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in BUS 2016H - Financial Mathematics by (1.1k points)
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In Tutorial 4, question 4 and 6, there are annuities where the time between each payment is more than a year. How do you work out the interest rate over a 4 year period? 

For my answers, I just used a summation of the present values of all the separate cash flows which got me the right answers but is there a formula so that I can get the interest rate and use the annuity formula? Here is my attempt at Q6.image

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by (2.6k points)
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If I understand your question correctly, you want to calculate an interest rate that is effective four-yearly, to use the general annuity formula. This is similiar to calculating the p-thly interest rate, and applying the p-thly annuity formula, but in this case \(p = 1/4\). So you would have:

\( i^{(1/4)}=((1+i)^4 -1)(1/4) \), where \(i\) is your effective annual interest rate and \(i^{(1/4)}\) is your nominal four yearly interest rate. Then you can apply the annuity formula for payments made regularly at four year intervals.

Just make sure that your interest is constant and not changing over the period of interest. I personally preferred to work from first principles rather than simply using formulae, as I was more certain of my answers, however there is nothing wrong with simply using the formulae.

by (2.6k points)

Also, when working from first principles, the geometric sum formula becomes very useful :)

by (1.1k points)

You understood my question perfectly thank you.I used your formula for the 4th yearly interest rate however, I have not gotten the right answer for the tutorial but if I use the method of summation it works. Would you mind looking at my work and seeing where I am going wrong with the application of that formula? Thank you for the advice on the geometric sum formula. I shall remember it. :)


by (2.6k points)

In your workings you have used the nominal interest rate convertible four yearly \((i^{(p)})\) when you need to use the effective interest rate compounded four yearly \((i^{(p)}/p)\). If you use the effective rate you will get the correct answer, whereas using the nominal rate gets you your answer :)

by (1.1k points)

I see, thank you very much. :)