the moeny compounded hourly, or even every minute, or perhaps every second? What if ultimately, it compounded continously??'
a). i) Write the value of an investment of $P compounding n times a year, after y years, if i is the interest rate per annum as a decimal.
A = P (1 + i/n) n * y
show that if $1 is invested under tehse conditions, its value (V) after one year is V = $(1 + i/n)^n
ii) P = $1 = PV, V = A = FV, y =1
A = P (1 + i/n) n * y
V = 1* (1 + i/n) n * 1
V = (1 + i/n) n
(NOT TO SURE about number ii)...seems a little stupid...)
b). Value [V = (1 + i/n) n ] of $1 invested for one year at different interest rates and increasingly frequent intervals – up to 501 times a year.
Interest rate (%)
Bill then noticed two trends in common as he looked down each column (n increasing) What are they?
c). i) The first, obvious trend is that, across the columns, the interest rate, as n increases, also increases; yet, the second trend, comes out when the intervals between each successive interest rate, in each column, is studied – the interval, or difference, between, for example, n = 501, and n = 451, where i = 0.01, have a difference of [1.010050066 – 1.010050055 = 0. 000000011], whereas between n = 451, and n = 401, the difference is [1.010050055 – 1.010050041 = 0. 000000014], and so on, the difference steadily getting larger as n decreases.
(ii) Bill discovered in a book that e^i was related to the numbers in this table. Comment on the vlaues of the last two rows, the ones with headings n = 501, and e^i. NO IDEA
(iii) He was alos able to predict a value for teh limit of (1+ i/n)^n as n approaches infinity and so give the value of an investment of $p after y years, if interest is compoudning continously and if i is teh interest rate per annum as a decimal. Explain how Bill found this value. NO IDEA
(iv) Bil alos found an expression for teh effective rate of interest when it is continously compounding. What is it? NO IDEA
And thats as far as i've come. I really need help with the last three things, in question c, about teh interest rate compounding continously. Please help in any way you can! Thankyou.