CM1-07, practice question 7.2 and 7.3

[Cam Bridger]

I'm really struggling to get my head round when I need to be calculating an effective rate vs a nominal rate.

For 7.2, my instinct was that I need to use the annual effective rate to calculate the nominal rate of interest payable two yearly,
i(p) = i(0.5) = p * ((1 + i) ^ (1/ p) - 1)
However the solution says the interest rate needed is the pthly effective rate of interest i(p)/ p = ((1 + i) ^ (1/ p) - 1)
Effective rates are where interest is payable once per unit time, which I thought is what my i(0.5) formula would be doing, once every 2 years.

Similarly for 7.3, instinct was to convert i(4) to i and then to i(12) for rates convertible monthly, but again we're wanting the effective monthly rate rather than the rate convertible monthly.

I've read through section 3 on effective rates and section 4 on nominal rates several times and I'm not quite getting it.

These questions are asking us to convert an annual effective rate into a pthly effective rate, how do I know to do that?

Thanks

 

[Richie Holway]

Hi Cam,

7.2

There are two alternative approaches to 7.2, and the rate required will differ depending on the approach we go with. The approach used in the main solution is to work in periods of two years, which we can appreciate by the term of the accumulation (S) function being 20, even though we have cashflows for 40 years. As the accumulation function uses periods of two years, it must be calculated at a two-yearly effective rate of interest, say i*, which is just (1+annual effective rate)^2 - 1, and this equals 25.44%. The denominator of the calculation of the Sdue function is d*, and this is worked out as i*/(1+i*) = 0.2544/1.2544.

The alternative approach, that is briefly mentioned at the bottom of the solution, is to work in periods of 1 year, which we can see by the term of the accumulation (S) function being 40. However, as we only have cashflows in every other one of those 40 years, we need a (p) value attached to the accumulation function of 0.5. An accumulation function with a (p) value attached is calculated in a different way to an S function without a p value: the important difference is that the denominator will be d(p) rather than just d. So I think your instinct was almost correct if you were following the logic of this alternative method. I say almost, because it is actually a d(p) needed, rather than i(p), and this is because the accumulation function is in advance rather than in arrears.

7.3

Once again, there are a couple of possible approaches. The main solution works in periods of 1 month, and so the annuity calculation requires the monthly effective rate of interest, which is (1+annual effective rate)^(1/12). And the annual effective rate is worked out as using the quarterly effective rate (i(4)/4) compounded 4 times, so (1+i(4)/4)^4 - 1.

The final alternative approach, which is more consistent with your instinct to use i(12), works in periods of a year, hence the term of the annuity is 0.5 as there are only cashflows for half a year. As the annuity is working in years, but we know we have monthly cashflows, the annuity has a (p) value of 12, and so the denominator of the annuity calculation will be i(12) as you mention above.


I hope this helps but do let me know if you have any further questions.


Richie