Methods to Calculate Historical Market Returns – Arithmetic Means versus Compound Annual Growth Rates

When I described previously how I was building my low charge investment portfolio and forecasting potential future retirement dates it turns out that in mathematics terms I have generally been using arithmetic means to calculate percentage returns from various data sets. That means I've taken the yearly percentage change for each entry of the dataset, summed these values and then divided by the number of items in the dataset.

It looks like this might be too bullish a method and instead I potentially should be using the compound annual growth rate (CAGR) which is the smoothed annualised gain. The formula is CAGR = (End Value/Start Value)^(1/number of years)-1.

Let me demonstrate with an example. At the end of year 0 your index is worth 100, by the end of year 1 your index has increased to 200 and then by the end of year 2 your index has decreased to 150. Using the arithmetic mean the mean annual return is (100% [year 1 gain] + -25% [year 2 loss])/2 = 37.5%. This can’t be correct as you don’t have 100 x 137.5% x 137.5% = 189 at the end of year 2. Now using the CAGR the return is (150/100)^(1/2)-1 = 22.5%. Checking this 100 x 122.5% x 122.5% = 150.

Let me now calculate a stock market example, the real (ie inflation adjusted) S&P 500 for the period January 1871 to January 2009. Using the arithmetic mean we get an average annual real price increase of 3.7% and an average annual real dividend of 4.5% for an average annual real return of 8.2%. Now using the CAGR (more complicated to calculate as I had to first calculate a real total return for the S&P 500) for an average real return of 6.7% which is significantly less than the 8.2% arithmetic mean calculation.

I'm now considering calculating real CAGR returns for my (S&P 500, ASX 200 and Gold) datasets and using these when projecting my retirement dates and amounts.

As always I would be very interested to hear others experience here.

You might also be interested in calculating portfolio year to date returns, annualised returns or multiple year returns.

How Much Am I Saving For My Retirement

My retirement investing strategy is focused on getting to a point where work can become optional as quickly as possible. Some would call this retirement. My thinking is that you never know what life is going to throw at you and by being financially independent at least it gives you choices.

To enable this to occur I do two things. Firstly, I live extremely frugally except occasionally I will also enjoying myself on a specific I consciously choose. Secondly, I always pay myself first. What this means is that as soon as my salary enters my account my retirement investing allocation is immediately moved to the appropriate location and I force myself to live on the remainder. If this means beans on toast at the end of the month then so be it. It would be so easy to spend first and then save later. However my thought is that there would be unlikely to be anything left to save at the end of the month as companies today have made it so easy to spend. I can really see how people get into this trap.

Additionally as pay rises are given (unfortunately usually only inflation matching) I always try and allocate this extra to my retirement strategy rather than increasing my standard of living. This means where I live is not as ‘nice’ as where my peers live, my television is not as large and I eat at home more but that’s the choice I have made in exchange for financial independence in the future.

Even where I am today, while not having yet reached retirement, I am happy knowing that if I lost my job tomorrow I would be ok. It’s amazing how liberating this is.

So what does this mean in terms of retirement savings per month? You hear about people saving 10 to 15%. If you save only this amount it’s going to be a long time before you get the retirement option. In my instance the UK government will take 23% (income tax and national insurance) and I will spend 17% on day to day living. This means I am left with about 60% of my earnings going to my retirement investing strategy. I am serious about financial independence and I intend to keep this up until I reach my goals. With a fair wind this is about 7 years away.

7 years is a short period of time meaning most of my final retirement pot will come from direct savings rather than the compound interest effect. If you are prepared to wait a lot longer for financial independence then compound interest may have time to work its magic meaning potentially less direct contribution from you.

Government Bond Yields are Rising

I monitor the government bond yields of three countries (Australia, United Kingdom and the United States) and they are all rising. My chart today shows the month end (except the last point which is the 08 January 2010) 10 year bond yields since 2007.

Why are they rising? Comparing Australia and the UK I think for different reasons.

As an outsider I think Australian bond yields are rising because the country is being run relatively prudently by the Reserve Bank of Australia and is raising interest rates as they are serious about keeping inflation at 2-3% over the cycle. The cash rate set by the RBA is now 3.75%. Savings account interest rates are also rising and without too much difficulty it is easy to find an instant access bank account paying 4.25%. So it makes sense for Bond yields to be rising to the 5.76% today.

I think United Kingdom bond (gilt) yields are rising for very different reasons:

Reason 1. I have shown previously that inflation is rising and it appears to me as though the Bank of England is going to hold interest rates at 0.5%, ignore their inflation target of 2% and start to let debts be inflated away which I discussed here. This seems to be reinforced by the Pension Fund of the Bank of England who have 88.2% of assets devoted to Index-linked gilts and other government guaranteed index-linked securities and 10.9% to fixed-interest gilts and other government guaranteed fixed-interest securities. This is up from 70.7% and 22.3% respectively in 2008. Buyers of government debt will however expect a real (after inflation) return on their investment and so if inflation rises then gilt yields must also rise.

Reason 2. The 2009 pre-budget report stated that UK government borrowing would be 12% of gross domestic product (GDP) in 2010/2011 and still a crazy 9.1% in 2011/2012. The Office for National Statistics reported on the 22 December that Q3 2009 GDP was £315.5 billion. Extrapolating this indicates that borrowing in 2010/2011 will be £150 billion and in 2011/2012 will still be £110 billion. To find buyers for all this debt (particularly if the Bank of England stops quantitative easing) you are going to have to attract them with increased yields.

Reason 3. The UK government are yet (and for that matter the Conservatives also) to explain how they are going to reduce the levels of borrowing. So far they have done nothing more than rearrange the deck chairs on the Titanic. If this continues the credit worthiness of the UK is going to be downgraded meaning yields will have to rise.

Reason 4. Those who already own government bonds and can see what’s happening will start to sell their holdings putting yet more gilts onto the market. This issue is real with the world’s biggest bond investor , Pimco, has started to sell off its holdings of gilts. More gilts coming to the market will mean gilt prices falling which will then mean rising gilt yields.
So what does this mean for my retirement investing strategy?

Firstly, if I owned gilts I’d be considering selling. As I’ve described previously I don’t own fixed interest gilts so I’m ok here. I do own index linked gilts but with inflation kicking off I’m comfortable with this.

Secondly, I’ll be watching house prices carefully. The interest rates on mortgages will have to rise as those wanting to borrow for a house will effectively be competing with the UK government for funds. I can’t see how house prices can continue to rise with increased borrowing costs and this could turn out to be the catalyst that brings on a reduction in house prices.

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The Bank of England Decides – Punish the Prudent

For the eleventh month in a row the Bank of England decided to keep the Official Bank Rate at 0.5%. This is the lowest rates have been even if I look back to the year 1694. Even during the Great Depression the Bank Rate only went as low as 2%. My first chart today shows the rates going back to 1948 showing just how low the Bank have set rates compared with recent history. Additionally, the Bank also decided to continue with its £200 billion quantitative easing program with £7 billion left to spend.

To me it looks as though the Bank of England decided today to punish the prudent and to reward the reckless. Who are the reckless? Well they are those who maxed out on as much credit as they could, whether to buy houses or plasma televisions. Who are prudent? Well they are those who didn’t extend themselves and decided to save for their future. I put myself into this category as I’ve decided to save for my retirement by investing with money I have earned.

What makes me think the bank has made this decision? As I’ve already identified previously inflation is starting to take off. Annualised UK Consumer Price Index (CPI) is currently 1.9% and the UK Retail Price Index (RPI) is currently 0.3%. The second chart however shows the true inflation story. As of November while the 12 month figure is 0.3% (as published by the ONS) disturbingly the 6 month figure is 3.6% and the 3 month figure is 4.1%.

By keeping the Bank Rate at record lows, quantitative easing and other factors like the return of VAT to 17.5% can only push inflation higher. I think the Bank has decided to take the inflation route. Allow those with debts to have them magically inflated away while those with assets see them devalued.

Why did I decide to take responsibility for my own future?

Australian Property Market – January 2009 Update

I intend to keep a close eye on Australian house prices as I build my retirement portfolio. This is because Australia is a very likely retirement possibility (if not sooner) for me.

The first chart shows the quarterly Real (adjusted for the Consumer Price Index) Brisbane and Real (again adjusted for CPI) Australian Eight Cities (Sydney, Melbourne, Brisbane, Adelaide, Perth, Hobart, Darwin & Canberra) House Price Index with data taken from the Australian Bureau of Statistics catalogue 6416.0 since 1991. This Index was reset in 2003/2004 and so I have “corrected” pre March 2002 data by taking the ratio’s of the pre and post September 2003 to June 2004 data as a multiplier. This chart carries data only until September 2009 and clearly shows a nice dip at the start of 2009. Could this be the Bull Trap phase of my second chart with us now nearing the Return to “Normal” phase?

My third chart shows Real Annual Changes in price from 1995 to present. In Real terms over this period Brisbane has seen average increases of 5.2% and the Australian Eight Cities has seen average increases of 4.8%. Unfortunately for me though the trend lines (particularly for Brisbane) continue to head upwards.

In non-inflation adjusted terms over the period Brisbane prices have seen average increases of 8.1% and the Australian Eight Cities prices have seen average increases of 7.6%. Unfortunately if you don’t already own a property (or three) you continue to be priced out when compared with average earnings. Using the Australian Bureau of Statistics catalogue 6302.0 which looks at average weekly earnings shows that Total Weekly Earnings has only increased by a yearly 3.8% and Total Full Time Adult Earnings by 4.3%.

My fourth chart shows what happens when house prices continue to rise at a rate greater than salaries. Over this period affordability of Brisbane houses when compared to Adult Full Time Weekly Earnings has gone from a low of 0.063 to 0.125 meaning affordability has halved and the Median Eight Cities houses have gone from a low of 0.064 to 0.113 which is a huge reduction. This type of shift is just not sustainable but when will the market turn? Will Australia raising interest rates and reducing first home buyer grants be the catalyst. Only time will tell...