Wikipedia:Reference desk/Archives/Mathematics/2018 June 11

= June 11 =

Abelian groups with trivial tensor powers
If A is a divisible torsion abelian group, then the tensor product $$A \otimes_\mathbb{Z} A$$ is trivial. Conversely, if A is an abelian group such that $$A \otimes_\mathbb{Z} A$$ is trivial, must A be a divisible torsion abelian group? Also, is there an abelian group A such that $$A \otimes_\mathbb{Z} A \otimes_\mathbb{Z} A$$ is trivial but $$A \otimes_\mathbb{Z} A$$ is nontrivial? GeoffreyT2000 (talk) 00:52, 11 June 2018 (UTC)
 * For the first question, pretty sure the answer is no. Let S be a subset of Z-{0} and say A is S-divisible if for any a∈A, n∈S, there is b∈A so that a=nb. Similarly, say A is an S-torsion group if for any a∈A there is n∈S so that na=0. You get the usual definitions of divisible and torsion groups by taking S=Z-{0}. It's easy to generalize the above to get that if A is an S-divisible S-torsion group then A⊗A is trivial. It's then a matter of finding a set S and a group A which is S-torsion and S-divisible, but not torsion and divisible. Taking S={2k: k≥0}, one such group is A={j/2k} / Z. The second question seems harder. --RDBury (talk) 15:35, 11 June 2018 (UTC)

calculation of returns
My question is pretty basic for a distinguished forum like this. However, I am seeking your help because I have not been able to find the answer. What is the method to determine the annualized return on investment in ULIPs? For e.g. Rupees twenty thousand has been invested since 2005 till 2018 (Total 14 installments ,out of which 11 have been made).Rs 20000 every April from 2005 till 2011. Payments of Rs 20,000 each in Jan 15, Nov 16, Jul 17 and May 18. A total of Rs 2,20,000 over 13 Years( Apr 2005- May 2018).Current value is Rs 452732.09. What is the annual rate of return? If someone can throw light on it, I would be grateful. PS: I have no idea how MS Excel operates. Sumalsn (talk) 04:38, 11 June 2018 (UTC)

The basic idea is to treat the value of the payment stream as the sum of a truncated geometric series. Say the rate of return is R. Let's simplify the problem slightly so there's just 1 payment per year, in May. So the current value of the May 2018 payment is 20,000 (since it just happened), for the May 2017 payment it's 20,000*(1+R), for the May 2016 payment it's 20,000&middot;(1+R)2, etc. Let's let Z=1+R. So the total is $$452732 = 20000\cdot(1+Z+Z^2+...+Z^{10})$$ Let $$U=1+Z+\cdots+Z^{10}$$, so $$UZ=Z+Z^2+\cdots+Z^{11}$$.

That means $$U-UZ = U\cdot(1-Z)= 1 - Z^{11}$$ since all the other terms cancel out (like a telescoping series), so $$U=(1-Z^{11})/(1-Z) = 452732/20000=22.6366.$$

Numerically solving for Z we get about Z=1.13675 or R=13.675% which is pretty good if there's not a lot of inflation in Rs (I don't know if there is). 173.228.123.166 (talk) 19:08, 11 June 2018 (UTC)
 * That's assuming that the payments were made on a regular schedule, but in this case they weren't. You could assume regular payments to get an approximation, but to get a more exact answer the equation you need to to solve is more like 1+Z+Z2+Z3+Z7+Z8+Z9+Z10+Z11+Z12+Z13=22.6366. Wolfram Alfa gives a value of Z=1.09725 or R=9.725% which is somewhat less; this makes sense because the length of investment is 13 years, not 10 years, and most of the amount was added near the beginning. --RDBury (talk) 20:13, 11 June 2018 (UTC)


 * Ah good point, I didn't notice some of the payments were spaced that far apart. Anyway OP, that's the general method of writing down the equation to solve for Z.  When it's complicated enough there's no closed formula for calculating the solution directly.  You instead use a root-finding algorithm to find an approximate solution.  Those algorithms are built into financial calculators and tools like Excel for doing this type of thing, and simple ones like the bisection method are easy to implement yourself, if you like writing computer code. 173.228.123.166 (talk) 21:26, 11 June 2018 (UTC)


 * I'd also add: depending on the type of investment, the rate of return might have fluctuated over time (2008 financial crisis etc). The averaged-out calculation tells you something, but looking at year-by-year returns might tell you more. 173.228.123.166 (talk) 21:35, 11 June 2018 (UTC)
 * I am deeply grateful to all those who have taken pains to answer my query. Thanks once again for the attention with which my question has been answered
 * Sumalsn (talk) 05:33, 13 June 2018 (UTC)

Suppose I payed you 100 Rs, and one year later you payed me 230 Rs, and yet one year later I payed you 132 Rs, and so we are even. What is the annual rate of return? Bo Jacoby (talk) 12:11, 16 June 2018 (UTC).
 * Belated courtesy link: Internal rate of return, which is what all the calculations in this thread are. Loraof (talk) 16:01, 16 June 2018 (UTC)