There’s a mental calculation trick that I use quite a lot at work when dealing with situations requiring compound interest. It's called the rule of 72, and it states that for interest rate $i$, under growth from annual compound interest, it takes approximately $\frac{72}{i} $ years for a given value to double in size. Why does it work? Here is a quick derivation showing why this works, all we need is to manipulate the exact solution with logarithms and then play around with the Taylor expansion. We are interested in the following identity, which gives the exact value of $n$ for which an investment doubles under compound interest: $$ \left( 1 + \frac{i}{100} \right)^n = 2$$ Taking logs of both sides gives the following: $$ ln \left( 1 + \frac{i}{100} \right)^n = ln(2)$$ And then bringing down the $n$: $$n* ln \left( 1 + \frac{i}{100} \right) = ln(2)$$ And finally solving for $n$: $$n = \frac {ln(2)} { ln \left( 1 + \frac{i}{100} \right) }$$ So the above gives us a formula for $n$, the number of years. We now need to come up with a simple approximation to this function, and we do so by examining the Taylor expansion denominator of the right have side: We can compute the value of $ln(2)$:
$$ln(2) \approx 69.3 \%$$
The Taylor expansion of the denominator is:
$$ln \left( 1 + \frac{i}{100} \right) = \frac{r}{100} – \frac{r^2}{20000} + … $$ In our case, it is more convenient to write this as: $$ln \left( 1 + \frac{i}{100} \right) = \frac{1}{100} \left( r – \frac{r^2}{200} + … \right) $$ For $r<10$, the second term is less than $\frac{100}{200} = 0.5$. Given the first term is of the order $10$, this means we are only throwing out an adjustment of less than $5 \%$ to our final answer. Taking just the first term of the Taylor expansion, we end up with: $$n \approx \frac{69.3 \%}{\frac{1}{100} * \frac{1}{r}}$$ And rearranging gives: $$n \approx \frac{69.3}{r}$$ So we see, we are pretty close to $ n \approx \frac{72}{r}$. Why 72? We saw above that using just the first term of the Taylor Expansion suggests we should be using the ‘rule of 69.3%' instead. Why then is this the rule of 72? There are two main reasons, the first is that for most of the interest rates we are interested in, the Rule of 72 actually gives a better approximation to the exact solution, the following table compares the exact solution, the approximation given by the ‘Rule of 69’, and the approximation given by the Rule of 72:
The reason for this is that for interest rates in the 4%10% range, the second term of the Taylor expansion is not completely negligible, and act to make the denominator slightly smaller and hence the fraction slightly bigger. It turns out 72 is quite a good fudge factor to account for this.
Another reason for using 72 over other close numbers is that 72 has a lot of divisors, in particular out of all the integers within 10 of 72, 72 has the most divisors. The following table displays the divisors function d(n), for values of n between 60 and 80. 72 clearly stands out as a good candidate.
The rule of 72 in Actuarial Modelling
The main use I find for this trick is in mentally adjusting historic claims for claims inflation. I know that if I put in 6% claims inflation, my trended losses will double in size from their original level approximately every 12 years. Other uses include when analysing investment returns, thinking about the effects of monetary inflation, or it can even be useful when thinking about the effects of discounting. Can we apply the Rule of 72 anywhere else? I used to use the rule of 72 quite a lot to estimate the effects of long term inflation, but I’ve since realised this doesn’t really work that well (though it’s not a problem with the rule per se). Say we are watching a movie set in 1940. If an item in the movie costs 10 dollars, can we use the Rule of 72 to estimate what that would be worth now? First we need to pick an average inflation rate for the intervening period (something in the range of 34% is probably reasonable). We can then reason as follows; 1940 was 80 years ago, at 4% inflation, $\frac{72}{4} = 18$, and we’ve had approx. 4 lots of 18 years in that time. Therefore the price would have doubled 4 times, or will now be a factor of 16. Suggesting that 10 dollars in 1940 is now worth around 160 dollars in today's terms. It turns out that this doesn’t really work though, let’s check it against another calculation. The average price of a new car in 1940 was around 800 dollars and the average price now is around 35k, which is a factor of 43.75, quite a bit higher than 16. The issue with using inflation figures like these over very long time periods, is for a given year the difference in the underlying goods is fairly small, therefore a simple percentage change in price is an appropriate measure. When we chain together a large number of annual changes, after a certain number of years, the underlying goods have almost completely changed from the first year to the last. For this reason, simply multiplying an inflation rate across decades completely ignores both improvements in the quality of goods over time, and changes in standards of living. There is no perfect method for comparing inflation over very long time frames – one useful reference point, which does gives information about relative affordability but still does not completely deal with the change in quality of goods is to compare a dollar in 1940 with average annual earnings. In 1940 average household earnings were something like 1,000 dollars pa, and in 2020 are now something something like 60,000 dollars pa. Which tells us that 10 dollars was approximately 1% of annual earnings, or equivalent to 600 dollars today.
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Pensions and tax you think to yourself  he’s managed to find the two most boring subjects possible and now he appears to be planning to write about a combination of them! Is this some sort of attempt to win an award for the most boring and tedious blog post every? Well what if I told you this blog post, whilst being about pension and tax was also about a burning social justice, about high finance, and about death? Would you find that interesting? Okay, so I may got slightly carried away in that last paragraph and over sold slightly. This is less of a 'burning social injustice', and more an 'unfair outcome' which happens to only affect people who are already well off  but someone has to write about this stuff right? And the current status quo is genuinely unfair I promise. Lifetime Allowance Before diving into the unfairness I mentioned earlier, we need to briefly set the scene. Let’s think about all the types of tax that the government levies on us; we get taxed on our income (income tax), we get taxed on our property wealth (council tax), we get taxed on our wealth when we die (inheritance tax), and we get taxed when we buy things (VAT and other sales taxes). Since the government taxes just about every financial transaction we make, they can tweak all these different types of tax to incentivise and disincentivizes certain behaviours. One thing governments have traditionally been very keen on incentivising is getting its citizens saving enough for retirement. The way this has historically been done is to allow income tax relief on the proportion of an employee’s gross salary which is put into a pension. i.e. If you earn £30,000 and put £5,000 of that into a pension, then you pay no tax on the £5,000 and are then taxed as if you only earn £25,000. You might think to yourself great, so now people are saving more for their retirement and everyone lives happily ever after, the end. Well not quite. The problem with this arrangement is certain unscrupulous individuals started to use this mechanism to avoid paying tax. Let’s say you’re 64, quite well off, and hoping to retire next year, why not chuck most of your money into your pension? You can live off your savings for a year and then all that money you would have taken as income and therefore been taxed on can instead be deferred for a year, taken as pension when you retire, and you would pay no income tax on that year's salary. Or let’s suppose an individual come from a very wealthy family, and doesn't really need to get paid a salary to live on, then they could just put their entire salary into a pension, not take anything until they retire (at 55), and then not pay any tax at all on their earnings from their job across their entire career. Okay, fine, it sounds like we’re going to need some caps on the maximum pension tax relief to stop people doing this. Let’s say – £225k per annum is the maximum that can be put into a pension pot per year, and £1.5m is the total size of a pension pot you can build up before we start making people pay tax. That’s a large amount of money, who’s going to complain about that? These were the levels set by Tony Blair’s government between 2004 and 2006. These caps were then slowly increased over time, reaching a height of around £250k pa, and £1.8m as a lifetime allowance in 2011. These are very big figures, and it’s hard to imagine anyone objecting to a tax on an annual pension contribution above £250k. In 2011, with a rapidly expanding budget deficit to deal with and a promise to balance the books, the Conservative government instigated their infamous austerity programme. The tories had pledged to ringfence certain areas of government spending (the NHS, and Education for example), they therefore started to look around for other areas to raise additional income which had not traditionally been taped. Tax relief for high earners seemed like an obvious target, and who can really say that the rich should not have to pay additional tax given the situation to help try to protect spending on areas such as policing or social care. By 2014 the annual allowance (the amount that can be saved per annum) had been reduced to £40k per annum, and the lifetime allowance was down to £1m. These values still seem high, however with the economy in the state it was, they are perhaps not as high as they first appear. This reduction came during a period of alltime low interest rates, I'm going to demonstrate which this caused issued by examining a series of graphs. The following graph displays the Bank of England base rate for the last twenty years, notice the significant reduction in 2007 following the global financial crisis. Note we have been at this very low level ever since. As a proxy for the investment return a pension or life insurance company could receive on a long term almost riskfree investment I’m going to use Base rate + 2%, I've plotted this as well on the graph below: Okay, so why is this important? The interesting idea is that the ‘price’ of an annuity can be approximated using quite a simple formula. It is (very) roughly equal to: $ A = \frac{1}{i} $ Where i is our investment return. What does this mean in real numbers? If I want to retire and get a pension of £10,000 per annum, Then the price of that is going to be 10,000 * A = 10,000 / i. If i=10%, then I will need to have a pot of £100,000. If i = 1%, then I will need a pot of £1m. Let’s plot the implied annuity price on the same chart as a secondary axis: So we see we have this reciprocal relationship, as interest rates go down, the cost of an annuity goes up. At the moment, due to the very low interest rates, it costs something like £37 to purchase an annuity of £1 per annum for life. Interesting you might (or might not) think. But this annuity price becomes useful if we combine it with the lifetime taxfree allowance. If we assume a potential retiree is going to use their pension pot to purchase an annuity, then to calculate the level of pension they will be able to get, we need to take their pension pot and divide by the cost of the annuity. The graph below shows the approximate taxfree pension one could expect if retiring with a pension pot equal to the full Lifetime allowance, using the prevailing annuity rates which are in turn based on the prevailing interest rates. So we see that through a combination of the LTA being reduced significantly and interest rates bottoming out, we are now in the position where the maximum taxfree pension (based on approximate annuity rates) is as little as £25k. A far cry from the approximately £100k when the rule was first implemented. What about that burning social injustice you mentioned? But at least we are all treated the same right? Well not exactly… all of the above glossed over one quite important point, it assumes that the individual in question has a defined contribution pension. For individuals with a final salary pension (which includes MPs, Judges, a majority of CEOs and senior public sector employees), the LTA is calculated with reference to a fixed annuity factor of 20. Given the actual average annuity price is more like 40, this has a massive impact. Let’s plot this on a graph as well. Hmmm, so we see that in 2006 when the LTA was first set up, this factor of 20 was quite generous to Defined Contribution pension holders, they could often retire with a pension of around £120k without paying any tax, whereas a defined benefit pension holder could only retire with a maximum of approx. £80k. This has now flipped, due to changes in annuity rates, and the fact that this factor has not been adjusted for so long. Under current conditions, the maximum pension that can be taken tax free is around £25k pa for a definied contribution pension, but around £50k pa for a definied benefit pension.
So what is the point of all this? What am I advocating? To me it seems clear that we need (approximate) equality of outcome in terms of taxation for defined contribution pensions compared to defined benefit schemes. This could be achieved in one of two ways, either we apply some sort of floor to how low the tax free pension can reduced to for defined contribution pensions, or we adjust the defined benefit annuity factor periodically to keep it roughly in line with market conditions and therefore apply a lower maximum tax free rate to defined benefit pensioners. This would correct the current system by which judges and MPs (and other people with DB pensions) get significant tax breaks at retirement compared to individuals with DC pensions. The Reinstatement Premium payable following a loss to an Excess of Loss contract, is related to the ceded loss to the contract by a simple formula. Therefore, it seems reasonable that we should be able to come up with a simple formula relating the price charged for the Excess of Loss contract to the price charged for a Reinstatement Premium Protection (RPP) cover. I was in a meeting last week with two brokers who were trying to do just this. They had come up with an indicative price for an Excess of Loss programme and were trying to use this to price the equivalent RPP cover. At the time I didn't have an answer for them, and when I did a quick Google, nothing came up. When thinking about it subsequently though, there are a couple of easy approximate methods we can use. Below I discuss three different methods for how you can price an RPP cover, two of which do not require any stochastic modelling assuming you already know the price of the Excess of Loss layers. Let's quickly review what we mean by all these terms so that we are starting from the same point. What is a Reinstatement Premium? If you already understand how a Reinstatement Premium works, then feel free to skip this section. Most Excess of Loss contracts will have some form of reinstatement premium. This is a payment from the Insurer to the Reinsurer to reinstate the protection in the event of a loss. In the London market, most contracts willl have either $1$, $2$, or $3$ reinstatements and generally these will be payable at $100 \%$. What is a Reinstatement Premium Protection Cover? The reinstatement premiums can be quite a large proportion of the overall premium paid for the Excess of Loss reinsurance. This is especially the case for lower level, working layers. Furthermore, the Reinstatement Premium will be payable when the insurer has just suffered a loss. From the point of view of the insurer, this additional payment comes at the worst possible time. The Insurer is being asked to fork over another large premium to the Reinsurer, just after having suffered a loss. To address some of these concerns, Reinsurers developed a product called a Reinstatement Premium Protection cover (RPP cover). This cover pays the Insurer's Reinstatement Premium for them, which gives the insurer further indemnification in the event of a loss. Here's an example of how it works in practice: Let's suppose we are considering a $5m$ xs $5m$ Excess of Loss contract, there is one reinstatement at $100 \%$ (written $1$ @ $100 \%$), and the Rate on Line is $25 \%$. The Rate on Line is just the Premium divided by the Limit. So here, the Premium can be found by multiplying the Limit and the RoL: $$5m* 25 \% = 1.25m$$ So we see that the Insurer will have to pay the Reinsurer $1.25m$ at the start of the contract. Now let's suppose there is a loss of $7m$. The Insurer will recover $2m$ from the Resinsurer, but they will also have to make a payment to cover the reinstatement premium of: $\frac {2m} {5m} * (5m * 25 \% ) = 2m * 25 \% = 0.5m$ to reinstate the cover. So the Insurer will actually have to pay out $5.5m$. The RPP cover, if purchased by the insurer, would pay the additional $0.5m$ on behalf of the insurer Now that we know how it works, how would we price the RPP cover? Three methods for pricing an RPP cover Method 1  Full stochastic model If we have priced the original Excess of Loss layer ourselves using a Monte Carlo model, then it should be relatively straight forward to price the RPP cover. We can just look at the expected Reinstatements, and apply a suitable loading for profit and expenses. This loading will probably be broadly in line with the loading that is applied to the expected losses to the Excess of Loss layer, but accounting for the fact that the writer of the RPP cover will not receive any form of Reinstatement for their Reinsurance. What if we do not have a stochastic model set up to price the Excess of Loss layer? What if all we know is the price being charged for the Excess of Loss layer? Method 2  Simple formula This was the situation I was in when I was asked to price the RPP cover last week. The broker had come up with some very approximate pricing for a programme of Excess of Loss layers. This pricing was driven by the burning cost, and other commercial factors rather than any actuarial modelling. The brokers wanted to come up with an approximate price for the RPP cover, just based on the price of the Excess of Loss layer. The two should be related, as they pay out dependant on the same underlying losses. So what can we say? If we denote the Expected Losses to the layer by $EL$, then the Expected Reinstatement Premium should be: $$EL * ROL $$ To see this is the case, I used the following reasoning; if we had losses in one year equal to the $EL$ (I'm talking about actual losses, not expected losses here), then the Reinstatement Premium for that year would be the proportion of the layer which had been exhausted $\frac {EL} {Limit} $ multiplied by the Deposit Premium $Limit * ROL$ i.e.: $$ RPP = \frac{EL} {Limit} * Limit * ROL = EL * ROL$$ Great! So we have our formula right? The issue now is that we don't know what the $EL$ is. We do however know the $ROL$, does this help? If we let $DP$ denote the deposit premium, which is the amount we initially pay for the Excess of Loss layer and we assume that we are dealing with a working layer, then we can assume that: $$DP = EL * (1 + \text{ Profit and Expense Loading } ) $$ Plugging this into our formula above, we can then conclude that the expected Reinstatement Premiums will be: $$\frac {DP} { \text{ Profit and Expense Loading } } * ROL $$ In order to turn this into a price (which we will denote $RPP$) rather than an expected loss, we then need to load our formula for profit and expenses i.e. $$RPP = \frac {DP} {\text{ Profit and Expense Loading }} * ROL * ( \text{ Profit and Expense Loading } ) $$Which with cancellation gives us: $$RPP = DP * ROL $$ Which is our first very simple formula for the price that should be charged for an RPP. Was there anything we missed out though in our analysis? Method 3  A more complicated formula: There is one subtlety we glossed over in order to get our simple formula. The writer of the Excess of Loss layer will also receive the Reinstatement Premiums during the course of the contract. The writer of the RPP cover on the other hand, will not receive any reinstatement premiums (or anything equivalent to a reinstatement premium). Therefore, when comparing the Premium charged for an Excess of Loss layer against the Premium charged for the equivalent RPP layer, we should actually consider the total expected Premium for the Excess of Loss Layer rather than just the Deposit Premium. What will the additional premium be? We already have a formula for the expected Reinstatement premium: $$EL * ROL $$ Therefore the total expected premium for the Excess of Loss Layer is the Deposit Premium plus the additional Premium: $$ DP + EL * ROL $$ This total expected premium is charged in exchange for an expected loss of $EL$. So at this point we know the Total Expected Premium for the Excess of Loss contract, and we can relate the expected loss to the Excess of Loss layer to the Expected Loss to the RPP contract. i.e. For an expected loss to the RPP of $EL * ROL$, we would actually expect an equivalent premium for the RPP to be: $$ RPP = (DP + EL * ROL) * ROL $$ This formula is already loaded for Profit and Expenses, as it is based on the total premium charged for the Excess of Loss contract. It does however still contain the $EL$ as one of its terms which we do not know. We have two choices at this point. We can either come up with an assumption for the profit and expense loading (which in this hard market might be as little as only be $5 \%  10 \%$ ). And then replace $EL$ with a scaled down $DP$: $$RPP = \frac{DP} {1.075} * ( 1 + ROL) * ROL $$ Or we could simply replace the $EL$ with the $DP$, which is partially justified by the fact that the $EL$ is only used to multiply the $ROL$, and will therefore have a relatively small impact on the result. Giving us the following formula: $$RPP = DP ( 1 + ROL) * ROL $$ Which of the three methods is the best? The full stochastic model is always going to be the best in my opinion. If we do not have access to one though, then out of the two formulas, the more complicated formula we derived should be more accurate (by which I mean more actuarially correct). If I was doing this in practice, I would probably calculate both, to generate some sort of range, but tend towards the second formula. That being said, when I compared the prices that the Brokers had come up with, which is based on what they thought they could actually place in the market, against my formulas, I found that the simple version of the formula was actually closer to the Broker's estimate of how much these contacts could be placed for in the market. Since the simple formula always comes out with a lower price than the more complicated formula, this suggests that there is a tendency for RPPs to be underpriced in the market. This systematic underpricing may be driven by commercial considerations rather than faulty reasoning on the part of market participants. According to the Broker I was discussing these contracts with, a common reason for placing an RPP is to give a Reinsurer who does not currently have a line on the underlying Excess of Loss layer, but who would like to start writing it, a chance to have an involvement in the same risk, without diminishing the signed lines for the existing markets. So let's say that Reinsurer A writes $100 \%$ of the Excess of Loss contract, and Reinsurer B would like to take a line on the contract. The only way to give them a line on the Excess of Loss contract is to reduce the line that Reinsurer A has. The insurer may not wish to do this though if Reinsurer A is keen to maintain their line. So the Insurer may allow Reinsurer B to write the RPP cover instead, and leave Reinsurer A with $100 \%$ of the Excess of Loss contract. This commercial factor may be one of the reasons that traditionally writers of an RPP would be inclined to give favourable terms relative to the Excess of Loss layer so as to encourage the insurer to allow them on to the main programme and to encourage them to allow them to wrte the RPP cover at all. Moral Hazard One point that is quite interesting to note about how these deals are structured is that RPP covers can have quite a significant moral hazard effect on the Insurer. The existence of Reinstatement Premiums is at least partially a mechanism to prevent moral hazard on the part of the Insurer. To see why this is the case, let's go back to our example of the $5m$ xs $5m$ layer. An insurer who purchases this layer is now exposed to the first $5m$ of any loss. But they are indemnified for the portion of the loss above $5m$, up to a limit of $5m$. If the insurer is presented with two risks which are seeking insurance  one with a total sum insured of $10m$, and another with a total sum insured of $6m$, the net retained exposure is the same for both risks from the point of view of the insurer. By including a reinstatement premium as part of the Excess of Loss layer, an therefore ensuring that the insurer has to make a payment any time a loss ceded to the layer, the reinsurer is ensuring that the insurer keeps their financial incentive to not have losses in this range. By purchasing an RPP cover, the insurer is removing their financial interest in losses which are ceded to the layer. There is an interesting conflict of interest in that the RPP cover will almost always be written by a different reinsurer to the Excess of Loss layer. The Reinsurer that is writing the RPP cover is therefore increasing the moral hazard risk whichever Reinsurer has written the Excess of Loss layer. Which will almost always be business written by one of the Reinsurer's competitors! Working Layers and unlimited Reinstatements Another point to note is that this pricing analysis makes a couple of implicit assumptions. The first is that there is a sensible relationship between the expected loss to the layer and the premium charged for the layer. This will normally only be the case for 'working layers'. These are layers to which a reasonable amount of loss activity is expected. If we are dealing with clash or other higher layers, then the pricing of these layers will be more heavily driven by considerations beyond the expected loss to the layer. These might be capital considerations on the part of the Reinsurer, commercial considerations such as Another implicit assumption in this analysis is that the reinstatements offered are unlimited,. If this is not the case, then the statement that the expected reinstatement is $EL * ROL$ no longer holds. If we have limited reinstatements (which is the case in practice most of the time) then we would expect the expected reinstatement to be less than or equal to this. Last week it was announced that UK Rail Fares were to increase once again at the maximum allowed rate  3.4%, corresponding to the RPI increase in July 2017. News article: www.bbc.co.uk/news/business42234488 When reading this it got me thinking  why is RPI even being used any more? Aren't we supposed to be using CPI now? In 2013 the ONS stated that: "Following a consultation on options for improving the Retail Prices Index (RPI), the National Statistician, Jil Matheson, has concluded that the formula used to produce the RPI does not meet international standards and recommended that a new index be published." So basically the ONS no longer endorses RPI as the best indicator of the level of inflation in the economy. The ONS instead supports the use of CPI. So why does it matter that some organisations are still using RPI? To see why, let's take a look at a chart showing the historic RPI and CPI increases in the UK: Source: www.ons.gov.uk/economy/inflationandpriceindices RPI has been greater than CPI in every single month since 2010. In fact, in this time period, RPI has been an average of 0.8% higher than CPI. This fact might go some way to explain why the Government is so slow to move rail increases from RPI to CPI. This way the Government and Rail Companies can claim that they are only increasing their costs in line with inflation, which seems fair, yet the index they are using is actually higher than the usual inflation index used in the UK. The Government also indexes some of it's outgoings by an inflation index, for example the State Pension, so at least this is also being consistently overstated right? Well actually no! Wherever the government is using an inflation index to increase payments, it seems to have already transitioned to a CPI increase. Let's look at the list of items which use the inflation index which is more beneficial to the Government: (remember that CPI is almost always lower than RPI): There are some pretty hefty items on the list, including, the State Pension, Benefit Payments, Rail Fares, Utility bills, Student Loans. The ones that decrease government spending seem to have already transitioned over, and the ones that increase the amount of tax collected, or the cost of regulated items all seem to still be using the higher RPI index. Now let's look at the list of items which use the inflation index which is to the disadvantage of the Government. The list is certainly a lot shorter, and the items on it are less substantial. Indexed linked Government Bonds are however quite substantial. The reason that the Government is not able to move these to a CPI index is that it would be considered a default to downgrade the index once the bonds have been sold. The Government has no choice but to continue paying the bonds at RPI. Also, the yield on the bonds will be set with an eye towards the yield on a fixed bond, and the expected level of inflation. Therefore the actual index used is not necessarily that important. It's nice to see though that at least stamps are increased at a CPI rate! Source: www.ons.gov.uk/economy/inflationandpriceindices/methodologies/usersandusesofconsumerpriceinflationstatistics I've got to admit, up until recently, I never really understood why anyone would buy Premium Bonds. You get an expected return of 1.15%, which is well below other asset classes, there is significant volatility in returns, and the 1.15% is not even guaranteed, i.e. it's possible you'll never get any returns at all. The risk and return profile just seem completely out of sync with the alternatives. But after speaking to my Grandad about them yesterday, I started to understand why some of the soft factors make them so attractive to savers. Then when I was mulling over it further, I realised that these soft factors would actually be seen as negatives when viewed through the lens of Modern Portfolio Theory. What are Premium Bonds? Premium Bonds were introduced by Harold Macmillan in 1956 in an attempt to encourage individuals to save more, and also to help raise money to finance Government spending. The bonds pay out prizes to bondholders based on a lottery system and are organised by the National Saving and Investment Agency. The Government uses the money to help finance the Public Sector Borrowing Requirement (the difference between the amount of tax the Government collects .every year, and the amount the Government spends  which has been negative for about 30 years now!) Every month, the NSI distributes a prize pool randomly among Bond holders, the pool is calculated so that the total annual amount paid out is equivalent to 1.15% of the total value of bonds held over the year. So as there are approximately £71bn of bonds owned in total, the prize pool will be £71bn * 1.15% = £817m per year, or £68.1m per month. This £68.1 million will then be distributed randomly to bond owners in various sized prices using the following table: Both the interest rate used to calculate the overall prize pool, and the distribution of prizes is updated from time to time, to account for changes in the overall market conditions. This table is correct as at the date of this post. Advantages of Premium Bonds In spite of their relatively low rate of return, Premium Bonds do a have a number of features that make them attractive to individual savers:
For many individual investors, the first five bullet points are extremely important. I think many savers would even say that are necessary requirements to get them to invest their money. It's the final bullet point which is particularly interesting though, the possibility of winning £1 million is a big selling point of Premium Bonds. This should interest any analyst who uses conventional theories of investment, as most investment theories would consider this a downside of Premium Bonds. To understand why this is the case, we'll need to take a brief look at Modern Portfolio Theory. Volatility of Investment Returns as a Risk Measure All the possible investment classes have different risk and return profiles. Some provide high returns, but also contain high levels of risk. Others offer lower levels of return, but also come with less risk. In order to compare different investment classes in a consistent way, we need a single definition of risk and return which we can then apply to each class. The obvious measure to use for return is simply the Mean Expected Return. The situation isn't so straight forward for our risk measure though. There is no perfect measure of risk, and all the possible measures have advantages and disadvantages. In practice, the most commonly used measure is variance of return and this is the approach used in Modern Portfolio Theory: en.wikipedia.org/wiki/Modern_portfolio_theory Under Modern Portfolio Theory, we calculate the expected return of each investment class, along with the variance of return and plot this on a graph. An investor is then assumed to want to maximise the expected return, while minimising the variance of returns (because this corresponds to risk). This allows us to construct an 'efficient frontier' corresponding to those investments, which for a given return, minimise the risk. The following graph demonstrates this analysis. By including the possibility of winning a £1 million prize, we are massively increasing the variance of the investment returns, but since the overall return is calibrated to a 1.15% expected return, the riskreward profile of premium bonds (using variance as our measure of risk) looks terrible. So we end up with the strange situation that one of the main selling points of Premium Bonds  the possibility of winning a £1 million prize each month, actually should make the investment less attractive under most theories of investment. I did a quick calculation of the variance of returns provided by premium bonds, and it comes out at a whopping 35,000%. Most of this volatility is coming from the fact that two of the prizes are for £1 million, if we take out all the prizes above £1000 and reallocate the money to the smaller prizes the volatility reduces by about 400 times, down to around 90%. But even this is still considered very volatile compared to most investment classes. For example, Burton Malkiel in A Random Walk Down Wall Street provides the following table of historic asset class returns with accompanying volatility (which I then copied from Wikipedia) So for comparison, we see that according to this analysis, investing in Small Companies Stocks provided an average return of 12.6%, much higher than premium bonds, yet only had a Standard Deviation of return of 32.9%. Under the assumptions of Modern Portfolio Theory, by using variance of investment returns as a risk measure, we would conclude that investors would prefer not to have the possibility of the £1 million prize, but instead would prefer to have more smaller prizes included in the payout. This is clearly not the case. Disadvantages of investing in Premium Bonds So far we've talked about the advantages of investing in Premium Bonds, and we've also talked about how some of these selling points go against the conclusions of Modern Portfolio Theory (though I think this might say more about the state of Modern Portfolio Theory than Premium Bonds). Are there however any clear disadvantages to investing in Premium Bonds? I think there are, and here are a few of them:
We often distinguish between real and nominal investment returns when analysing investment classes. What this means is that some investment classes provide returns which are expected to be above inflation, other classes provide returns which are of a fixed amount and may be greater than or smaller than the level of inflation in the economy, these are called nominal returns. To see what we mean by this, let's say you invest £1,000 in premium bonds in a given year and you get £25 in prizes, leaving you with £1,025 at the end of the year. Let's assume also that prices (i.e. inflation) went up by 5% during the year, which means that in order to buy something which would have cost £1,000 at the start of the year, you would now need £1,050. So your £1,025 which you got back from the Premium Bonds at the end of the year, can now buy less than the £1,000 you put in originally. So that even though you've now got more money, the real return on your investment was negative. Premium Bonds are currently paying out at 1.15%, whereas inflation in July 2017 was 2.6% according to the Office of National Statistics. This means that you are in effect losing money by holding Premium Bonds.
Due to the lottery structure of Premium Bond returns, it's possible (and in fact highly likely unless you hold large quantities of bonds) that you will get no investment returns at all. For example, someone who holds £100 of Premium bonds has a 96% chance of not getting any prizes at all in a given year. This is in contrast to a fixed interest bank account, for example if it guarantee to pay 3% pa there would be no possibility of winning a £1 million prize, but which you could be certain would always give you your 3% return at the end of the year. If you are willing to go through the hassle of shopping around for a savings account which does guarantee 3%, then you'd probably be better off from a financial perspective investing in the savings account and then buying yourself a lottery ticket every week rather than investing in premium bonds.
The expected return from Premium Bonds is much lower than the returns we could expect from other investment classes. To see how much of a different this can make, let's take two hypothetical savers, both of whom have been saving £100 per month for the last 30 years, one has been investing in Premium Bonds (and getting an average return of 3.5% pa) and another has been investing in a Property Fund (and getting a 10% average return). Over the years, the difference between these two expected returns can have a significant impact on the overall amount that they save. I ran a quick simulation of this with some volatility producing the following graph: So we've talked about the positives and negatives of Premium Bonds, and also touched on some inconsistencies between the marketing of Premium Bonds, and how they would be viewed using traditional finance techniques. Are there any other features of Premium Bonds which are interesting? One quirk of Premium Bonds which seems to have captured people's imaginations is ERNIE. So who is ERNIE? E.R.N.I.E ERNIE stands for Electronic Random Number Indicator Equipment, and is the name of the machine that is used to randomly select which bonds win, Due to the fact that Premium Bonds were first set up in 1957, creating a computer to generate random numbers was a nontrivial problem at the time. ERNIE was developed in the Post Office Research Station in North West London. Despite its quaint sounding name, the Post Office Research Station was actually at the forefront of computing research during the 40s and 50s. The Colossus, the world's first electronic programmable computer, which had been developed during the second World War to help break one of the codes used by the Axis was developed at the Post Office Research Station. The details of Colossus were classified until the 1970s so it had limited impact on most other computers built around in the 50s and 60s, however the design of ERNIE was heavily influenced by the work that had been done on Colossus during the war as the same design team worked on both. Unlike most computers today that are pseudorandom number generators, ERNIE was a true random number generator. It contained neon tubes, through which an electric current was passed. Due to the electrons bumping into the neon atoms as they passed through the neon tube, the current leaving the neon tube varied randomly. This randomness was due to millions of tiny interactions between the electrons and the neon tube and was a source of true statistical randomness. This randomness was then calibrated so that ERNIE would select a collection of random numbers between 1 and 100 million, which corresponded to the Premium Bond numbers that people had purchased. I struggled to follow a lot of the technical details of how ERNIE works, due to an insufficient knowledge of electrical engineering, but if you are interested the following link contains a technical document describing how ERNIE selected random numbers: www.tnmoc.org/sites/default/files/Ernietechnology.pdf I might do some reading up on Electrical Engineering at some point, and then write up exactly how ERNIE worked, because I couldn't find a decent nontechnical description online. Conclusion Premium Bonds were a massive success when they were launched. They managed to get people excited about saving, and they gave the Government a cheap source of borrowing. I think it's a shame they have fallen by the wayside in recent years. They are conceptually simple, they offer a lot of the guarantees that savers look for in an investment opportunity  protecting their capital, not locking them in for a fixed period  they exploit the popularity of lotteries in a positive way (by tricking people into saving more in order to play), and they don't require complicated decisions to be made by savers (who has time to trawl through the bewildering array of different options currently available  Cash ISAs, Stocks and Shares ISAs, NSI bonds, Fixed interest bank accounts....) All too often, it seems Government savings schemes are designed by individuals steeped in financial theory who are too far removed from the savers who will be using the products. We should rethink how we design the kind of products offered by the NS&I and take more account of Behavioural Economics, and Marketing Strategies rather than relying on microeconomic models which assume rational investors, or relying on models of investment returns like Modern Portfolio Theory. What even are Ogden Rates anyway? The Ogden tables are tables of annuity factors, published by the Government's Actuary Department, which are used to calculate court awards for claimants who have had life changing injuries or a fatal accident and are eligible for a payout from their insurance policy. For example, consider a 50 year old, male, primary school teacher who suffers a car accident which means that they will not be able to work for the rest of their life. The Ogden Tables will be used to calculate how much they should be paid now to compensate them for their loss of earnings. Suppose the teacher is earning a salary of £33,000 when they have the accident, then under the Ogden Rates prior to March 2017, the teacher would be paid a lump sum of £33,000* 20.53 = £677,490 where 20.53 is the factor from the tables. How did the Government's Actuary Department come up with these factors? The factors in the table are based on two main pieces of information, how long the person is expected to live, and how much money they can earn from the lump sum once they are given it (called the discount rate). It's this second part which has caused all the problems between the Ministry of Justice and the Insurance Industry. The discount rate should be selected to match the return generated on assets. For example, if the claimant puts all their money in shares then on average, they will generate much more income than if they put the lump sum in a savings account. So what should we assume our school teacher will invest their lump sum in? Since the school teacher will not be able to work again, and therefore will need to live off this money for the rest of their life, they will not want to risk losing all their money by investing in something too risky. In technical terms, we would say that the claimant is a risk adverse investor. In order to mimic the investment style of this risk adverse investor, when the Ogden tables were first set, it was decided to assume that the investor would put all their money in indexlinked bonds. There are a couple of reasons to assume this, many risk adverse institutional investors do purchase a lot of indexlinked bonds, and also, the average discount rate for these bonds is readily available as it is already published by the UK DMO. At the time the tables were set up, this seemed like a great idea, but recently it has made a lot of people very angry and been widely regarded as a bad move. What are Indexlinked bonds again? In the 1981 the UK government started issuing a series of gilts which instead of paying a fixed coupon, paid a floating coupon which was a fixed percentage above the rate of inflation. The UK Debt Management Office is responsible for issuing these bonds, and the following website has details of the bonds that are currently in issue. It's quite interesting to see how it all works: www.dmo.gov.uk/reportView.aspx?rptCode=D1D&rptName=50545854&reportpage=D1D The basic principle is if you purchase a bond that pays 2% coupons, if inflation is 3%, they would pay 3%+2%, if inflation was 5%, then they would pay 5% + 2%. Due to the fact that these bonds always gave a fixed real return (2% in this case), institutional investors really like them. Because there is no inflation risk, on average indexlinked bonds cost more than fixed coupon bonds once you account for the effects of inflation. Pension Schemes in particular purchase a lot of these bonds, Why do Pension Schemes like these bonds so much? Most pensions are increased annually in line with inflation, due to this Pension Schemes like to hold assets that also go up in line with inflation every year. In order to get real returns on their investments, Pension Schemes traditionally held a mix of shares and indexlinked bonds, the shares gave better returns, but the bonds were more safe. This all started to go very wrong after the financial crisis . A huge drop in interest rates and investment returns, combined with soaring life expectancy lead to more and more pension schemes winding up and the remaining ones have funding issues. As the schemes started winding up they became more and more risk adverse and started to move away from the more volatile assets like shares and moved towards indexlinked bonds instead. This table from the PPF's Purple Book shows the move away from shares into bonds. We can see that back in 2006, prior to the financial crisis, Pension Schemes were on average holding around 61% of their assets in equities. When we look again at 2014 this percentage has dropped to 33% and the slack has largely been taken up by bonds. Pension Schemes like these assets so much in fact that Schroders estimated that 80% of the long term indexlinked gilts market is held by private sector pension schemes as the following chart shows. Source: www.schroders.co.uk/en/SysGlobalAssets/schroders/sites/ukpensions/pdfs/201606pensionschemesandindexlinkedgilts.pdf Does it matter that Pension Schemes own such a high proportion of these gilts? The problem with the indexlinked gilt market being dominated by Pension Schemes is one of supply and demand. The demand for these bonds from Pension Schemes far outweighs the supply of the bonds. Another chart from Schroder's estimates the demand for the bonds is almost 5 times the supply. Source: www.schroders.co.uk/en/SysGlobalAssets/schroders/sites/ukpensions/pdfs/201606pensionschemesandindexlinkedgilts.pdf As you might expect with such a disparity between supply and demand, Pension funds have been chasing these assets so much that yields have actually become negative. This means that Pension Schemes on average are paying the government to hold their money for them, as long as it's protected against inflation. Here is a chart showing the yield over the last 5 years for a 1.25% 2032 indexlinked gilt. Source: www.fixedincomeinvestor.co.uk/x/bondchart.html?id=3473&stash=F67129F0&groupid=3530 So what does this have to do with Ogden Rates? So now we are in a position to link this back to the recent change in the Ogden Rate. Because the yield on indexlinked bonds has traditionally been used as a proxy for a riskfree real return, the yield is still used to decide the discount rate that should be used to calculate court award payouts. Because Pension Schemes have been driving up the price of these bonds so much, we have the bizarre situation that the amount that insurance companies have to pay out to claimants has suddenly jumped up considerably. In the case of a 20 year old female for example, the amount that would be paid out has almost tripled. As these pay outs are already considerable, the financial impact of this change has been massive. So what should the Government do? There is no easy answer, if the Government doesn't use the yield on indexlinked gilts to calculate the Ogden rate then there is no obvious alternative. I think the most reasonable alternative would be to use a weighted average of returns on the types of assets that an average claimant would hold. For example, we might assume the claimant is going to hold 50% of their lump sum in cash, 30% in shares, and 20% in bonds, and we would then calculate the weighted return from this portfolio. The issues with doing nothing is that the additional cost from these increased pay outs will inevitably be passed on to the policyholders through higher premiums. So ultimately there is an issue of fairness whereby people who are receiving payouts are being paid a disproportionate amount of money, and this is being subsidised by policyholders other policyholders. 
AuthorI work as a pricing actuary at a reinsurer in London. Categories
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