Why is i(p)/p called the "effective pthly rate"?

[Gerry]

I do not exactly understand why i(p)/p is the "periodic effective rate" but i(p) is the nominal rate per annum.

How does dividing i(p) by p turn it into an effective rate? Shouldn't the rate change for each compounding period once compounding is taken into account for??

 

[Joe Hook]

Hi Gerry,

i(p) is a slightly artificial interest rate which comes about from trying to value an annuity annually that makes payments pthly.

Keeping it simple, if we have a 3 year annuity making payments of 2 per annum in arrears half-yearly then we could value as:

(1) PV = v^0.5 + v + v^1.5 + v^2 + v^2.5 + v^3. Multiply both sides by (1+i)^0.5 and we get:
(2) (1+i)^0.5 * PV = 1 + v^0.5 + v^1 + v^1.5 + v^2 + v^2.5
(2)-(1) = ((1+i)^0.5-1)*PV = 1 - v^3
Hence PV = (1-v^3) / (1+i)^0.5-1

The denominator here is the half-yearly effective rate of interest. Since v^3 at the annual effective rate is equal to v^6 at the half-yearly effective rate we can rewrite the numerator as 1-v^6 at the half-yearly rate. Hence we can value this as a 6 period annuity making payments of 1 evaluated at the half-yearly effective rate.

Now imagine I wanted to use the annual rate of payment instead. I'd have to multiply this expression by 2 but also divide by 2.

So i'd get 2* [(1-v^3) / 2*((1+i)^0.5-1)]

The numerator in the brackets is the same as we'd get for a 3 year annuity but the denominator is 2 x the half-yearly effective rate. To make notation easy we write the denominator as i(2) and call it the nominal interest rate convertible half-yearly. It's simply twice the half-yearly effective rate so equivalently the half-yearly effective rate is 0.5*i(2).

Does this help?
Joe