**The Parks plan to take three cruises, one each year. They will take their first cruise 9 years from today, the second cruise one year after that, and the third cruise 11 years from today. The type of cruise they will take currently costs $5,000, but the expect inflation will increase this cost by 3.5% per year on average. They will contribute to an account to save for these cruises that will earn 8% per year. What equal contributions must they make today and every year until their first cruise (ten contributions) in order to have saved enough at that time for all three cruises? They pay for cruises when taken.**

Originally, I figured that I could solve by treating this as a “fund an annuity due” problem. However, we are told that the cruse price will increase 3.5% per year ever year, meaning the cruises taken in years 10 and 11 will cost more than the cruse taken in year 9. So, I solved out for these three prices and got:

Cruise (T=9): $6,814.49

Cruise (T=10): $7,053.00

Cruise (T=11): $7,299.85

Of course, the problem with this is finding the PV of an annuity requires that it be an annuity (i.e. a series of even cash flows) whereas this problem has a series of uneven (i.e. increasing) CF’s. I can’t figure out how to PV these to T=9 – when I do as I normally would, I get the PV for all 3 = 6,814.48. If I sum them up I get 20,443.44. Now, the PV (20,433) becomes the future value – what I need in 9 years and at 10 payments. So, in my calculator (TI BA II) I put: [N]=10, [I/Y]=8, [FV]=20433.44, [CPT] [PMT] and get 1,306. This is close to the answer of 1,353, but obviously it is not correct.

The book solves by getting the FV of each cruise then dividing by the rate of savings to get the PV for TODAY (i.e. T=0) as opposed to the PV when the first outflow starts. For example:

Cruise (T=9): (6,814.49) / [(1+8%)^9] = 3,408.94

Cruise (T=10): (7,053.00) / [(1+8%)^10] = 3,266.90

Cruise (T=11): (7,299.85) / [(1+8%)^11] = 3,130.78

The sum of all 3 at T=0: 9,806.02

From here I set calculator to [BEG] (because all payments are made at the beginning of the year) and I input [N]=10, [I/Y]=8, [PV]=9806, [CPT] [PMT] = 1,353, the correct answer.

*Why can’t I solve the way I originally did? I believe that it is because my method is not taking into account the 8% return on the account during the years of the cruise, and by dividing the inflation rate by the savings rate, you get an adjusted rate of return. Is this correct?*

*Why can’t I solve for regular annuity fund problems like this? My guess is because unlike an ordinary “fund an annuity due problem” which assumes a 10% rate of growth during savings and during the life of the annuity, this problem has rates of inflation affecting the cost and a different rate for savings. Is this correct?*

Thanks!