Simple, compound & super compound

[Sinead Mc Laughlin]

Hi,

I imagine this to be a simple answer but want to check as I couldn't find it in the notes.

In April 2010 Q1 (iii) the question is

"Over the past three years the company has declared regular bonuses of 1% p.a. using the simple approach. The actuary has determined that over the same period they could have declared 0.5% three years ago, 1% two years ago and 1.5% last year using the compound approach. (iii) Determine the benefit amount for a single premium policy written three years ago with a sum assured of 10,000, under both approaches. [2]"

And the solution is:
Simple Approach: = 10,000 + (1% * 10,000 * 3) = 10,300.00
Compound approach: = 10,000 * (1 + 0.5%) * (1 + 1%) * (1 + 1.5%) = 10,302.76

Are these derived from formulae? And what would the super compound answer be? Just not sure where the 3 comes from in the simple approach or why the compound adds 1 to each percentage?

Thanks in advance.

 

[Mark Willder]

Hi,

I imagine this to be a simple answer but want to check as I couldn't find it in the notes.

In April 2010 Q1 (iii) the question is

"Over the past three years the company has declared regular bonuses of 1% p.a. using the simple approach. The actuary has determined that over the same period they could have declared 0.5% three years ago, 1% two years ago and 1.5% last year using the compound approach. (iii) Determine the benefit amount for a single premium policy written three years ago with a sum assured of 10,000, under both approaches. [2]"

And the solution is:
Simple Approach: = 10,000 + (1% * 10,000 * 3) = 10,300.00
Compound approach: = 10,000 * (1 + 0.5%) * (1 + 1%) * (1 + 1.5%) = 10,302.76

Are these derived from formulae? And what would the super compound answer be? Just not sure where the 3 comes from in the simple approach or why the compound adds 1 to each percentage?

Thanks in advance.

Rather than memorising a formula, it's probably best to go back to first principles.

A simple bonus is added to the sum assured only (ie it is not added to the attaching bonuses). So in the first year we add a bonus of 10,000 x 1% = 100. In the second year we add a bonus of 10,000 x 1% = 100. In the third year we add a bonus of 10,000 x 1% = 100. As the bonuses are the same every year, the solution just takes a shortcut and calculates the bonus as 3 lots of 100. So the total guarantee (sum assured plus bonuses) is 10,300.

A compound bonus is added to the sum assured and the attaching bonuses. So in the first year we add a bonus (b) of 10,000 x 0.5% = 50, and the guarantee becomes SA x (1+b) = 10,050. In the second year we add a bonus of 10,050 x 1% = 100.5, and the guarantee becomes (SA+RB) x (1+b) = 10,050 x 1.01 = 10,150.5. In the third year we add a bonus of 10,150.5 x 1.5% = 152.26, and the guarantee becomes (SA+RB) x (1+b) = 10,150.5 x 1.015 = 10,302.76.

Super compound bonuses are declared at different rates, eg 0.5% on sum assured and 1.5% on attaching bonuses. So in the first year we add a bonus (b) of 10,000 x 0.5% = 50, and the guarantee becomes 10,050. In the second year we add a bonus of 10,000 x 0.5% + 50 x 1.5% = 50.75, and the guarantee becomes 10,050 + 50.75 = 10,100.75. In the third year we add a bonus of 10,000 x 0.5% + 100.75 x 1.5= 51.51, and the guarantee becomes 10,100.75 + 51.51 = 10,152.26.

If you would like more practice with these calculations, April 2015 also included lots of calculations of bonuses.

Best wishes

Mark

 

[Sinead Mc Laughlin]

Perfect that makes sense. Thanks very much Mark.

Thanks,
Sinead