What does Risk do to your Investment Returns? Is there a free lunch?
With increasing longevity, leading to a longer time spent in retirement, we are all keenly aware that it is no longer enough to save for our retirement nest egg. Instead, we are told that we have to make investments to make the nest egg grow, not only in order to outlast the many years we will spend in retirement, but also to outpace the inflation in the years before and after retirement. To get higher investment returns, we will have to take higher investment risk. But over a long period of time, we are also told that it is more likely, based on history, there will be a higher chance that our savings will grow more than if we just left them at the bank.
Risk and Investment Returns
How true is this advice? Well, the first part is definitely true. Risk is usually defined as volatility of prices or the value of our investments. Stocks, which have higher risk, tend to have higher returns compared to bonds, which have lower risk.
For most of us, our retirement savings are mainly our CPF, which yield between 2.5% to 4% a year, riskfree. If we invested these savings in riskier investment funds and the like, we can earn a higher return, say 6%. Over a period of 30 years, $100,000 invested at these different returns will yield very different outcomes:
Value of $100,000 invested in different options after 30 years
CPF Ordinary Account 2.5%  CPF Special Account 4%  Investment 6%  

Value in 30 years  $209,757  $324,340  $574,349 
Graphically, it looks like this:
Value of $100,000 invested in different options after 30 years
Taking more risk for more returns
A 6% annual return is usually marketed for a balanced fund of 60% in stocks and 40% in bonds. What if we took on more risk? For example, investing in a 100% stock fund for returns of 10% annually. Or maybe even throwing in some derivatives and active stock picking or management to get 12.5% annual returns? What would we get after 30 years?
Value of $100,000 invested in different options after 30 years
Investment 6%  Investment 10%  Investment 12% 


Value in 30 years  $574,349  $1,744,940  $3,424,330 
Graphically, it starts to look really interesting, like this:
Value of $100,000 invested in different options after 30 years
Why don’t we just take more risk in our investments?
The returns after 30 years shown certainly look tantalizing! Taking more risk means that we can fund our retirement quite easily. We usually measure risk by the volatility of annual returns, as a standard deviation. The Sharpe Ratio, which is the excess returns (over the riskfree rate) divided by volatility, gives the relation between risk and return.
Assuming that the investment options we have have a Sharpe Ratio of 0.50 (which is quite good, actually), and also assuming (wlog) that the riskfree rate is 0%, this means that our different investment options may have these different levels of risk:
Risk and Returns of Different Investment Options
Annual Return  Volatility  

Balanced Fund  6%  12% 
Stock Fund  10%  20% 
Actively Managed Investment  12.5%  25% 
As a comparison, Asian stock market indices like the Straits Times Index, or the Hang Seng Index have volatilities between 20% to 25%, while a larger market like the US S&P500 Index has volatility of between 15% to 20%. So we are not overstating the risks of the investment options here (although we may be overstating the returns!).
But you can argue that with a 30 year investment horizon, the risk of our investments shouldn’t matter much. It is true that volatility of the investment affects the yeartoyear value of our investments. But since we are cashing in our investments in the short term, we can stomach the risk.
So, shouldn’t we go for the highest risk investments available? After all, we are long term investors. This gives us the highest returns and best outcomes. Is there really a free lunch?
There is no free lunch. Sorry!
As you may expect, there is no free lunch, least of all in finance. Taking higher risk in our investments does eat into the expected returns over a long period. In fact, over 30 years, the reduction in the expected returns (see the dotted lines below) can be very significant.
Effect of risk on returns over 30 years
Expected Value without Volatility  Expected Value with Volatility  

6% Balanced Fund  $574,349  $468,141 
10% Stock Fund  $1,744,940  $1,006,266 
12.5% Actively Managed Investment  $3,424,330  $1,470,761 
Graphically, at the highest risk levels, the effect of risk cuts the expected return by half over 30 years!
Effect of risk on returns over 30 years
What is happening here to our expected returns? We can answer this with a simple example. Suppose you have two investments. The first has a return of 0% every period. The second can goes up 10% and then falls 10% alternately, so the average return is 0% as well. What happens after a while?
Value of two 0% investments, with and without risk
Without Risk  With Risk  

Period  Return  Value  Return  Value 
1  0%  $100,000  +10%  $110,000 
2  0%  $100,000  10%  $99,000 
3  0%  $100,000  +10%  $108,900 
4  0%  $100,000  10%  $98,010 
5  0%  $100,000  +10%  $107,811 
6  0%  $100,000  10%  $97,030 
The risky investment ends up with a lower value over time. This is even though both investments have the same average annual return. Risk and volatility creates a drag on returns over time. This is not noticeable at first, but becomes larger over time.
The mathematics of compounded returns in investment
The answer to this phenomenon comes from Brownian Motion, which won Einstein his Nobel Prize. Brownian Motion studies the behaviour of particles which have random movements over time, much like how share prices behave. Applied to finance, an annual return of x% and a volatility of y%, gives an average annual compounded return over time of:
This is the same as the geometric return. Applying this to our three investments previously, we find that risk has reduced the return on the investments by:
Risk and Returns of Different Investment Options
Annual Return  Volatility  Annual Compounded Return  

Balanced Fund  6%  12%  5.28% 
Stock Fund  10%  20%  8.00% 
Actively Managed Investment  12.5%  25%  9.38% 
These effects of risk on return look small on an annual basis. But over a long period, they can reduce the value of an investment by half!
Does dollar cost averaging help?
The effects of risk on investment returns for a lump sum investment can be huge. Does the same apply for investments made over time, using dollar cost averaging? The answer is yes, but to a lesser extent. We show this below:
Investment returns on $7,000 invested annually over 30 years
Investing $7,000 annually over 30 years in an investment with a 6% return will still give us around $600,000 at the end. This is just like investing a lump sum of $100,000 will. For the investments which yield 10% and 12.5% annually, the reduction in total returns due to volatility are smaller as well, compared with a lump sum investment.
When we apply dollar cost averaging to a comparison of investments versus CPF savings, we can appreciate the conventional wisdom that we should leave our Special Account (SA) balances untouched. Instead, only make investments with our Ordinary Account (OA) balances. The increase in returns over the Special Account rate of 4% is likely to be small.
Investment returns on $7,000 invested annually over 30 years
Concluding Thoughts
A large and growing industry of insurers, banks, financial advisors, and asset managers have been hard at work persuading us to invest to grow our nest eggs for retirement. An oftenheard argument is that as long term investors with 30 years to go before retirement, we can afford to be more aggressive in investing, taking more risk.
While it is true we need to take more risk for more return, time diversification of risk doesn’t really work. Paul Samuelson showed back in 1963. Risk creates a drag on returns over time, which is why our long term returns are often lower than promised by fund managers based on their 3 or 5 year performance track records.
This drag on returns is also why an investment in the Straits Times Index with an annual return of 8% cannot to sustain a 4% inflationadjusted withdrawal rate for a retirement period of 30 years.
We show this in Does the 4 percent rule work in Singapore? The 25% annual volatility of the Index returns has a 3% drag on it over time, lowering it to a 5% compounded return, which is insufficient to sustain a high withdrawal rate.
So, a takeaway is not to ramp up risk as high as possible to get those 10% to 15% returns. These expectations are not realistic over the long term due to the high volatility of the investments.
So what can an investor investing for retirement do? Keeping money in the CPF Special Account is a good way to grow your nest egg. Otherwise, take some risks to get better returns with your savings, such as a 6% to 8% annual return, but do not overreach on the risk taken to do so.
Ultimately, here is no free lunch, least of all in finance.
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