April 2015 exam question 9 - Help

[Salma]

Hi

can someone please help me with April 2015 past paper, question 9.

For the rental income, i chose to work in quarters, so i used an interest rate of (1.12)^.05 -1 = 0.0287 and income of 0.36/4 = 0.09.

So my level annuity was 0.09 * (a-dot dot 4). I then worked in quarters through the calculations for the PV of the income, so my eqn looks like this:

(0.09 * (a-dot dot 4)* v^4) (1+ v^4 * (1+b)^4 + .... + v^16 * (1+b)^16) all @ i = 0.0287

for the 6.8 at time 6, I used the annual int rate (so 6.8 v^6 @12%).

I used 12% for the outgo (-4-0.9v^0.5).

Then, for the final answer, i got:

4.85042 - 3.445 = 0.30827 (1-((1+b)/1.0287))^20)/(1-(1+b)/1.0287))

Looking at the soln, i plugged in b as (1.06835)^0.25 -1 = 0.01666, but the 2 sides dont match up.

Help in working in quarters for this question would be appreciated!

 

[Lucy England]

The denominator (1-(1+b)/1.0287) in your geometric progression on the RHS should be (1-((1+b)/1.0287)^4) as the common ratio is ((1+b)/1.0287)^4, not (1+b)/1.0287.

I think it's much easier to work in years for most of this question. The one-year annuity in the solution can be subsitiuted with a four-quarter annuity without too much trouble, but you're less likely to get muddled up if you stick to years for the rest of the equation of value.

Using a four-quarter annuity give you this alternative equation of value:

\[
\require{enclose}
4+0.9*1.12^{-0.5}-6.8*1.12^{-6}=0.09*1.12^{-1} \ {}\ddot{a}_{\enclose{actuarial}{4}2.87\%} \ (1+\frac{(1+b)}{1.12}+...+\frac{(1+b)^{4}}{1.12^{4}})
\]

Here you're not trying to convert between quarters and years when working out what b is. I think this makes the calculation less confusing!

 

[Salma]

Thank you Lucy this has helped