Actual interest rate calculation question

masonic

subjecttocontract said:

Let's not loose sight of the other facts which include:-
* Its a bit of fun...... and I suspect for many bond holders the fun factor each month far outweighs the return.
* There are plenty of wealthy bond holders who will view their £50K holding as of little importance. The £50K represents such a small part of their overall wealth that they won't loose any sleep about its return.

It's also important not to lose sight of the tax treatment, which makes the rates we've been discussing somewhat competitive for basic rate taxpayers, but a very good deal for higher rate taxpayers. So the wealthy bond holders are probably getting a good deal unless they are very unlucky.

slinger2

zagfles said:

Obviously the prize rate can and will change over the longer term but so can rates in the short term, so any projection can only be based on current rules/rates. 

The median return over set periods particularly shorter periods can be lumpy, in fact I reckon the median return for a couple with the full holding over one year will be over 4%. PB prizes are heavily biased towards lower prizes, so with 2x the full holding you can pretty much expect to get the prize rate % of the lower prizes. The current prize rate is 4.4% and 80% of the prize fund is lower value prizes. So on £100k you can expect to win 3.52% ie £3520 in small prizes.

But it'd be impossible to win £3520 because prizes are in multiples of £25, therefore the median will likely be £3525. In fact not "likely" but almost certainly. 

But in addition, there is more than a 50% chance of winning a £500 prize in a year. So the median couple with full holding will win £4025, ie a return of 4.025%. 

I'm sure the graphs will smooth this lumpy effect, over 2 years the median would be 3.775% as the median couple would still only win one £500. Then it'd go up a bit over 3 years...

You need to be careful here. Seems that (close to) 6% of the prize money goes into the £500 prizes, so you'd expect 6%*4.4%*100000 = £264 in such prizes over the year. That might make you think that the chances of winning a £500 prize was 264/500 = 52.8%. However that's not true since you might win 2 or more of these prizes. In fact, with such a large sample, the winnings will tend to follow a Poisson distribution with an expected value of 0.528 (as I remember). Plugging the numbers in, tells you that the chance of winning 0 £500 prizes is 59%, 1 prize 31%, 2 prizes 8.2%, 3 prizes 1.4%, etc. So the most likely outcome is 0 wins, even though the mean is over half a win.

EthicsGradient

No, a step change is not the definition of a median. They have nothing to do with each other. For clarity, I mean a step change when another draw is taken - with respect to time, effectively. After 11 draws, the chances of £100,000 holdings getting a £500 win are 49%; after 12, 52%. The median total winnings will not suddenly change in a step of £500 at that point.

Think of it this way: if you guaranteed all the million people the extra £500, and you had calculated the median had been £3525, then it would indeed become £4025 - everyone moves up by exactly the same amount, and stays in the same order.

But if only 520,000 get it, then the order changes - some people who get it were already in the upper half, and they'll remain there by definition; some people who were in the upper half and don't get it will get overtaken by people who do (some of whom have come from the lower half); some people in the lower half still won't make it into the new upper half because their winnings from the £100 and below prizes was relatively poor, and so on. The exact movement of the median point will depend on the exact shape of the distribution due to the lower prizes.

zagfles

slinger2 said:

zagfles said:

Obviously the prize rate can and will change over the longer term but so can rates in the short term, so any projection can only be based on current rules/rates. 

The median return over set periods particularly shorter periods can be lumpy, in fact I reckon the median return for a couple with the full holding over one year will be over 4%. PB prizes are heavily biased towards lower prizes, so with 2x the full holding you can pretty much expect to get the prize rate % of the lower prizes. The current prize rate is 4.4% and 80% of the prize fund is lower value prizes. So on £100k you can expect to win 3.52% ie £3520 in small prizes.

But it'd be impossible to win £3520 because prizes are in multiples of £25, therefore the median will likely be £3525. In fact not "likely" but almost certainly. 

But in addition, there is more than a 50% chance of winning a £500 prize in a year. So the median couple with full holding will win £4025, ie a return of 4.025%. 

I'm sure the graphs will smooth this lumpy effect, over 2 years the median would be 3.775% as the median couple would still only win one £500. Then it'd go up a bit over 3 years...

You need to be careful here. Seems that (close to) 6% of the prize money goes into the £500 prizes, so you'd expect 6%*4.4%*100000 = £264 in such prizes over the year. That might make you think that the chances of winning a £500 prize was 264/500 = 52.8%. However that's not true since you might win 2 or more of these prizes. In fact, with such a large sample, the winnings will tend to follow a Poisson distribution with an expected value of 0.528 (as I remember). Plugging the numbers in, tells you that the chance of winning 0 £500 prizes is 59%, 1 prize 31%, 2 prizes 8.2%, 3 prizes 1.4%, etc. So the most likely outcome is 0 wins, even though the mean is over half a win.

Yes, although if you use £500+ prizes, the chances are over 50%. There are 76311 £500+ prizes, so the mean number of these prizes won per year with a full couple holding would be 100,000*12*76311/125 billion = 0.73.

However I've spotted another mistake in my PP. You can't just consider small prizes evenly distributed separately to bigger prizes. Because if you start with the assumption that the median small prize win will be £3525, which I'm pretty sure is correct, but then you add a scattering of bigger prizes, this will result in some people who were below the median leapfrogging the mid point and being above the median. This will push the median up, so even prizes which there's no way the median person will win in a year (ie £1000+) will have an effect on the median. 

slinger2

Yes, although if you use £500+ prizes, the chances are over 50%. There are 76311 £500+ prizes, so the mean number of these prizes won per year with a full couple holding would be 100,000*12*76311/125 billion = 0.73.

Perhaps you're still missing my point: Just because the mean number of prizes is over 0.5 doesn't necessarily mean that your chances of winning are more than a half. It would be the case if you could only win 1 prize but in this case it doesn't always follow.

zagfles

slinger2 said:

Yes, although if you use £500+ prizes, the chances are over 50%. There are 76311 £500+ prizes, so the mean number of these prizes won per year with a full couple holding would be 100,000*12*76311/125 billion = 0.73.

Perhaps you're still missing my point: Just because the mean number of prizes is over 0.5 doesn't necessarily mean that your chances of winning are more than a half. It would be the case if you could only win 1 prize but in this case it doesn't always follow.

I got your point, which is why I said "Yes..." 

Lets do it properly then. The chances of winning a £500+ prize is 76311/125 billion per £1 per draw = 0.000000610488

So the chances NOT winning one is 1-0.000000610488 = 0.999999389512. So over 1.2 million draws (100,000*12) the chances of NOT winning a £500+ prize is 0.999999389512^1200000 = 0.48066 

So the chance of winning a £500+ prize is about 52%. 

Nebulous2

subjecttocontract said:

Let's not loose sight of the other facts which include:-
* Its a bit of fun...... and I suspect for many bond holders the fun factor each month far outweighs the return.
* There are plenty of wealthy bond holders who will view their £50K holding as of little importance. The £50K represents such a small part of their overall wealth that they won't loose any sleep about its return.

What do you  mean by- 'won't lose any sleep about its return?' 

One of the points often made is that NS & I is extremely safe.

subjecttocontract

Nebulous2 said:

subjecttocontract said:

Let's not loose sight of the other facts which include:-
* Its a bit of fun...... and I suspect for many bond holders the fun factor each month far outweighs the return.
* There are plenty of wealthy bond holders who will view their £50K holding as of little importance. The £50K represents such a small part of their overall wealth that they won't loose any sleep about its return.

What do you  mean by- 'won't lose any sleep about its return?' 

One of the points often made is that NS & I is extremely safe.

That was a poor choice of words on my part. What I mean is that the wealthy bond holders won't be concerned that their potential 'prize' money may be a little less than the interest they could get elsewhere.