Big trouble with END/BGN mode calculations

I’m confused when it comes to selecting values for “n” when you’re doing PV/FV calculations in begin/end mode. As I understand it: a) When working in END mode, your calculator will spit out a PV value that is as of one period before the period of the first payment. b) When working in BGN mode, your calculator will give you an FV value that is as of one period after the period of the last payment. Now here’s the trouble I’ve been having with these two, respectively: 1. END MODE: The way I understand it, the practical application of the above point is that if you are calculating, say, $100 to be received (at the ends of) t3 and t4, and then you want to find out the PV of that amount today, you would first calculate PV as of t3, and then discount back from t2 and not t3 (n = 2, not 3) back to t0. You would do this because the PV you calculate as of t3 is actually as of t2. However, lets say you’re getting a payment of $300 to be received at the end of 3 years from now - according to a Qbank question that involves exactly this situation, the n used is 3, NOT 2. If we are saying that the PV we calculate is already as of one period before the payment, shouldn’t we discount back two periods here too? Why is n=2 in the example above, and not here? 2. BEGIN MODE: Similarly, the application here is that if you are making a payment of $50 today and in the beginning of the next two years, you are making three payments in T0, T1, T2. THE FV VAL of that will be as of T3, not T2. If you were to bring this amount another two years into the future, your n would be 1, not 2. However, lets say you make 10 equal annual deposits of $1,000 into an account earning 9%, starting today. And you wanted to know how much you’d have in 20 years. First, you’d get the FV of the 10 payments. Then, I would think that *given how the FV val is one period FORWARD from the last payment*, you would use the previously calculated FV as the PV, and bring the amount forward another *9* years. However, in this exact question in the book, they discount forward 10 years. So, why doesn’t the rule we stated just above being applied here? I’m sorry for the lengthy question but I’ve had trouble with this pretty much from day 1! The cliffnotes version is: can anyone explain exactly when and why the “n” values are changed when doing END/BGN mode problems? Thanks, I appreciate it!

you are over-thinking this… Slow your roll. When you are discounting any series of payments, you are concerned with the timing. Traditionally, and if not stated - your going to leave your calculator alone and discount your stream with the payments coming at the end of the period. If you read a question that says, BEGINNING of period. or some language to indicate this. then you want to change your calculator mode to begin of period mode. That’s all you really need to know. You don’t need memorize the transformation or that nasty formula for converting an beginning of period annuity stream to end of period. Waste of brain power. Just be on the look out for the timing of the cash flows on any stream you need to value.

Infinitesun is right, it’s really just about identifying which direction you are moving on the timeline. Just pinpoint your variables and work your way backwards/forwards. Typically your just looking for a few key words in the question. Keep practicing it will come naturally after awhile.