September 2008 - Question Number 6

[wasif_hyder]

Hi,

I have one confusion regarding part 1.

The problem is, I assumed this to be a question asking the expected value of the accumulation, and therefore proceeded to solve it by calculating it over all possible values of the interest rates, like this

(in millions)
1 * (1.07*0.5 +1.03*0.5) * (1.02*0.3 + 1.04*0.4 + 1.06*0.3 )

This gave the answer of 1,494,477.

The marking scheme calculates it using a single expected value of the interest, and I feel you are aware of how it goes.

The problem here is, both approaches are different, and should not yield the same answer. But they do, i.e 1494477. As a matter of fact (1.07*0.5 + 1.03*0.5) results in the same rate of interest as (0.07*0.05 + 0.03*0.5) i.e 5%

This raises a number of questions. Firstly, these are two distinct approaches that are not supposed to be equal. This may be a rather unusual case however.

Apart from that, the question mentioned the expected value of the fund, which, to me, implied calculating over all possible rates of interest. Why, then, is a single expected rate of interest being used. And how would I know which of the two approaches to use, given my confusion to this situation.

Considering that I interpreted the question correctly, or am wrong, but happen to get the correct answer. Would I lose any marks ?

This is a rather confusing thing, and I would greatly appreciate any help.

Am I missing something ???

Thank you for your help.
Kind Regards.

 

[John Lee]

Hi,

I have one confusion regarding part 1.

The problem is, I assumed this to be a question asking the expected value of the accumulation, and therefore proceeded to solve it by calculating it over all possible values of the interest rates, like this

(in millions)
1 * (1.07*0.5 +1.03*0.5) * (1.02*0.3 + 1.04*0.4 + 1.06*0.3 )^9

This gave the answer of 1,494,477.

The marking scheme calculates it using a single expected value of the interest, and I feel you are aware of how it goes.

The problem here is, both approaches are different, and should not yield the same answer. But they do, i.e 1494477. As a matter of fact (1.07*0.5 + 1.03*0.5) results in the same rate of interest as (0.07*0.05 + 0.03*0.5) i.e 5%

This raises a number of questions. Firstly, these are two distinct approaches that are not supposed to be equal. This may be a rather unusual case however.

They will always be equal. Take the 9 year part:

(1.02*0.3 + 1.04*0.4 + 1.06*0.3 )^9

= ((1+0.02)*0.3 + (1+0.04)*0.4 + (1+0.06)*0.3)^9
= ((1*0.3 + 1*0.4 + 1*0.3) + (0.02*0.3 + 0.04*0.4 + 0.06*0.3))^9
= (1 + 0.04)^9

You are perhaps confusing the "fixed" model where you get the accumulations under each interest rate THEN average them:

1.02^9*0.3 + 1.04^9*0.4 + 1.06^9*0.3

 

[wasif_hyder]

They will always be equal. Take the 9 year part:

(1.02*0.3 + 1.04*0.4 + 1.06*0.3 )^9

= ((1+0.02)*0.3 + (1+0.04)*0.4 + (1+0.06)*0.3)^9
= ((1*0.3 + 1*0.4 + 1*0.3) + (0.02*0.3 + 0.04*0.4 + 0.06*0.3))^9
= (1 + 0.04)^9

You are perhaps confusing the "fixed" model where you get the accumulations under each interest rate THEN average them:

1.02^9*0.3 + 1.04^9*0.4 + 1.06^9*0.3

Oh. I see.

With that I did achieve the correct answer. But my method wasn't as lengthy as the one in the marking scheme. Would I have needed to do it the way in the marking scheme to score full marks ? Or is this sufficient.

I'd also like to ask how do you identify which method to use ? Calculating the fund over a single expected value or the expected value of the fund calculated over all possible values of the interest rate (like you mentioned above). I saw no keyword or any distinguishing aspect of the question that would aid me in this, so I'd like to ask how we decide which model to choose.

And thank you very much John. Your help is greatly appreciated.

Kind regards
Wasif Hyder

 

[John Lee]

With that I did achieve the correct answer. But my method wasn't as lengthy as the one in the marking scheme. Would I have needed to do it the way in the marking scheme to score full marks ? Or is this sufficient.

Off the top of my head I think you may have problems with the variance - but it'll be fine to use in the exam for the expectation.

I'd also like to ask how do you identify which method to use ? Calculating the fund over a single expected value or the expected value of the fund calculated over all possible values of the interest rate (like you mentioned above). I saw no keyword or any distinguishing aspect of the question that would aid me in this, so I'd like to ask how we decide which model to choose.

The "fixed" model will say something like the interest rate is the same over the whole term.
The "variable" model will say that interest rates are independent (this is the key word) in each year.

 

[wasif_hyder]

Off the top of my head I think you may have problems with the variance - but it'll be fine to use in the exam for the expectation.



The "fixed" model will say something like the interest rate is the same over the whole term.
The "variable" model will say that interest rates are independent (this is the key word) in each year.

That really cleared it up for me, and saved me for the exam.

Thank you very much