# Extra tut question number 2.2: annuity-certain payable half-yearly

+1 vote
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edited Apr 11, 2016

The question states:

"Find the present value, at 10% per annum effective, of an annuity-certain of R100 p.a., payable half-yearly in advance for 10 years."

By converting $$i=0.1$$ into an effective rate, compounded half-yearly, and then using the normal annuity in advance formula i.e. $$\frac{1-V^{n}}{1-V}$$ I get the correct answer of R660.18. However, when attempting to use the annuity payable p-thly formula I run into issues.

First, I get a nominal interest rate, convertible half-yearly using $$i^{(2)}=2(\sqrt{1+i}-1) =0.09762$$.

I then use the formula $$\frac{1-V_{j}^{np}}{1-V_{i^{(2)}}}$$ where $$j=\frac{i^{(2)}}{2}$$ and land up with the answer R345.25 and I'm not sure where I've gone wrong.

Thanks for any help :)

commented Apr 11, 2016 by (2,610 points)
edited Apr 12, 2016

You could be entering it incorrectly into your calculator. I just attempted the question using the p-thly annuity formula and got the correct answer.

To be honest I am not sure what method you are referring to in the first method you describe, as the p-thly annuity formula is essentially the only way I am aware of of solving a problem like this (obviously you can do this from first principles by using the annual effective interest rate and the geometric sum formula, but you will end up with the p-thly annuity formula).

Hope this helps

Edit: So I just reread your question as it was bothering me. I found the error. In your denominator for the p-thly annuity formula, you $$v$$ should be with respect to $$j$$ again and not $$i(2)$$.

commented Apr 12, 2016 by (2,850 points)
edited Apr 12, 2016 by Pandy

Great, thanks very much!

I'm still a bit confused though as the slides say the formula for an annuity payable p-thly in advance is $$\frac{1-V_{j}^{np}}{d^{(p)}}$$.

My understanding is that $$d^{(p)}=1-V^{(p)}=1-\frac{1}{1+i^{(p)}}$$. So i'm unsure as to where you got the $$j$$ from. Am I missing a simplifying step somewhere?

EDIT: Sorted, thanks. Answer below from Dean_Bunce got me to the right answer.

Find $$d^{(p)}$$ from 1.1 and then apply the normal formula for an annuity paid pthly in advance (1-$$v^{(n)})/d^{(p)}$$
sorry find $$d^{(p)}$$ from:
$$(1-d^{(p)}/p)^{(-p)}=1.1$$