# When does it make sense to buy an annual rather than a monthly pack?

My TV provider offers me a monthly package, but lets me prepay for a longer duration at lower price. Here are the options:

• ₹100 monthly
• ₹567 semi-annually
• ₹1,098 per annum
• ₹2,100 biennial

All these are prepaid, not postpaid.

Assume that I have an investment available that generates a post-tax return of 12%. Which of these options works out to be the cheapest, and in what order do they rank?

Put differently, how much return does my investment need to have for it to be profitable to go for (say) the monthly pack over the annual one?

I understand that by prepaying, I will lose money if I decide to switch providers before the duration I've prepaid for is up.

PS: I googled for "NPV calculator" and "discount rate calculator" but they seem to handle a different situation from this one — they assume that you're investing a certain amount of money in (say) a factory that will generate regular cash flows. That's not the case here. Besides, I never know whether to enter numbers with a positive or negative sign, and they then tell me that at least one of the numbers must be negative, etc.

• You may also 'lose' money by prepaying if the rate drops over the next few years - but I don't know how likely that is. Commented Feb 21, 2016 at 10:11
• That never happens in India, with high inflation. The most we can expect is that prices remain the same. Commented Feb 22, 2016 at 4:38

A quick Excel calculation tells me that, if you are earning a guaranteed post-tax return of 12% in a liquid investment, then it doesn't matter which one you pick.

According to the following Excel formula:

=PV(0.12/12,24,-100)

You would be able to invest ₹2,124 now at 12% interest, and you could withdraw ₹100 every month for 24 months. Which means that the ₹100/month option and the ₹2100/biennium option are essentially the same.

This, of course, is depending on that 12% guaranteed return. Where I come from, this type of investment is unheard of. If I was sure I'd still be using the same service two years from now, I would choose the biennial payment option.

You asked in the comments how to change the formula to account for risk in the investment. Risk is a hard thing to quantify. However, if you are certain that you will be using this service in two years from now, you are essentially achieving 13% in a guaranteed return by pre-paying your fee. In my experience, a 13% guaranteed return is worth taking. Trying to achieve any more than that in an investment is simply a gamble.

That having been said, at the amount we are talking about, each percent difference in return is only about ₹22. The biggest risk here is the fact that you might want to change services before your term is up. If these amounts are relatively small for you, then if there is any chance at all that you will want to drop the service before the 2 years is up, just pay the monthly fee.

• I changed the question to use 12% rather than 14%. The former is more realistic. But it's still not guaranteed — it's the return I expect to get from a mixture of equity and debt investments, of the sort I generally make. How would you modify the NPV calculation to take the risk into account? If 100/month equals 2100/biennium in the presence of a risk-free investment that returns 12%, how much discount should I demand in the presence of an investment that has some risk and is expected to return 12%? Should I use the rate of return from a debt investment, like 8%, in the NPV calculation? Commented Feb 21, 2016 at 9:37
• I edit your answer to use 12%, so that it remains valid for the updated question. Commented Feb 21, 2016 at 9:38
• How do I invert this? What's the effective interest rate if I'm being offered a ₹1000 annual plan in lieu of a ₹100 monthly plan? In other words, what interest rate should I derive from my risk-free investment for these two plans to be equally good? Commented Oct 9, 2016 at 3:33