### Personal finance 7: investment return

#### by George Hatjoullis

One of the most important aspects of managing personal finance is to understand how well you are doing. It also helps to understand how well those trying to lure you are doing and what they might be offering. The issue was partly covered in Personal Finance 4 on interest rates. What about portfolio returns and the returns quoted on investments and funds that you might consider?

First, consider a simple holding period return. You invest £10 and get back £11 at the end of the period. You have made 10%. We divide 11 by 10 and get 1.1. This is the return ratio. It is the rate of return (10%) divided by 100 and added to one. When working with returns this ratio is the basic unit.What if you take your first period return out and just reinvest 10 again? Assume you lose 1 in this period so that, at the end of the second period,you have 9, which divided by 10 equals 0.9. You have lost 1 and made 1. Does that mean your average rate of return is zero? Apparently not. Your compound return is derived from 1.1 x 0.9 = 0.99. If you add one and multiply by 100 you get -1%. To annualise this you take the square root of 0.99 (as there are two years), subtract 1 and multiply by 100 and you get -0.5% per annum. This counterintuitive result should be borne in mind when using compounding. It arises because compounding involves **geometric averages** and relates to the difference between **geometric and arithmetic averages**. Both are ‘correct’ but may or may not be appropriate depending on the context.

Now consider investing £10k each year for 3 years. The investments are made at the start of each year. The value of the portfolio is £11,000 at the end of year one, £21k at the end of year 2 and £35k at the end of year 3. So how well have you done? Your total investment is £30k and you have made £5k. You could say you have made 16.67% over 3 years, the same as 5.3% per annum. Or you could calculate the ** time weighted rate of return**, which is more common in the financial services industry. To do this you simply calculate each holding period return ratio and compound the rates of return.

The holding period return ratio is the simple ratio return up to the point of you next cash inflow (or outflow). The return ratio is the value of fund at the end of the period but before the next cashflow, divided by the value of the fund at the start of the period including new cashflows. In this case the first holding period return ratio is £11000/10000= 1.10. The second holding period return ratio is £21000/21000= 1. The value of your fund was £11k at the end of the first period and you have added £10k. The third holding period return ratio is £35000/31000= 1.129. The value of your fund was £21k at the end of period 2 and you added £10k. The compounded return is 1.1×1.0x1.129=1.2419. To get an annual rate of return just take the cube root, subtract 1 and multiply by 100. The result is 7.41%. This is somewhat better than the crude method used above and is most likely to be the basis of the returns reported to you by various funds. For those comfortable with a little maths this link should be interesting (http://bit.ly/1ghuYIa). It explains why this is the preferred method of calculating portfolio returns.

For many even the above example may be a struggle so a little elaboration. The time weighted rate of return merely calculates each return up to the next big cashflow, combines the returns by compounding and then expresses this total return as an annual figure. The return periods need not be equal nor annual. The return is just the value of the fund at the end of the period plus any dividends or interest, but before the next cashflow, divided by the value of the fund at the start of the period which will include any new cashflow. The cashflow could also be withdrawals. When comparing returns try to establish how they have been calculated and make sure the method is consistent. Comparing apples and oranges is likely to result in nuts. If the return is not a time weighted rate of return then you need to look deeper.

The other important consideration in calculating returns is the matter of inflation adjustment. Money is the unit of account in most investment calculations and calculating money gains is the common practice. However, money has value as a claim on goods and services. Money has a ‘GDP’ or commodity price (see Economic Series if this issue is interesting). Even though you account in money you are interested in what you can buy with it. If the commodity price of money changes it should alter your investment calculation. If inflation is positive then the amount of goods a unit of currency will buy falls over time and it is worth factoring this out to get a *real return. *

The first consideration is which inflation measure? Each official country statistical office calculates some measure of inflation but it may not measure ** your** inflation rate. Strictly speaking you should adjust for a basket of goods in which you have interest and the official basket may not reflect your consumption interest. Students and pensioners have different consumption baskets. Those saving to buy a home may have different consumption plans to those that habitually rent. Planned consumption should be a part of any investment strategy and a good place to start is by incorporating it in the real return calculation. Is it entirely appropriate to save for a house deposit in a savings account if the rate of house price inflation is higher than the deposit interest? If today I need a deposit of 10% for a 500k house then I need to save 50k. If house prices rise by 5% per annum then in ten years I will need 81.44k as a deposit. If I have saved 10k this year this amount will be worth 6.14k in ‘house price units’ in ten years. I need to make at least 5% return in nominal or money terms in order to maintain my initial 10k’s value as part of a 10% deposit. In reality everyone just measures inflation using official statistics but the principle should be borne in mind.

So how does one incorporate the chosen measure of inflation? The purist might convert all nominal values into a stream of ‘* constant price*‘ values and work with the derived constant price values. Price indices have a base year, usually 100, and the values represent cumulative inflation from the base year. They can also go backwards by setting the base year to the current year. However, the more common but equivalent approach is to calculate the rate of inflation over the holding period (using the index) and then adjust the holding period return ratio by the inflation ratio. If the holding period return is 6% and the inflation rate is 5% then our adjustment is 1.06/1.05=1.0095. The real return ratio is 1.0095 and the real return is 0.95%. The time weighted rate of return is then calculated as before but using the real return ratios.

The main purpose of this, yet again, dry personal finance lesson is to make clear that return calculations are not as simple as they may seem and that it is important to compare like with like. The time weighted rate of return is industry standard but always check how the returns you are being touted have been calculated and what is actually on offer. Take nothing at face value and if in doubt, ask. The secondary purpose is to encourage all not just to think about real returns but to make sure the real returns are real from the individual’s perspective. Money ultimately must be judged by what it will buy. The following in the series will start to look at some specific personal finance issues.