April 2016 question 12 part (i)

hi,

I am incorrectly calculating the EPV(Death Benefit) compared to the solutions and cannot work out why.

The DB is a whole life assurance to life aged 45, with SA of 150,000 payable immediately on death. Bonuses of 1.92308% vest immediately. Use Am92@6%.

I understand the bonuses and discount factors of 6% cancel out, thus leaving a discount rate of 4% so we end up with:

150,000 x A(bar)(45) @4%.

Which can then be further simplified to:
150,000 x ((1.04)^0.5) x A(45) @4%

However in the solution it is
150,000 x ((1.06)^0.5) x A(45) @4%

I don't understand why this is,

surely; A(bar)(x) @4% = ((1.04)^0.5) A(x) @4%

why would you instead use an interest rate of 6% instead of 4% to turn this into a payable at YE Assurance, when it is being valued using i=4%?


please advise

Thanks in advance

 

Slightly correction required
If DB is paid immediately then multiply by (1.06)^0.5
Now SA increase compound @ 1.92308 % so ..
(150000/1.0192308) * (1.06)^0.5 * A(45) ( A(45) is @ 4%)

 

Okay that makes sense, but why could you not alternatively choose to 'not make that immediate correction' and instead valuate it as a payable immediately Assurance to begin with and then correct it to a payable at YE in the final step :

DB =150,000[(v^0.5)(1.0192308^0.5)q45 + (v^1.5)(1.0192308^1.5)p45 q46 +.......] @6%

=150,000[(v^0.5)q45 + (v^1.5)p45 q46 +.......] @4%

= 150,000 [A(bar)(45)] @4%

= 150,000 ((1.04)^0.5) A(45) @4%

I understand the method you have said in your reply but I don't understand what is incorrect about this approach, apart from the fact it gives the wrong answer

 

Increase is annually.
DB = \(150000 \) \(\left[v^{0.5} 1.0192308 q_{45} + v^{1.5} 1.0192308^{2} p_{45} q_{46} + ...\right]\) v @ 6%
now multiplying by \( 1.06^{0.5}\ / 1.0192308\) cancels non integer powers of v inside the square bracket and makes it ...
\(\left(1.0192308/1.06\right)..\left(1.0192308/1.06\right)^{2}...\)
so finally
\(\left(150000 / 1.0192308 \right)\) \(1.06^{0.5}\) \(\left[ v q_{45} + \left(1.0192308/1.06\right)^{2} p_{45} q_{46} + ...\right]\)
(150000/1.0192308)(1.06)^0.5 A45 (@ 4%)

 

Okay I see - I assumed the bonus increase could be considered to be increasing continuously throughout the year, hence you could evaluate at the half year point of each year e.g. evaluating the HY1 bonus as ((1.0192308)^0.5).

I am sure I have seen questions where expenses inflate at for example 1.0192308% pa, and we can evaluate these at the half year point, i.e. (Expense x (1.0192308^0.5))

So can such bonuses can only be evaluated at the year end point ? Whereas inflation applies to expenses can be evaluated at HY? Or is there some specific wording in question to look for.

Thanks

 

As per current study material, bonus are increase either at constant rate or compound
Also please note that some time examiner tells time of inflation effect. i.e immediate or at time of renewal. This gives two different answers.
To calculate expenses with inflationary effect same principle applies ( i.e annuity function calculated @ i')