Hi All.
Here is a real life math puzzle just for a little fun and to check out some feedback. 
Check out the image below and fill in the (answer) the four missing answers to the projections. I.e. What are the answers to ? ANS 1, ? ANS 2, ? ANS 3 and ? ANS 4?
The only other info I can give is the that interest is compounded daily.
I have also been told that a person can use this website to help find the answer:
This depends on the daycount convention, there are lots of them. Let’s assume that act/act is used, so a year has 365 days. So there are 3x365 = 1095 interest periods.
The Interest rate per day is 5.45/365 = 0.0149315068493151
Using the formula K(n)=K(0)*(1+p/100)^n should give a result of 1177,61$ in column B (Sorry, don’t have pounds on my keyboard)
Annual payment (without compounding) will net 163,50, so the compounding effect is not that great.
------------------------- edit
Corrected my formula, and the symbols are german.
K(n) = Capital at maturity
K(0) = Capital invested (initial deposit)
p = Interest rate per Interest period
n = Number of interest periods.
Thank you for participating … 
So, your answer for ? ANS 2 is ÂŁ1177.61.
I will wait for some more people to try answering the questions before showing what I think the answer is. (I used that calculator, which I hoped worked correctly.)
Out of curiosity, did you have an answer for ? ANS 4?
4: 1173,02
See chapter " Periodic compounding"
Actually the formula to convert from yearly to daily interest is (1 + q)^(1/365) - 1, which gives a daily interest rate of 0,014539885%
The ÂŁ1,000 after 1095 days would then give ÂŁ1,172.57 (but no need to calculate daily, obviously the result is the same with an interest rate of 5.45% per year and a duration of 3 years): K x 1,0545^3 = K x 1,014539885^1095 
Assuming the 5,32% per month is actually per month, the same formula gives ÂŁ6 462,35 after 3 years (or 36 months)
To summarize:
ANS1 = ÂŁ172.57
ANS2 = ÂŁ1,172.57
ANS3 = ÂŁ5,462.35
ANS4 = ÂŁ6,462.35
By the way, banks would get lower results, as they work with years of 360 days only…
Also, thanks for responding …
It is interesting to see a different result.
I was surprised by both your results (for different reasons), but will PM with a response so as to give others a chance to respond. But to say for now…
Your annual interest was not what I made it using the daily interest calculator, and so am interested in this some more. The monthly interest appears to be extremely high to what I was expecting, but I suspect that is because it is not clear from the table format… which is part of the problem I also face.
I’ll PM you now.
My apologies, my formula for compounding interests was wrong, I corrected it.
Your daily interest is completely different from mine, I simply divided the 5.45 through 365. Where did you get this formula?
Nobody will pay 5.32% per month. (Maybe somebody will offer this, but if you ever will see this money is a different question). I think the monthly interest rate is a bit lower because the investor can terminate the contract early.
Mathematically
For a yearly interest r, after one year we will increase the initial deposit by multiplying it by (1 + r).
For a daily interest r’, after one year there would have been 365 increases by (1 + r’) i.e. the capital would have been multiplied by (1 + r’)^365.
To calculate the equivalent daily interest rate r’, we will then need to resolve (1 + r’)^365 = (1 + r) <=> (1 + r’) = (1 + r)^(1/365) <=> r’ = (1 + r)^(1/365) - 1
Not easy to write equations on a forum, I hope it’s understandable though.
I just wanted to clarify that the second monthly figures were supposed to calculate a return as if the compounded interest for the month had been paid to the customer, leaving a lower capital to start growing again the following month until the next monthly pay out
In brief, the annual assumes compounded interest for each year before a payout, the monthly each month before a payout. The overall investment lasts three years.
Logically, we can assume that over the course of the total three years investment, the monthly pay out arrangements should provide a slightly lower return (reflected in the lower rate) than the annual payout, as we have the more money compounding over the course of the year if left there compared to being released in stages in previous month’s.
I know @4760 has another value of ÂŁ1,168.24 (@ 5.32%) for the year return at monthly payments. This compares to the ÂŁ1,172.57 (@ 5.45%) year return on a year payout.
I am just trying to grasp some of the maths you use here. Please correct my assumptions and errors below.
From what I have searched, I believe <=> means “implies”, and so am I correct that the above line is a way of saying one formula step implies another?
Using the formula above, I tried doing these, so why do I have a different result?
What BODMAS have I done wrong?
looks like rearranging the equation (aka. implies in a mathematical sense)
(1 + r')^365 = (1 + r)
(1 + r') = (1 + r)^(1/365)
r' = (1 + r)^(1/365) - 1
Aha, I think I may have found my mistake … it seems my rate percentage value should be used differently to how I thought…
STAGE 1:
STAGE 2:
STAGE 3:
STAGE 4: (I had to multiply by 100 for the percentage thing again)
Yes (rather means “implies in both ways” = “is equivalent to”)
Careful - in Boolean Algebra Implies and is equivalent to are different and have different truth tables.
The truth table for is equivalent to is exactly the same as for Not Exclusive Or -
TR
Yes, but the sign was <=> (is equivalent) and not => only (implies)
Mathematically
“The natural numbers are made by god, any other numbers are made by the devil …” 
Your calculation is correct. There is a simple check:
1000 + 1000 * 5.45 / 100 = 1,054.50 (Capital after one year)
With your daily interest: 1000*1.00014539885^365 = 1,054.50
Under the premise the interests are payed every year (reducing the capital to the initial deposit this way), the total interest after 3 years should be 163.5. Which is quite different to the given figure of 172,57. The latter would be the result, if there is no interest payout at all.
So I’m puzzed, what the correct terms are … 
I think I may have finally understood why and how these figures are confusing, but I am waiting for @4760 to confirm my latest understanding in a PM.
Suffice to say, I currently believe that these so called rates are all in fact relating to just compounded interest, even though the two different rates imply they are for annual or monthly payouts.
So, what this table actually shows is the same A.E.R. of 5.45% compounded interest, as if the account was being paid at maturity, rather than showing two rates for different types of payout terms.
Basically, I cannot see the recalculated monthly interest as serving any purpose in this sense, as it is simply recalculating the same amount of compounded interest 5.45% but as if calculated being monthly returned or annually returned (to the same savings account). I mean, what is the point of that?
Therefore, I believe it to be deliberately misleading, as the normal point of showing two rates (one for annual and one for monthly) is to show the return you would receive for taking either an annual payout or a monthly one.
But, to reiterate, the illustration was just showing the same On Maturity final payment, but playing around with the figures so they could make a table to show it in annual and monthly % terms, which, in my opinion was aimed at confusing people who were looking for comparisons for if they took annual or monthly payouts. Those, however, are actually NOT part of the illustration.
For all those following, here are the answers/figures given at the bank in question:
It now appears that the “Monthly Interest” figures are simply a recalculation of the exact same 5.45% “Annual Interest” for an account that is selected to pay out on maturity. I can see no point in doing this, except that the illustration being setup this way, can confuse people who are looking for the differences in the annual and monthly returns as if receiving monthly or annual payouts, which this account can also be setup to do. I hasten to add that I have never (to date) ever seen any other bank illustrate such a similar product this way.
I’m quite sure you saw price tags at £9.99 and the vendor telling you “it’s a bargain - you see, less than £10!” - it’s the same here: in order to make you opt for the monthly plan, you see a slightly better return on investment, but it’s only a rounding artefact. In the end, you’ll get 5.45% per year, and the bank will keep whatever profit they got above this percentage. On the other hand, you’ll still get something, so it’s not a bad deal per se, except you won’t be able to use the money for three years, and if you’re in finances (which I’m not!) you’ll most certainly get better deals elsewhere.
Thierry,
Your maths was what I needed to solve the problem, and so from that perspective, your information was golden!
I’m not sure how bonds work in France, but in the UK, while the initial investment is locked away (£1000 in this example), interest payout can be claimed either annually or monthly over the course of the term, be it 1 year to five years term.
However, when you apply for the bond, you need to state how you want the interest to be paid back to you, either every year or month of the term, or at the end only (at maturity).
From your maths, we were able to recognise that the illustration they give on their website was only for the On Maturity option. Therefore, anyone who might be looking for an illustration for opting for the annual or monthly payouts until the return of the initial investment may well be confused by this illustration due to it mentioning annual and monthly rates… which we now know only illustrates one of the options a person can take with this account.
It would be interesting to see what the return rate would equate to for receiving either monthly or annual payouts during this term. I don’t think it would lower the rate much, but I still think the bank should have offered the extra info to allow the customer to have an informed choice, as opting for a monthly payout option (and not the monthly compounded On Maturity option as illustrated) would be different.
Thanks again ! Have a great weekend!
It sure is different! But why would they shoot themselves in the foot? I don’t know all bankers, but the ones I had to make deals with didn’t fit in the philanthropist category
