Ch7 exam-style question; query

[Tim.Sullivan]

Hi - I'm having difficulty understanding the solution for the chapter 7 exam-style question. I've attached a scanned copy highlighting the specific part that I have the issue with.

I attempted the question myself first and thought I had it right. Where I differ is on the calculation of the interest rate j. I calculated j as (0.12-0.03)/1.03 giving 0.087379 (8.74%). Applying this gives a solution of 488,022.

In the solution in the notes, j appears to be calculated as 1.087379^5-1= 52.021%. I can't understand why.

My steps were:
1) calc the PV of the first 5 years = 77426.33
2) The PV of the 10 x 5-years increasing by 1.03 each time can be represented as:
PV= 77426.33V + 77426.33V^2(1.03) + 77426.33V^3(1.03)^2 etc
or PV = 77426.33V [1 + 1.03V + 1.03^2V^2 + 1.03^3V^3 etc
3) So this is 77426.33V * a_10 (annuity due calc) at rate j. Using the (i-e/1+e) calc I get j = (0.12-0.03 / 1.03) = 0.087379.

Since we are working in 5 year periods and the annuity due calc in step 3 effectively works in 10 x 5 year periods, why does j need to be increased by a factor of (1+j)^5?

Thank you!

Tim

 

[KrisZ]

Hi Tim

I am on Ch5 at the moment but when I am on Ch7 I will pay particular attention to your question. Hopefully, I will be able to get back to you with some explanation in the next couple of weeks if no one posts before hand.

Will get back to you soon

KrisZ

 

[Tim.Sullivan]

2)The PV of the 10 x 5-years increasing by 1.03 each time can be represented as:
PV= 77426.33V + 77426.33V^2(1.03) + 77426.33V^3(1.03)^2 etc
or PV = 77426.33V [1 + 1.03V + 1.03^2V^2 + 1.03^3V^3 etc

I can see where my representation of the PV above differs from the notes, which shows as:
PV= 77426.33(1+ 1.03^5V^5 + 1.03^10V^10 + 1.03^15V^15 etc)

So I guess it makes sense to calculate j at ^5 too, but I'd like to see the logic behind it. Can someone show me how the calc would fit into the Sum of a Geometric Progression calculation? And can anyone explain the logic for j?

Thanks

Tim

 

[econkong]

I can see where my representation of the PV above differs from the notes, which shows as:
PV= 77426.33(1+ 1.03^5V^5 + 1.03^10V^10 + 1.03^15V^15 etc)

So I guess it makes sense to calculate j at ^5 too, but I'd like to see the logic behind it. Can someone show me how the calc would fit into the Sum of a Geometric Progression calculation? And can anyone explain the logic for j?

Thanks

Tim

i guess its because every year we need to take into account the inflation, so for each period(5 years) we need to times it 5 times, regarding the geometric progression, if we only look at the formula in the bracket, each term(from the second of course) would be the previous one times by: 1.03^5 * v^5, applying the geometric formula would give the result, hope this helps

 

[Slumpy]

i guess its because every year we need to take into account the inflation, so for each period(5 years) we need to times it 5 times, regarding the geometric progression, if we only look at the formula in the bracket, each term(from the second of course) would be the previous one times by: 1.03^5 * v^5, applying the geometric formula would give the result, hope this helps

This, basically.

The inflation is 3% p.a., and you're working out the inflationary effects of 5 years' worth of inflation, that is, 1.03^5.

 

[bystander]

For the progression, if you let say w=1.03^5v5 you will get it into the usual geometric format of

1+ w + w^2......