Actual interest rate calculation question

masonic

zagfles said:

In this context, "luck" is about how much you win, which is absolutely quantifiable. But that's a semantic argument that there's no point getting into, if you disagree so be it. 

But your maths is completely off above. Your PP was based on a full holding, so how do you reckon there's only 1.2 million draws in 100,000 years? There's 50000x12x100000 = 6x10^10 ie 60 billion !!

So based on total PB holding of as you say above of about 125 billion, and 2 £million jackpots per draw, the chances of winning the jackpot per draw is about 1 in 62.5 billion. So in 60 billion draws, someone with average luck (however you define it) would win a jackpot. And the median for wins of the other prizes will be bang on the mean. 

https://www.moneysavingexpert.com/banking/savings/premium-bonds/#tips-3

I don't personally agree or disagree with the definition of this term. I am just tailoring my language to be consistent with that used elsewhere in these parts. I do the same when making a hard distinction between 'saving' and 'investing', and several pieces of MSE-jargon that don't necessarily get used the same way elsewhere. The terminology of average luck gets used frequently on the forum in relation to premium bonds, so I didn't think it would need further explanation, but hopefully now the meaning is crystal clear.

You are correct that I goofed over the 'draw' terminology. It was late and I conflated the different meanings, then didn't follow through to any actual calculations. I probably would have spotted the issue if I had. There is a monthly prize draw (in NS&I's terminology), so 12 of these per year and therefore 1.2 million over 100,000 years. In each prize draw, as you say, only two of the ~125bn bonds will win the jackpot, so the probability of not winning the jackpot in one draw are approx 124,999,999,998/125,000,000,000 per bond held. The probability of not winning the jackpot in a single monthly draw with a full holding is 124.9999bn/125bn (100,000 chances per monthly draw) and the probability of not winning in 1.2 million monthly draws is (124.9999/125)^1.2m, which is a little over 38% according to Wolfram Alpha. So agree you are more likely to win the jackpot at least once than not and the median would include a single jackpot win in 62% of 100,000 year periods. So better than a coin flip, but far from certain.

My gut feeling about it being more effective to converge on the mean through capturing a greater proportion of the bonds in issue was, however, wide of the mark. An equivalent scenario would be 100,000 people with full holdings forming a syndicate and entering 12 monthly draws. This would give them a probability of not winning a jackpot prize of 120/125 = 24/25 per draw and so 24/25^12 = 61%, so a little bit less likely than a single individual over 100,000 years.

zagfles

masonic said:

zagfles said:

In this context, "luck" is about how much you win, which is absolutely quantifiable. But that's a semantic argument that there's no point getting into, if you disagree so be it. 

But your maths is completely off above. Your PP was based on a full holding, so how do you reckon there's only 1.2 million draws in 100,000 years? There's 50000x12x100000 = 6x10^10 ie 60 billion !!

So based on total PB holding of as you say above of about 125 billion, and 2 £million jackpots per draw, the chances of winning the jackpot per draw is about 1 in 62.5 billion. So in 60 billion draws, someone with average luck (however you define it) would win a jackpot. And the median for wins of the other prizes will be bang on the mean. 

https://www.moneysavingexpert.com/banking/savings/premium-bonds/#tips-3

I don't personally agree or disagree with the definition of this term. I am just tailoring my language to be consistent with that used elsewhere in these parts. I do the same when making a hard distinction between 'saving' and 'investing', and several pieces of MSE-jargon that don't necessarily get used the same way elsewhere. The terminology of average luck gets used frequently on the forum in relation to premium bonds, so I didn't think it would need further explanation, but hopefully now the meaning is crystal clear.

You are correct that I goofed over the 'draw' terminology. It was late and I conflated the different meanings, then didn't follow through to any actual calculations. I probably would have spotted the issue if I had. There is a monthly prize draw (in NS&I's terminology), so 12 of these per year and therefore 1.2 million over 100,000 years. In each prize draw, as you say, only two of the ~125bn bonds will win the jackpot, so the probability of not winning the jackpot in one draw are approx 124,999,999,998/125,000,000,000 per bond held. The probability of not winning the jackpot in a single monthly draw with a full holding is 124.9999bn/125bn (100,000 chances per monthly draw) and the probability of not winning in 1.2 million monthly draws is (124.9999/125)^1.2m, which is a little over 38% according to Wolfram Alpha. So agree you are more likely to win the jackpot at least once than not and the median would include a single jackpot win in 62% of 100,000 year periods. So better than a coin flip, but far from certain.

Yup. Obviously holding PBs for 100,000 years is theoretical nonsense but it was just to illustrate the point that quoting a return for "average" (meaning median) luck of 3.85% is very specific to the time period they're held for, in your case it was a year.

But over 10 years "median luck" would return more than 3.85% because higher value prizes on a full holding become more likely, so as the time period held gets bigger the median return gets bigger. 

masonic

zagfles said:

Yup. Obviously holding PBs for 100,000 years is theoretical nonsense but it was just to illustrate the point that quoting a return for "average" (meaning median) luck of 3.85% is very specific to the time period they're held for, in your case it was a year.

But over 10 years "median luck" would return more than 3.85% because higher value prizes on a full holding become more likely, so as the time period held gets bigger the median return gets bigger. 

It would be fair to state that over a longer period like 10 years, the median would be a little more than 87.5% of the mean prize rate over the holding period. But within a realistic timescale, the chance of winning a higher value prize (£1m to £5,000) that make up 10% of the fund is rather low (it's about evens over 50 years and highly weighted towards £5k prizes). So a dedicated holder perhaps should use a median in the low 90s% of the prevailing prize rate. That would edge up the median rate just north of 4% if the current prize rate remained static over the whole period. Of course the prize rate does not remain static, and the distribution of prizes is fiddled with regularly, along with more occasional meddling with the winning odds, so a long term % return figure would be nonsensical. 

In the end I settled for a year as a compromise, since it converged enough to be a meaningful comparison, and someone with an interest in the comparison is probably comparing because they are an active saver who would consider moving out of PB if other savings options are more attractive, which they frequently are. I took an interest in premium bonds during the period they were offering median returns well above the derisory savings rates of the past. At that time, NS&I were not being so responsive to changes in the savings market, so you could consider a shortish holding period of a year or so. More recently, they seem to have been tracking the savings market more closely, so I suppose a longer term median return expressed as a percentage of the prize rate could be meaningful where it wasn't before.

zagfles

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...

masonic

zagfles said:

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.

From the 100,000 1-year simulations I did yesterday, the median return for a couple each with a full holding over a year was 3.92% with 25th and 75th percentiles at 3.42% and 4.47% respectively. So the median return was very close to breaching 4%. Of course stochastic methods like mine will generate slightly different results each time they are used. That said, it suggests a single holding over 2 years would have a median return of about 4%, since time seems to be ever so slightly better than holding size at improving returns. But the curve flattens out considerably, given the median would reach 4.4% only after tens of thousands of years.

slinger2

My "thought experiment" about this:

Consider a syndicate of 600 people with the maximum holding of £50k. This is roughly equivalent to one person holding the maximum for 50 years and nothing changing.

So this syndicate has 30 million PBs and, with winnings at an annual rate of 4.4%, can expect to win £110,000 each draw (30 million*4.4%/12).

As noted above, the winnings from the lower and medium prizes should be pretty close to the expected value. This is 90% of the winnings, so £99,000. The question then is about the other 10% (£11,000) in higher prizes.

Now 2% of the prize money goes to each of the £5k, £10k, £25k, £50k and £100k prizes (I'm ignoring the £1 million prizes). That's £2,200 for our syndicate. So ballpark-wise you'd think the chances of winning these prizes was 44% (2200/5000), 22%, 8.8%, 4.4% and 2.2%. This is all very approximate since it ignores the fact that you might win 2 or more of these prizes. These add up to about 80%, so clearly there's a fair chance of not winning any of these prizes. Anyway seems to me that the median prize the syndicate will get (in higher prizes) is £5k.

So I'd say that the median winnings for the syndicate would be about £104,000, which works out at about 94.5% of the expected £110,000, and so someone having the maximum holding for 50 years could hope for that sort of return.

EthicsGradient

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...

I don't think you can add on the whole £500 to the median, when the chance of winning £500 is only just over 50% (I make the chance of winning £500 or over for a max holding couple over a year to be 52%, and 41% for £500 specifically). If the chance of winning £500 were just under 50% for a period, you wouldn't then say "don't add anything on at all". It'll be a gradual rise as the chances increase as the period lengthens, not a step change.

If you take the sum of all prizes of £100 or under, over a year, you get the 3.52% rate. If you take the sum of all prizes of £500 or under for a year, you get 3.78%; for £1,000 or under, 3.96% (and there's only a 19% chance of a max holding couple getting £1,000 or over).

masonic

slinger2 said:

So I'd say that the median winnings for the syndicate would be about £104,000, which works out at about 94.5% of the expected £110,000, and so someone having the maximum holding for 50 years could hope for that sort of return.

FWIW, I've run some quick simulations to test out different time periods...

1 year holding period (100k simulations):
full holding: 3.85% (87.5% prize rate); couple with two full holdings: 3.92% (89.1% prize rate)

5 year holding period (50k simulations):
full holding: 3.98% (90.5% prize rate); couple with two full holdings: 4.02% (91.4% prize rate)

10 year holding period (20k simulations):
full holding: 4.03% (91.6% prize rate); couple with two full holdings: 4.07% (92.5% prize rate)

50 year holding period (10k simulations):
full holding: 4.14% (94.1% prize rate); couple with two full holdings: 4.20% (95.5% prize rate)

Although I had to reduce the number of simulations over longer time periods to fit into reasonable computational time, I'd say that's pretty good agreement.

EthicsGradient said:

I don't think you can add on the whole £500 to the median, when the chance of winning £500 is only just over 50% (I make the chance of winning £500 or over for a max holding couple over a year to be 52%, and 41% for £500 specifically). If the chance of winning £500 were just under 50% for a period, you wouldn't then say "don't add anything on at all". It'll be a gradual rise as the chances increase as the period lengthens, not a step change.

I think when using a median instead of a mean, you would expect a step change. If there are an odd number of participants, the one representing the 50th percentile must win one of the possible combination of prizes (i.e. an actual outcome). This is why those with small holdings held over a short period of time would have a median return of exactly zero.

zagfles

EthicsGradient 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...

I don't think you can add on the whole £500 to the median, when the chance of winning £500 is only just over 50% (I make the chance of winning £500 or over for a max holding couple over a year to be 52%, and 41% for £500 specifically). If the chance of winning £500 were just under 50% for a period, you wouldn't then say "don't add anything on at all". It'll be a gradual rise as the chances increase as the period lengthens, not a step change.

If you take the sum of all prizes of £100 or under, over a year, you get the 3.52% rate. If you take the sum of all prizes of £500 or under for a year, you get 3.78%; for £1,000 or under, 3.96% (and there's only a 19% chance of a max holding couple getting £1,000 or over).

It is a step change - that's the definition of the median. If you lined everyone up in order of their winnings then it's the person who's in the middle whose winnings is the median winnings. 

If there's a 52% chance of winning a £500+ prize, out of say 1,000,000 people, 520,000 will win it and 480,000 won't. The person in the middle will be the 500,000th and therefore will have won £500. 

In the case of PBs it's a bit more complicated because there are other prizes, but with a full holding small prizes as above are going to be fairly evenly distributed according to the odds, and bigger prizes are going to be too rare for the middle person to have won any. 

That's the whole reason why the median is lower than the mean, the step change when bigger prizes start having an effect. 

subjecttocontract

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.