CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
3.3 Choosing a Proxy for the Risk-Free Rate
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
N Rt =N Rt−1× P It P It−1
×
1 + DY 100×n
(3.1) WhereN Rtis the return index for dayt,P Itis the price index on dayt,DY is the net dividend yield of the price index and n is the number of days in a financial year (normally 260).
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 Real Risk-Free Rate
20 40 60 80 100 120
PV of 1 unit in Perpetuity
Beta=0 Beta=0.1 Beta=0.5 Beta=1
Figure 3.1: CAPM’s sensitivity to the risk-free rate in perpetuity
The figure shows the evolution of the present value under CAPM for an asset that pays one real unit yearly in perpetuity, when the real expected market return is 7% and we let r vary along the X-axis. Note how the present value of the low-beta assets become increasingly higher as r approaches zero from above.
According to both ECB and Consensus Economics, the current 10-year real government bond yields are negative for most leading European economies (see ECB(a) and CEconomics(a)).
Consequently, the Gordon growth formula cannot be applied to value a risk-free asset if real 10-year government yields are used as proxies forr. This is also the case for our proposed model, which breaks down when valuing perpetual cash flows if the risk-free rate is zero or negative.
Note that this problem can be disguised, but not solved, by using nominal rates. At the time of writing, the nominal 10-year government yields are in fact mostly positive. The reason being that the expected inflation is more positive than the real risk-free rate is negative. However, nominal yields and discount rates require nominal cash flows. Hence, if we are consistent in our choice of risk-free rate and estimation of cash flows, our nominal cash flows should grow with the same inflation that is included in the nominal risk-free rate. Since the inflationary effect in the cash flows cancel out the inflationary effect in the risk-free rate, we are back were we started, i.e. we cannot use Gordon’s growth formula to price the risk-free asset paying cash flows in perpetuity. Negative r also contradicts the assumption made in section 2.1.2, which is that the sum over all state prices is less than one. I.e. we have the following
B0 =
S
X
i=1
vi >1if r < 0
Since the sum of the state prices is larger than one when the real risk-free rate is less than zero, we know that the sum of a geometric series ofB0 will not converge, in fact it will diverge. This is
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
of a geometric series introduced in section 5.1.7, can be used to value a perpetual stream of cash flows. Conceptually, the problem is linked to what the real risk-free rate actually represents. A textbook idea in finance is the concept of time value of money, which is that money is worth more today than tomorrow due to its earnings capacity. Following this intuition, negative r implies that money is worth more tomorrow than today, i.e you receive a premium from borrowing money. In neoclassical economics, the rate of time preference captures the tradeoff between consumption today and consumption in the future. This is thus the underlying determinant of r. In this setting, causality runs from the aggregate propensity individuals have to consume in the present (versus the future) to r, not the other way around. Thus, it is not the fact that individuals can earn positive r that results in the time value of money, it is the fact that individuals generally prefer to consume today versus tomorrow that results in a positive r.
Taking on this view implies that today’s individuals prefer not spending today. In fact, they would rather spend tomorrow. In section 2.1.5 we derived a utility based expression for r in a one-period economy
r= ln 1 exp(−δ)
u0(g0) E[u0(g1)]
!
The expression states that r is a function of impatience, current marginal utility and future expected marginal utility. Furthermore, the impatience factor exp(−δ) is restricted to being positive, i.e. individuals generally prefer spending today rather than tomorrow. The risk-free rate can in this setting become negative if the expected marginal utility in the future compared to today is high enough to offset the impatience effect, at least in the short run. Nonetheless, we should not go too far with interpreting the formula above. After all, it is derived from the strictly theoretical Arrow-Debreu framework. For example, it should by no manner be under-stood as a description of the dynamics of real government bond yields, although it can offer some useful insights, as shown above.
We argue that the issues related to negative real yields on government bonds, are general problems for economists and practitioners, and that there exists no perfect solution as of today.
Furthermore, the goal of this thesis is in no way to solve these problems (although we will provide an imperfect remedy). It is nevertheless important to highlight these issues, as the level of the real risk-free rate highly influences the present value of any project valuated using our proposed model. As shown, the effect is increasingly strong when the real risk-free rate approaches zero. Therefore, the user of our model will strongly benefit from understanding the intuitive implications of changes in the parameter r.
3.3.1 Separation of the Risk-Free Rate
Based on the previous discussion, we choose to use two risk-free rates in our proposed model.
One in the forecasting period and one in the terminal period. We will use an observed yield in the forecasting period, to avoid making assumptions about the appropriate level of the risk-free rate. In the terminal period, we will constrain the risk-free rate to be strictly positive. This is both a mathematically and economically consistent solution, which builds on the idea that time value of money should induce a positive real risk-free rate in the long run. Thus, we have established a lower bound for the long-term risk-free rate. In the next section, we will establish an upper bound and present a logical argument for the long-term risk-free rate.
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE
3.3.2 A Logical Argument for the Long-Term Risk-Free Rate
As one of the main themes of this thesis is applied capital budgeting, we seek to choose the long-term risk-free rate in a simple manner, while preserving as much economic validity as possible. To do so, we begin by establishing a basic economic premise for the real risk-free rate, based on the discussion in the previous section. If we assume that causality runs from aggregate propensity to consume to the risk-free rate, and that causality runs from aggregate propensity to consume to economic growth, then we have the following propositions:
1. An increase in the propensity to consume today, implies a decrease in the demand for the risk-free asset, which then again implies an increase in the risk-free rate
2. A increase in the propensity to consume today, implies an increase in aggregate consump-tion in the present, which then again implies an increase in current economic growth Although this is a great simplification of reality, the above provides some intuition to how economic growth and the risk-free rate are related. Furthermore, it is natural to conclude that we need economic growth to support the yield on the risk-free asset, at least in the long run.
We can use this conclusion to define an upper bound. Since in the long run, the promised yield on any investment must be supported by actual economic growth, the promised real long-term risk-free return cannot be higher than the real long-term growth in the economy. Moreover, if the real risk-free return is equal to the real economic growth, there is no compensation for risk in the long run, which there should be. Hence, the real risk-free rate is bounded to be lower than the real economic growth in the long run. Together with our previously defined lower bound of r > 0, we now have a reasonable interval of choice for the long-term risk-free rate.
Obviously, the exact choice is guesswork, but as stated previously, we do not claim or aim to solve the problem of a negative real risk-free rate. However, we do believe that we have settled on a reasonably prudent approach for choosing it. We note that the exact choice of our risk-free rates is presented in chapter 5.
CHAPTER 3. CHOICE OF UNDERLYING AND RISK-FREE RATE