Simple Pound

I’m back from holiday and it’s time to start my new series on portfolio returns (as discussed in my previous post) with some basics on averages, limits and other interesting information you can extract from your data without too much effort. Since I’m working on a spreadsheet alongside these posts, I will also include some Excel references that you can use to shortcut the calculations.

Based on my outline, this is what I’m looking to cover in this post:

• arithmetic mean
• geometric mean
• frequency distribution / histograms
• maximum and minimum value
• positive / negative number of weeks / months / years

Arithmetic Mean

The arithmetic mean is more commonly referred to as the average (or just mean) and is calculated by summing all numbers of a list and subsequently dividing the result by the number of items in the list. Finding the average of three numbers A, B and C would involve forming the sum A + B + C and dividing by 3. It answers the question “If all quantities had the same value, what would that value have to be in order to achieve the same total?” and the calculation is relevant whenever the quantities concerned add together to produce a total. For instance, if you buy milk, bread and butter you will be expected to pay a total sum to cover the price of these three items and calculating the average would tell you what price a different product X would have to have so that you could buy three instances of X and end up paying the same amount.

The trouble with the simple average is that is greatly influenced by outliers, i.e. extreme values at either end of the spectrum. If you calculate the average of ten 5s and 1000 you will end up with (10 * 5 + 1000) / 11 = 95.45 which is unproportionately large and could give the impression (assuming the individual data points are not known) that the data is concentrated around values just short of 100. This problem is a simple example of skewed distributions where the mean of a data series doesn’t coincide with the median (the “middle value” if you were to list them all).

Finally, the arithmetic mean is not suitable for calculating stock performance since you’re looking for the product of the yearly performance, not the sum of those values. Consider an investment that loses 10% of its value in the first year and gains 30% in the second year. Now the annualised performance since you started investing is not [(-10%) + 30%]/2 = 10% but in fact 8.2% since the 30% increase only affects the 90% of your investment that’s left after the first year. To correctly calculate annualised performance rates like this we will have to use the geometric mean.

Excel Function: AVERAGE(number1, [number2], …)

Geometric Mean

The geometric mean is a value that gives an indication of the central tendency of a set of numbers (i.e. of a distribution) regardless of whether this distribution is normal or skewed. In other words, it shows the “typical value” of a list of values. Note that this is different from the most common value which is a mathematical value called “mode”. It answers the question “If all quantities had the same value, what would that value have to be in order to achieve the same product?” and hence is relevant whenever quantities multiply together to produce a product. The geometric mean is always less than or equal to the arithmetic mean.

To calculate the geometric mean you need to multiply all the values in your set and subsequently find a scientific calculator to take the nth root of the product where n is the number of numbers you previously multiplied. In the simplest case when calculating the geometric mean of two values A and B you would need to take the square root of A*B. Unfortunately, this also means that the geometric mean in its purest form can only be applied to positive values as (in conventional mathematics and without exploring imaginary values) you cannot take the root of a negative number.

While we all hope that our investments are going to increase year after year, we can’t simply assume this and happily apply the geometric mean without a little work first. To take negative growth into account, you will need to look at your percentage values from a slightly different angle. Let’s go back to our original example of a return of -10% in the first year and 30% in the second year. After the first year, only 90% of your original investment will remain - that is, your portfolio will stand at 0.9 instead of 1 (= 100%). In the second year, you’re more lucky and see a 30% growth of that remainder. If you would have had a fresh portfolio (i.e 100% = 1) you’d now be left with 130% of your money or 1.30. Since you only had 90% remaining we need to multiply 0.9 by 1.30 to get 1.17. This means that in relative terms to the amount you started with, your investment is worth 17% more after 2 years. To now calculate the annualised growth rate of your investment we need to take the square root of 1.17 which yields 1.0816 or a growth of 8.16% per annum.

The Excel function given below does not convert your percentages accordingly so the restriction with regards to positive values still applies. So far, I haven’t been able to find an in-built function that does the work for you which I find rather frustrating. It is extremely annoying to represent a 10% drop in my portfolio as 0.90 and if I don’t come across a more workable solution soon, I will probably end up writing a function myself If you know something that I don’t please let me know before I spent hours writing an Excel Add-In from scratch.

Excel Function: GEOMEAN(number1, [number2], …)

Frequency Distribution

If you’re a visual thinker like myself, you will appreciate that I often find it more helpful to look at a graph instead of raw numbers. An obvious choice in the case of portfolio returns is a histogram which is based on a frequency distribution. At this stage, I won’t dwell on this topic for long as I haven’t got sufficient data yet to experiment with histograms and the associated frequency distributions yet. As soon as that is the case, I will almost certainly give you a little more insight into this topic.

For the moment, imagine that you have two buckets in which you need to throw little cards that represent your portfolio performance - whether you’re looking at weekly or monthly returns or any other values is fairly irrelevant as long as you have enough data. Your task now is to separate the positive returns from the negative ones by throwing them into their respective buckets. Once you’re done with that, you can draw a simple bar graph with the number of items per basket representing the height of an individual bar (assuming one unit width per bar). Voila - there is your first histogram

If you imagine that you could have as many buckets as you like, then you only need to come up with a criteria to decide which bucket contains what data to create whatever histogram you like. You could distinguish between negative and positive values, determine 5% steps or pick any other classification you might find useful. The only rule is that categories must not overlap - i.e. there is always one unique bucket for every single piece of data and at no point would any value (whether in your sample or not) match two or more buckets.

The interesting thing about histograms is that it’s not technically the height of a bar that determines the value it represents (unlike in a simple bar chart where only the height matters) but it’s the area that is covered by it. Hence you could draw a bar that’s one inch wide and three inches high and that would represent the same value as a bar that’s three inches wide and one inch high. For this reason the bars of a histogram have to be adjacent with no gaps in between.

Because you are free to choose however many buckets you want to use when constructing a histogram (the official term for my made-up bucket is “bin”) much research has been done to determine the optimal number of bins as too small a number might hide valuable insights while too many bins render the diagram useless. The level of granularity is crucial. One formula that has been put forward is k = [(max(x) - min(x)) / bin width h] where k represents the optimal number of bins and max(x) and min(x) denote the maximum and minimum values of your data respectively. Whatever value you calculate, you should round up to the nearest full value and use this to create your categories.

Excel 2007 includes an Add-In that creates a histogram from your data if presented in the correct way (data values in one column, bin limits in second column) and hence extends the basic FREQUENCY function that simply returned an array of values. While I have Excel 2007, the data arrangement it requires to execute properly doesn’t work very well if you’re planning to have more than just a histogram graph in your worksheet. Further, from the previews I have seen of the functionality, the histogram looks wrong as there are in fact gaps between the bars (which violates the rules given above). Hence, once more I find myself wondering whether I should invest the time and effort to create something a little more elegant myself. Stay tuned

Maximum and Minimum Value

Since I assume you graduated from primary school long ago, I don’t really need to explain the meaning of maximum and minimum value to you. I simply include it for completeness’s sake and to create a central place for all important Excel functions

Excel Function: MAX(number1, [number2], …) and MIN(number1, [number2], …)

Positive and Negative Number of Weeks

This is an interesting little insight I came across in the book I was reading about Asset Allocation not long ago. While it doesn’t tell you that much per se, it gives you valuable perspective and a sense of patience when you’re tempted to give up your long-term investing for short-term speculation. If you can see that your investment is up most weeks / months / years and only down in value occassionally, you might find it easier to sit out the rough patch - rest assured that other (better) times will come.

To avoid having to count these items manually every time you update the performance, you can use the Excel function COUNTIF that looks at a list of values and only counts the instances that fit a criteria given as part of the formula. By using the criteria “>0″ you will end up with a count of all positive values, while “<0″ returns a total of negative occurences.

Excel Function: COUNTIF(range, criteria)

I hope you enjoyed this first introduction into maths and basic statistics as I very much look forward to researching the upcoming posts! If you have any questions or concerns, just leave a comment and either myself or another reader will surely be able to help.

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Admittedly, it’s been getting very quiet on the blog front and you’ve all heard the excuses of working late and being busy, so I won’t bore you for very long before moving on to more interesting stuff. But can I just say that I also walked 20 miles for charity and that took away some blogging time! I bet that’s a new one for most of you…

Anyway, as I promised in my last post (yes it’s been a while) I have had an idea about a series of posts that might be of interest to you, especially if you really really like numbers (like I do). While reallocating my pension across several funds, I kept wondering how to best measure how well (or badly) my choices were performing compared to the market in general. For a start I have split money fairly evenly across index funds and actively managed funds in a way that will hopefully allow me to compare apples with apples - i.e. simply speaking, for each asset class I have picked an index fund as well as an actively managed fund to compare their performance against each other.

Somehow, that didn’t seem good enough and so I did some research on what else I could do. As I said, I like Maths and numbers because they have an inherent logic and beauty… great, now I sound like a total geek. Or idiot. Your choice

Hence, here’s my line-up of posts that will hopefully appear throughout June and July (bear with me as I’m also going on holiday in three days). I’m not going to explain much (or anything) at this stage, because otherwise there’s not much point in writing separate posts about each. I simply hope you’ll be excited to read what’s coming up and all the things you might be able to learn soon(ish).

1. Post : The Basics
- arithmetic mean (average loss, average gain)
- geometric mean
- frequency distribution
- maximum value
- minimum value
- positive # of years / months / weeks
- negative # of years / months / weeks

2. Post : Risk
- VAR: value at risk
- M-squared

3. Post : Statistics
- standard deviation
- semi-variance (semi-deviation)
- downside variance
- below-target probability

4. Post : All About Interaction
- covariance: degree of variability of returns between two assets
- correlation coefficient
- units of annual return per unit of std. dev.
- expected final value of $1.00 / £1.00

5. Post : First lesson in Greek
- beta coefficient or an assets’ degree of responsiveness to market movements

6. Post : Advanced Greek
- alpha or superior returns

7. Post : More Jargon
- sharpe ratio

Some of these will be rather obvious, or aren’t even really mathematical but I thought they were nevertheless useful when analysing your portfolio. I’m going to try my best to explain even the more complex concepts in a straight-forward way that will make you (and me) understand why exactly a particular formula might in fact be useful.

Knowing myself, all of the above will eventually end up in a big spreadsheet, which I naturally will also make available to you so you don’t have to play with Excel for hours to get it all onto one page (surprisingly enough not everyone finds that sort of stuff fascinating).

I’m getting rather excited while writing this, so hopefully my energy won’t be wasted and you’ll enjoy it as well! Let me know if you think I’ve omitted anything absolutely obvious that shouldn’t be missing from a series of posts on portfolio calculations.

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A year ago I wrote about Zopa, a (then) new loan concept that wanted to bring people together in order to facilitate a lending and borrowing market that wouldn’t require banks. The concept is remarkably easy - on the one hand we have people with spare cash looking for a good and (reasonably) safe return while at the same time we have people out there who are looking for some spare cash (i.e. a loan). Why not make these people talk directly, instead of forcing the former group to deposit their savings into a bank account with a mediocre interest rate and subject the latter to bank’s exorbitant fees (exceptions apply)?

After I had been watching the Zopa site grow for quite some time, I decided that it was time to join the fun and have a go at it myself. With the house purchase one of my major short-term goals at the minute, I didn’t want to tie up a lot of capital for a long time, hence the amounts I’m allowing myself to use at Zopa are fairly small (£ 25 in total at the minute, increasing by about £ 10 a month). Nevertheless I figured I had seen enough to share my experience with the site so far…

Signing up: Ideally this should have been a fairly easy process, especially since I was intending to become a lender, not borrower. However, due to money laundering regulations Zopa is still required to verify your identity and address. Since we’ve moved to our current flat only a few months ago, this identity check could not be carried out online and I had to submit the usual proof of identity and address documentation. This is nothing unique to Zopa and I’ve encountered the same issue several times before with banks, credit card and loan companies. In the end it took only 2 days for them to process my documents and I could sign up successfully! Overall impression: good.

Customer Service: Apart from the registering process I’ve had several other encounters with the Customer Service department relating simple queries as well as a functionality problem at one point. The usual way of contacting them is by sending an email and the response time is always within the promised 1 - 2 business days. All requests were dealt with swiftly and the team is very helpful and always friendly. Overall impression: excellent.

Transferring money: There are three major ways you can transfer money into your Zopa account: Debit card by phone (for instant transferral) or online (for transferral within the same business day), standing order (similar to the way you’d set up the standing order for a savings account) or by bank transfer (longest of all methods as it takes about 3 days to reach your account). With either option you will receive a confirmation email when your funds reach the account and are ready to be used. To transfer money out, you will need to have your bank account confirmed with Zopa. To do that, you simply need to transfer £1 by bank transfer once for them to be able to verify the account belongs to you. At this stage, you cannot transfer the money in your Zopa account to anybody else but yourself. Overall impression: very good.

Lending in Zopa Markets: With Zopa you have two major lending options - Markets
or Listings. If you allocate your money to the Markets section, most of the work matching your lending offer with a borrower request will be done behind the scenes for you. You merely see your money moving between the stages of being offered (currently available), processing (matched to a borrower, loan verification in progress), lent out and late payments (hopefully very few in the latter category). To determine your rate of return you can either give Zopa your desired rate of return and the longest amount of time you’re willing to tie up your capital or you can fine tune your offer by indicating an exact rate of return per market segment. These segments are determined by the borrowers credit rating and the duration of the loan and range from A* for 12 months to C for 60 months. Zopa is helping you to offer realistic rates by quoting you a range of rates that other lenders are offering.

My experience with the Markets section is thoroughly positive. I’ve got all my lending offers at the higher end of the market range and yet I find that my available money is usually processing within a time span of about 2 days. I’ve only had one slight hiccup so far that was explained to me and hence resolved by the Customer Service team within 2 days (my Zopa account contained a little less than the shown £ 10 due to the Zopa fees being earmarked but not deducted every month). Overall impression: good

Lending on Zopa Listings: Zopa Listings are essentially an eBay-like reverse auction system where borrowers advertise their borrowing needs together with an explanation of their finances and lenders can quote how much they’d be willing to lend to this one borrower and at which rate. All quotes get aggregated throughout the duration of the listing. When the borrower’s desired loan amount has been reached (i.e. funding is at 100%), lenders can continue to quote and hence will start outbidding each other with lower rates. Eventually only the minimum number of lenders with the lowest rates will be kept in the listing and hence will be able to lend their money to the borrower. The advantage of the Listings is that you might be able to get a higher rate than you’ve quoted in the Markets section by bidding at the last minute - similarly to how you can get a bargain at eBay through sniping (or old-fashioned pressing of refresh and bidding on the last second).

I’ve only (actively) participated in one Listing so far which ended at 4.20am in the morning. I waited to submit my quote until half past midnight and went to bed hoping lots of people would have already done the same. By the time I submitted my quote, I was about 50 offers (out of 130) away from being excluded so I felt pretty safe and happy as I had a good impression of the borrower. Unfortunately I was out-bid just 15 minutes before the end of the Listing… In any case, the entire process was certainly exciting and I’m intending to look out for other Listings as soon as I fund my Zopa account with more money (waiting for the paycheck, anyone?). Overall impression: excellent

Total verdict: For me, Zopa turned out to be everything I expected and wanted it to be. Obviously I can’t really comment on the bad loan rate at this stage, but then I don’t think it is a major part of evaluating Zopa itself. Every lender can adjust the risk he or she is willing to take by only offering money in certain (high-quality) markets or reducing the term of the loan they’re happy to accept. I believe that people might be less likely to default on their loans when they know that they owe their money to individual people, not big face-less institutions - if you had the choice, whom would you pay back first? Your neighbour or the bank? I might be wrong, but this is what I would like to believe and Zopa’s low bad-loan quota might prove just that.

If you’re intrigued by the concept and would like to explore alternative ways of making money / earning a return on investments, I urge you to give this a go. Sign up here to get £30 when you start lending (minimum amount applies) and become part of the Zopa Community. Trust me, it’s fun!

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Purchasing power risk is one of the most fundamental constraints you should always bear in mind when deciding how and where to invest your money. As much as compounded interest works in favour of any investor, inflation will always work against you and erode the real value of your money over time - if you don’t do your best to protect it.

The following table is taken from the book I’m currently reading (“The Art of Asset Allocation” by David M. Darst) and shows vividly how important capital protection is and what growth rates it requires over the years.

It is quite frightening to see that inflation at the 15% level would erode 96% of your money’s purchasing power if you kept it “safe” under your mattress. Obviously, most developed countries don’t face double-digit inflation rates anymore, but it is nevertheless an economic force that cannot be ignored. The list below (alphabetical by country name) should give you a reasonable idea of how much various countries are currently affected.

All figures express the year-on-year percentage change in inflation for March 2008, unless otherwise stated.

• Australia: 3.00% (December 2007)
• Bulgaria: 13.20%
• Canada: 1.40%
• China: 8.30%
• Colombia: 5.93%
• France: 3.50%
• Germany: 3.30%
• Iceland: 6.80%
• India: 5.20%
• Japan: 1.00% (February 2008)
  
• New Zealand: 3.44%
• Mexico: 4.25%
• Russia: 5.60%
• Spain: 4.60%
• South Africa: 9.8%
• Switzerland: 2.50%
• Turkey: 9.10%
• United Kingdom: 2.50%
• United States: 3.98%

It is immediately obvious that the inflation threat is more real in some countries than others - compare Japan and South Africa. To give you a better idea what sort of growth rates you need to achieve in order to simply maintain the purchasing power of your money, I have adapted the table above to show growth rates.

These are calculated by simply dividing 1 by the fraction representing the real value of your money after x years. For instance, if we assume that your money’s real value after 20 years at an inflation rate of 3% is equal to 0.54, then you need to nearly double your money to maintain purchasing power: 1 / 0.54 = 1.85.

So what can you do to achieve these growth rates?

• make sure the return on your savings account is positive in real terms (after inflation & tax)
• take advantage of tax-free savings and investments (ISAs, pensions etc.)
• if you want to play safe, inflation-linked gilts will always give you a fixed real return as they are linked to the current RPI (which includes mortgage costs and is hence usually a lot higher than the CPI)
• diversify your investments to reduce the impact of one asset class underperforming or showing negative growth rates
• avoid mutual funds with high entry and/or exit fees (where possible) as you will need even higher growth rates to just restore your purchasing power

A year ago on Simple Pound: Investment Choices - Bonds (I)

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I was just watching one episode of Channel 4’s documentary “Dispatches” that examines the root causes of the subprime crisis and the financial meltdown that followed.

Close to the message the documentary was supposed to bring across, it was called “How the banks bet your money” and I somehow knew there were going to be a number of things I would disagree with.

In one way or another, the documentary blames banks, credit rating agencies, regulators, governments and banks again for what has been an enduring theme across the public press for the last 6 months. The banks are blamed for creating the product that made the crisis possible and for becoming a victim of their own greed for higher and higher margins and the profits that were hoped to follow. The credit rating agencies are criticised for rating these financial products with one of their best ratings available in the market (S&P’s AAA-rating) and hence creating the fallacy that those products were inherently low-risk while at the same time understanding little of how they operated and apparently receiving payments from banks for these individual ratings. Gordon Brown is reproached with decoupling the financial regulation from governmental interference and thus having created a tripod of financial supervision. The FSA is accused of not acting in a way it was expected to when Northern Rock became a problem. And the list goes on…

Overall, I’m not disagreeing with many of the rational arguments that are being made in this documentary. I am, however, disagreeing with how it is presented and what misconceptions it will cause with people who don’t have a thorough understanding of how financial markets really operate (i.e. the majority of the population).

The suggestion that the invention of collateralised-debt-obligations (CDOs) was purely evil and driven exclusively by the bank’s greed is an immense exaggeration that fails to own up to the positive effects that financial innovation brings. The banks merely created the vehicle that was subsequently abused - and I’m not disputing that greed led to this abuse. Without financial innovation we wouldn’t be as developed a nation as we are. We wouldn’t have access to all the financial products we take for granted if it wasn’t for banks’ natural strive to create newer, better and ultimately more profitable products.

Credit cards are the best example - they’re not inherently evil. It’s just people’s misjudgement of their own abilities that can turn them into a harmful invention. Technically speaking, credit cards provide you with a 50-day interest free loan every month. It is not their fault that you think your cash supply is limitless.

Similarly, nobody forced people to take out mortgages they couldn’t afford. Of course the bank will try to sell you something they’ll eventually make a profit of. That doesn’t mean you’re supposed to blindly trust everything your banker is telling you. If you don’t feel you fully understand what you’re signing, then just don’t sign it.

The important side effect that people seem to ignore is that it also gave those people that didn’t have a long credit history the opportunity to own property while a few years back no bank would have even bothered to see that they might be eligible for a mortgage. It gave people who understood their limitations (!) the opportunity to buy their own place despite never having owned a credit card or never having taken out a loan. It’s not a flaw of the product that made people take out mortgages that were far to expensive for them in the first place. It’s also not the product’s fault that people wanted to own properties that were way out of their realistic budget.

If you expect that your bank will lecture you about what you can and cannot afford, then you’ve missed the fundamental principle of capitalism. At the end of the day, a bank is a company like any other that can only exist if it returns a profit at the end of the year. After all, Walkers doesn’t tell you to stop eating their crisps because they’re bad for you either. Don’t expect banks to operate as a super-human institution that would rather see it’s customers be happy than return a profit.

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I was grocery shopping the other day when the woman in front of me in the queue handed the cashier a piece of paper and got a payout of £100. Lottery ticket – as you might have guessed. I immediately felt the urge to buy some tickets myself to try out my luck, maybe do the unthinkable – get an immediate return from just one random lottery ticket. I resisted.

Most people would surely not consider playing the lottery a sound financial decision as the odds are phenomenally against you. Yet so many people seem to do it regardless and while fully aware of the fact they’re effectively throwing their (hard-earned) money out of the window.

Technically, premium bonds belong to the same category unsound investment you would associate lottery tickets with, nevertheless roughly half the UK owns them. An estimated £36 billion (that’s £36,000,000,000 !) is held in these “investment” vehicles that are guaranteed by the government.

The idea is that you buy a number of bonds of £1 nominal value (minimum £100, maximum £32,000) each of which is entered into a prize draw once a month. Depending on whether you’re lucky or not, you’ll get a cash prize of between £50 and £1,000,000. The catch is there is no guarantee that you will receive any payout whatsoever… so in the worst case, you could be owning £32,000 worth of bonds for years and never see a single pound return on your investment. In the best case you buy 100 bonds tomorrow and win £1,000,000 in the prize draw next month… I’ll leave it to you to work out which scenario is more likely to happen.

But to be honest with you, I’m still thinking about buying a few of those bonds – just to see whether I can win anything. It might not be the wisest thing to do with my cash, but then I already have all of my savings in accounts that earn between 6.30 – 6.76% AER which is quite possibly the highest return you can get in the market at the moment. So why not be just a tiny bit irresponsible?

I guess the moral of the story is that you should always fully understand the investments you pursue and only proceed if you are completely happy to accept the risks that come with it.

If you are considering joining the crowds to put your money to work in a Premium Bond, I strongly suggest you read the following article (twice): Premium Bonds - Are they worth it? If you’re still convinced Premium Bonds are the way forward, I’ll keep my fingers crossed that you’ll experience lots of happy months with decent returns accompanied by the excitement of having “beat the system”.

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