Advanced Math/maths


A moneylender charges interest at the rate of 10% per month payable in advance. What is the effective rate of interest per annum?

Find the nominal rate compounded monthly which is equivalent to 5% compounded semiannually.

The Cost of a machine is Rs.12000. The machine can be resold for Rs.2500 only, at the end of 10 years. Find annual depreciation on the assumption that it remains constant in various years.
(use Compound Interest method)

Hi Ayaz,
1) In short term loans the interest may be based on the final amount instead of the present value. This means that the interest is paid in advance and the borrower receives the future value with the interest deducted. The associated rate here is referred to as discount (or bank discount) and is different from interest since the rate is applied to the future value and not the present value.
The present and future values are related to the discount in one conversion period according to the formula;
P = F(1 - d)
Say, for example, you want to borrow Rs.1000 at a rate of 10% payable in advance or discount then the amount you will receive is
P = 1000(1 - 0.1)
= Rs.900
This is quite different from borrowing Rs.900 at an interest of 10% where you'll only pay back Rs.990
Obviously, discount rates are more beneficial to lenders than interest rates.
Now, interest rate r and discount rate d in one conversion period are equivalent to each other according to the formulas;
r = d/(1 - d)
d = r/(1 + r)
and so, a 10% discount rate per month would be equivalent to
r = 0.1/(1 - 0.1) = 1/9 (about 11.11%)
The effective annual rate of interest R is the interest rate that yields the same amount of interest in one year as r, therefore on a present value P
P(1 + r)^12 = P(1 + R)
(1 + r)^12 = 1 + R
(10/9)^12 = 1 + R
R = (10/9)^12 - 1
= 2.541
= 254.1%

2) The two rates have to yield the same amount of interest in a year, so
[1 + (r/12)]^12 = [1 + (0.05/2)]^2
[1 + (r/12)]^12 = [1 + (0.025)]^2
[1 + (r/12)]^12 = [1.025)]^2
[1 + (r/12)]^12 = 1.051
r/12 = 0.0041
r = 0.0495
= 4.95%

You should note here that r is the nominal rate 'compounded monthly' and NOT rate 'per month' which is r/12.

3) For a present value P that depreciates to a future value F in n conversion periods at a compound rate of r,
F = P(1 - r)^n
2500 = 12000(1 - r)^10
(1 - r)^10 = 0.2083
1 - r = 0.8548
r = 0.1452
= 14.52%

I hope its all clear.


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Ahmed Salami


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