This article offers some insight into determining long-term expected returns on “risk-free” assets which are then used as a basis for risky asset forecasts and strategic allocations. It first examines the notion of “risk-free” assets in themselves in order to demonstrate that, even if regularly renewed short-term investment does present some relevant characteristics, there is in fact no such thing over the long term. It then analyses the impact of different economic and financial theories on short-rate forecasts and includes a series of tests over a long-term period from 1930 to 2013 in the United States. The simple operational solution derived from these findings consists in a compromise between:

- the current short rate given the strong autocorrelation between short rates (
**25%**), - inflation, which avoids endless extrapolation from either very restrictive or extremely accommodating monetary policies linked to a specific backdrop (
**25%**), - long rates over the full horizon as the yield curve provides some information on future short rates (
**50%**)

## I - Why this question?

When building a strategic allocation over the long term, investors assess an asset’s appeal according to its risk-return trade-off. To do so, they often reason in terms of its expected excess return in excess of a “risk-free” investment. One could, therefore, be forgiven for thinking that risk premiums on assets are the only thing that matter and that the risk-free rate is of little importance.

This is not, however, always the case since long-term solvency forecasts or accurate simulations for a product or portfolio very often require a hypothesis in terms of cash returns which is then used as a basis in determining the profitability of assets by adding a risk premium. This report focuses on long-term horizons of at least 5 to 7 years.

Normalized returns are taken to mean returns that are coherent but that are not based on any macroeconomic forecasts. The aim is to establish a set of rules that use market information that is readily available when determining normalized returns. First we analyse the notions of cash and risk-free rates, then we review the most commonly-used approaches to establish the normalized returns on cash. We use historical data from the US to ascertain how accurate these different approaches are, and offer a compromise that is easy to implement.

## II - Cash, a “risk-free” rate that really does mean no risk?

Determining whether or not cash is entirely risk-free depends on the investment horizon and instruments in play, and should prompt us to question the notion of risk itself which can take a number of forms over a long investment period. For investors who choose not to spend in the immediate term but to invest over the long term, the risk is perceived as the uncertainty linked to what the money they have invested will be worth in the future. This uncertainty can have several causes. The first thing we naturally think of is the volatility of the markets. However, over the long term, volatility is not the only source of risk. There is also the risk of erosion via inflation, the risk of default, systematic risk and, even in the absence of these elements, the uncertainty linked to the future level of rates means there is a risk of opportunity at the time of the reinvestment of coupons, dividends, or repayments. Over longer horizons, the hierarchy of risks can also differ from over the short term.

**Over the long term, the risk of reinvestment is far from negligible when it comes to cash.** The graph below shows the standard deviation in the annualised return on cash over rolling periods of 1, 3, 7, 10 and 30 years. The calculations apply to the US between 1929-2013.

It is clear that, over long and forward-looking timeframes, cash is not without its risks. If, for each horizon, we determine risk in the form of an equivalent normal volatility which offers the same dispersion (i.e. by multiplying it by the root of the horizon), we see that equivalent volatility increases.

This means that, over a 30-year horizon, the risk of reinvestment linked to cash is high. In fact, dispersion is similar to that of an asset with a volatility of 12% (namely around 60-80% of equities). While the natural response, to avoid this risk of reinvestment, is therefore to define zero-coupon investments with a nominal rate over the horizon in question as risk-free, this definition poses several problems.

First, although it is true that nominal returns are known, this is no longer the case if we reason in real terms. **A zero-coupon investment over the long term is seriously exposed to the risk of erosion by inflation**. As such, a regularly renewed short-term investment (i.e. for which there is a certain adjustment in short rates in relation to inflation) is therefore less risky in real terms.

Second, depending on the issuer, **such an investment may carry a credit risk.** Before the crisis, and at least for strong currencies, the tendency was to consider government bond rates as risk-free. Within the eurozone, the sovereign debt crisis led to a serious discrepancy in government borrowing rates depending on the country, much in line with the drop in ratings for peripheral countries. Unless we accept the highly debatable view that returns on assets in euros would be different according to country, a common reference curve is a fundamental requirement. Swap rates are of course an option as their collateralisation means they incur very little credit risk. Having said that, the crisis in confidence in the banking system and the increase in Libor rates led to a distortion in swap rates, linked not to the risk of these instruments, but rather to the very strong distortion of the Libor 3-month rates used as a reference for variable rate swaps. One remedy was to replace the Libor 3-month rate by a daily rate that was therefore less distorted. OIS rates (overnight interest swaps) do indeed appear to bear less credit risk even if there are no long-term time series on OIS rates.

Lastly, if we compare the risky nature of a cash investment in relation to a zero-coupon investment, the latter is exposed to a capital risk in the event of early redemption.

These findings suggest that **there are in fact no universally risk-free assets over the long term**. An investment decision must factor in the known or unknown horizon and the outlook in nominal or real terms. Insofar as our goal is to use risk-free investment returns as a reference to which we can then add a risk premium on assets, we think it preferable to define this “risk-free” reference as what the markets** appear ****to consider as less risky as they are less demanding in terms of returns: cash**. In practise, proxies can be used as long as they do not present any substantial risk in relation to a day-to-day investment, and they correspond more to what would be achieved via a regularly renewed short-term investment. It is nonetheless prudent to remember that, while this asset may bear the least risk, it is not entirely risk-free, except over short-term horizons. Wherever possible, the rate for cash investments must be close to an OIS rate over a short horizon or to a government rate as long as the country has an AAA rating or genuine control over its currency.

## III - Approaches

### 3.1 Macroeconomic equilibrium model

In a balanced economy, the interest rate at which the economy finances itself is equal to the sum of the equilibrium values for (real) GDP growth and inflation. This approach is particularly useful in establishing a reasonable target value for a short rate associated with a short-term investment in the far distant future. This perfect equilibrium is obviously not achieved today which is why this approach cannot be used to directly forecast future cumulative returns for cash starting immediately. Furthermore, the rate at which the economy finances itself is a mix of short and long rates. Ascertaining the cumulative return on cash over a long period starting today using this method means establishing a path between the current and target rates. Finally, this method also requires an estimation for growth and inflation at equilibrium.

### 3.2 Yield curve

The great merit of methods that use the yield curve is that they are based on observable data and not on forecasts. A first solution is a pure and simple extrapolation of the current short rate. The virtue of this solution is its simplicity, but it is only justified if investors anticipate that rates will remain stable which will mean a flat yield curve. The slope between long and short rates is, however, much too variable over time to make it an acceptable hypothesis. Another approach is to use the zero-coupon rate over the investment horizon as a normative forecast for the return on cash. This approach is based on two arguments. The first is to say that, in nominal terms at least, the return on a buy and hold bond investment is known in advance and can therefore be considered as risk-free. We have seen that this point of view is debatable and, in any case, in no way guarantees the link between the initial long rate and the return on cash as the investment is not made on the same asset. The second argument is to tie the return on cash to long rates on the basis that long rates are an indication of the future change in short rates (Lutz [1940]), i.e. to say that the best forecast for the behaviour of short rates is that they will evolve towards forwards. This also enables us to define a coherent trajectory for short rates. Despite its simplicity, this model is criticised for various reasons. The first is that it implicitly supposes that short rates evolve in a deterministic manner and fails to factor in their random nature. In actual fact, there is a risk premium between long and short rates which explains why the slope of the yield curve is most often positive. The second is that short rates are essentially administrated and do not factor in agents’ future forecasts. Long rates, on the other hand, are market data which do factor in short rate forecasts but which are also affected at certain times by other criteria (fly to quality on the German curve, regulatory changes which affect the appetite of buyer pension funds for very long bonds).

### 3.3 Arbitrage pricing theory

This theory was presented for the first time by Black and Scholes to value options on stocks. Over the last 30 years, countless variations have been applied to different asset classes. The first to apply the theory to interest rates was Vasicek [1977] for short rates, after which adjustments over the entire yield curve were introduced by Heath, Jarrow, Morton [1989] and Hull and White [1997]. This theory is used as the basis for the valuation of option derivatives and explicitly takes into account the uncertainty linked to future rates. **It also provides a framework which allows for the clear separation, for long or forward rates, between what is linked to rates forecasts and what constitutes a risk premium**. The risk premium depends on the implicit volatility of rates and a coefficient in terms of risk aversion which can be mapped suing the Sharpe ratio for the asset class. For our purposes, this theory is used to establish that future short rates are lower than forward rates. Correction in absolute terms is the more significant so as volatility is high and risk aversion is strong. As illustrated in the graph below, when it comes to current standard levels, the size of the correction is significant, particularly for long horizons.

Over a given horizon, the level of volatility affects the risk premium. The graph below shows how the level of volatility on long rates affects the correction. All other parameters were kept constant for these calculations. Over a 7-year horizon, the effect becomes really significant when volatility of long term rates is high. Today, for a volatility of around 0.5%, the correction is around 35 basis points.

While these approaches are interesting, they require complex calculations and the calibration of a substantial number of parameters. It is nonetheless useful to keep in mind the implications of this theory.

## IV - Historic analysis of long series in the US

We have examined, over a long period of time, the relationship between long and short rates in the US which is the country for which we have the longest historical series, and whose long and short rates are not too distorted by credit risk and the government administration of foreign exchange rates. This is more delicate when it comes to the eurozone where, in the 1980s and 1990s, short rates were affected by German reunification and the currency crises in the European monetary system.**Forecasts based on the initial short rate alone are not very credible, except ****over short horizons**. We looked at several horizons ranging from 3 months to 30 years. For the 7-year horizon, for example, we calculated the cumulative performance over 7 years of a short-rate investment and compared it with the short rate at the start of the period - the difference constituting the forecast error. To ascertain whether the use of the initial short rate is actually relevant, we compared the dispersion of the forecast errors with the actual unconditional dispersion of the annualised returns on cash over the same horizon.

The following graph shows the difference between dispersion figures.

Unsurprisingly, over short horizons, the dispersion on forecast errors is much lower than the dispersion on the actual unconditional returns for cash. In fact, it is virtually nil which is how cash earned its reputation as a “risk-free” asset. This is not verified over long horizons for which it is therefore much less important to know the initial short rate.

We examined whether it is more useful to know the initial long rate, and the long rate for which we have very long series is the 10-year bond yield. However, because a coupon bond with a 10-year maturity has a shorter duration (around 7 years), we examined the cumulative returns on a short-rate investment over 7-year periods, and analysed the forecast errors made using the initial long rate for each sub-period.

The dispersion is a little lower than with the initial short rate but is still strong.

Here, we employ a 7-year horizon as an approximate duration for the long rate used. Our aim is to identify the relationships between the forecast errors using the short rate or the initial long rate and other variables.

In the first instance, the graph below shows that** when we use the initial short rate alone, this forecast error is linked to the slope.**

While this relationship is significant and robust, it is also variable over time with a sensitivity of around 50% on average.

More specifically, we limited our analysis to periods where initial conditions are not fundamentally different (initial inflation between 0% and 5%). The graph below shows the relationship between the forecast errors and the slope depending on whether we use the long rate or the initial short rate.

What we find is that there is a positive relationship between the forecast error using the initial short rate and the slope. Using the initial long rate on its own creates an error with a negative and often downward bias in relation to the slope. This means that the **initial long rate globally overestimates the future return on cash, all the more so when the slope is steep.**

The graph below shows that there is also a link with the volatility of rates.

Using the initial long rate means we overestimate future returns, all the more so when volatility is high. The scope of the phenomenon is also significant since, for 1% volatility, using the initial long rate overestimates the return on cash by an average 0.8%. While this is consistent with the arbitrage pricing theory, it is even higher than what is predicted by the theory with a Sharpe ratio of 0.3. There are two possible explanations here. The first is that the decline in rates during the period in question meant that the Sharpe ratio was higher. The second is that the increase in risk aversion during periods of strong volatility appears to accentuate the phenomenon.

Given today’s extremely positive curve and the low volatility in rates, using short and long rates together therefore seems more effective than using either one or the other on their own. The correction linked to volatility, which involves complex calibration, is less useful.

We can also see that the forecast error is linked to the real short rate.

Mitigating the short rate with inflation means we can improve the forecast as we avoid having to extrapolate a short rate level, distorted due to a very restrictive or accommodating monetary policy, which, in the long term, is not consistent with the fundamentals.

Our forecast model mixes the short and long rates and initial inflation. This enables us to eliminate a substantial part of the dependency between the forecast error and the slope and real initial rate, even if it does remain highly sensitive to a change in inflation regime. The graph shows that there is a strong relationship with the variation between initial and cumulative inflation.

The preferred alternative, where available, is forward inflation.

We have found no link between the forecast error and growth.

## V - Operational implications

In order to marry operational simplicity and coherence with theory and observation, we recommend using **a combination of short rates (25%), inflation (25%) and zero-coupon long rates (50%) over the horizon as a normative forecast for returns on short-term investment.**

The initial long rate segment (50%) factors in the data on the future trend in rates extracted from the yield curve.

The initial short rate segment (25%) reflects the fact that, on average, short rates are lower than their forecast using long rates and present a strong level of autocorrelation.

The inflation segment (25%) is used to correct the distortion between short rates and the macroeconomic fundamentals linked to a temporary episode of very accommodating or very lax monetary policy.

Today, we recommend using the OIS curve rates or, failing that, government bond rates where governments have an acceptable rating or are in control of their currency.

We then suggest extracting the forward short rates from the curve obtained for the future performance of cash using the same mechanism as that used to calculate forward rates.

This rule should however be reviewed in the event of:

- a notable change in inflation forecasts,
- a significant increase in the volatility of long rates,
- an inversion in the yield curve.

## Acknowledgements

I would like to thank my colleagues Jean Gabriel Morineau, Gianni Pola and EricTazé-Bernard at Amundi for their valuable input in improving this paper.

### References

**Black F, Scholes M** [1972] “The Valuation of Option Contracts and a Test of Market Efficiency”, Journal of Finance, Vol. 27.

**Heath DC , Jarrow BA, Morton A,** [1992] “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation”, Econometrica, Vol. 60.

**Hull JC, White A** [1997] “Options, Futures and Other Derivatives”, 3rd Edition, Prentice-Hall.

**Musiela M, Ruthowski M** [2004] “Martingale Methods in Financial Modelling”, 2nd Edition, Springer.

**Lûtz FA** [1940] “The Structure of Interest Rates”, Journal of Economics, Vol 55.

**Modigliani F, Shiller R** [1973] “Inflation, Rational Expectations and the Term Structure of Interest Rates”, Economica, Vol 40.

**Vasicek O** [1977] “An Equilibrium Characterization of the Term Structure”, Journal of Financial Economics, Vol 5.

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