1999 April exam Question 19

I keep getting the wrong answer for the DPP and i can not work out what i am doing wrong, the solution says the answer is 11 years and 9 months but when i calculate the NPV at this time it is clearly :

- (200 + 600a_(0.5)^(12) + 15(v^(1/2))a_(11+(3/12))^(12)) + 96(v^(1/2))(adue_(11+3/12)^(12))

= - (200 + 290.283 + 89.6922) + 579.480833

= -0.4944 ????????

The only way i can see this is possible is if you include the rental payment due at the start of the month at time t=11+(9/12) but can you actually do this

 

I think you need to include the income at t=11 9/12 for the start of the next month (which you get an instant after the expense you did include).

This is the most sensible approach. The (discounted) payback period is the time to recover your money. As time goes by, you'll collect the rent payment at the start of the month and stop. It seems silly (not that I'm saying you'll never find a case where the official method for some actuarial tasks is silly) to not stop there but wait a month, make an expense payment and then check whether to stop.
At a stretch, the only reason I could think of doing it like that is possibly if you did it by an accruals basis for accounting (which typically ignores time value of money anyway)

I believe there is a typo in the report where they have the A(t) as the first expression omits this "final" rental payment but it is included thereafter.

Re doing it quickly, firstly they somewhat spell out what they want you to do (and you should do it that way), ie find the value at t=11.5 and work from there. The recursion approach to add other payments is quite useful.

If this approach wasn't mandated I might have broken it down into a PV of the expense plus a income of 6750 per month.
Then either work in years and see what your a_n needs to be and scroll down the 12% tables to find a close enough value.
Otherwise you could solve the value explicitly, but remember your answer always should be right after a specific income payment (since this is what reduces the amount outstanding).

Sometimes working in months (or quarters, etc) might be easier.
No one way to do things fast, it all depends on what you need to get and what you start with, then decide a way to do the whole thing efficiently. Eg you might choose to spend "too long" on one part so that it speeds up the next.

 

Thank you for your reply didster, i realise now what is going on, but technically shouldnt the DPP be stated as 11 years 9 months and one day, since time does actually have to tick over into the next day until the payment is made?

 

Lol, well perhaps, but perhaps not.

I wouldn't go into that much detail. In fact, in real life and where there is uncertainty, I might very well round to nearest quarter year.

We don't know when the payment is made. In practice the payments aren't going to be paid on the same day. You may say start/end of month, but this may be intentionally paid a couple of days early/late and then weekends/holidays may come into play.