**The long-term discount rate**

**Philippe Raimbourg**

Director of the Ecole de Management de la Sorbonne (Université Panthéon-Sorbonne)

Affiliate professor at ESCP Business School

If since Irving Fisher we know that the value of an asset equals the discounted value of the cash flows that it can generate, we also know that **the discounting process significantly erodes the value of long-term cash flows and reduces the attractiveness of long-term projects.**

**THIS RESULT IS THE CONSEQUENCE OF A DUAL PHENOMENON:**

• **the passage of time**, which automatically whittles down the present value of all remote cash flows;

• **the shape of the yield-to-maturity curve**, which generally leads to the use of higher discount rates the further in the future the cash flows are due; indeed, we usually observe that the yield curve increases with **the maturity of the cash flow considered.**

**THE DISCOUNTING PROCESS SIGNIFICANTLY ERODES THE VALUE OF LONG-TERM CASH FLOWS**

For this reason, the majority of companies generally invest in **short-term and medium-term projects** and leave long-term projects to state bodies or bodies close to public authorities.

We will try to explain here the potentially inevitable nature of this observation and under what conditions long-term rates can be

less penalising than short-term ones. **This will require us to explain the concept of the ‘equilibrium interest rate’ as a first step.**

**THE EQUILIBRIUM INTEREST RATE**

We are only discussing the risk-free rate here, before taking into account any risk premium. In a context of maximising the inter-temporal well-being of economic agents, **the equilibrium interest rate is the rate that enables an agent to choose between an investment **(i.e. a diminution of his or her immediate well-being resulting from the reduction of his or her consumption at moment 0 in favour of savings authorising the investment) and a **future consumption, the fruit of the investment made.**

**WE CAN EASILY SHOW THAT TWO COMPONENTS DETERMINE THE EQUILIBRIUM INTEREST RATE:**

• economic agents’** rate of preference for the present**;

• **a potential wealth effect that is positive** when consumption growth is expected.

The rate of preference for the present (or the impatience rate) is an individual parameter whose value can vary considerably from one individual to another. However, from a macroeconomic point of view, this rate is situated in an intergenerational perspective, which leads us to believe that the value of this parameter should be close to zero. **Indeed, no argument can justify prioritising one generation over another.**

The wealth effect results from economic growth, enabling economic agents to increase their consumption over time. The prospect of increased consumption encourages economic agents to **favour the present and to use a discounting factor that is ever higher the further into the future they look.**

In parallel to this potential wealth effect, we also understand that the equilibrium interest rate **depends on the characteristics and choices of the agents.** They may have a strong preference for spreading their consumption over time, or on the contrary, they may not be averse to possible inequality in the inter-temporal distribution of their consumption.

Technically, once the utility function of the consumers is known (or assumed), it is the degree of curvature of this function that will provide us with the consumers’ R coefficient of aversion to the risk of inter-temporal imbalance in their consumption.

**If this coefficient equals 1**, this means that the consumer will be ready to reduce his or her consumption by one unit at time 0 in view of benefitting from one additional unit of consumption at time 1. **A coefficient of 2** would mean that the consumer is ready to reduce his or her consumption by two units at time 0. **It is reasonable to think that R lies somewhere between 1 and 2.**

From this perspective, **in 1928 Ramsey proposed a simple and illuminating formula for the equilibrium interest rate.** Using a power function to measure the consumer’s perceived utility, he showed that the wealth effect in the formation of the equilibrium interest rate was equal to the product of the nominal period growth rate of the economy and the consumer coef ficient of aversion R. **This leads to the following relationship:**

**r = δ + gR**

where r is the equilibrium interest rate, δ the impatience rate, g the nominal period growth rate of the economy and R the consumer’s coefficient of aversion to the risk of inter-temporal imbalance in his or her consumption.

Assuming a very low value for δ and a value close to the unit for R, we see that the nominal growth rate of the economy constitutes a reference value for the equilibrium interest rate. This equilibrium interest rate, as explained, is **the risk-free rate that must be used to value risk-free assets;** if we consider risky assets, we must of course add a risk premium.

**In the current context, Ramsey’s relationship makes it possible to appreciate the extent of the effects of unconventional policies put in place by central banks,** which have given rise to a risk-free rate close to 0% in the financial markets.

**THE LONG-TERM DISCOUNT RATE**

Now that we have established the notion of the equilibrium interest rate, we can move on to the question of the structure of discount rates based on their term.

We have just seen that the discount rate is determined by the impatience rate of consumers, their coefficient of aversion R and expectations for the growth rate of the economy. If we consider the impatience rate to be negligible and by assuming that the coefficient of aversion remains unchanged over time, **this gives a very important role to the economic outlook:** the discount rate based on maturity will mainly reflect the expectations of economic agents in terms of the future growth rate.

Therefore, if we expect economic growth at a constant rate g, the yield-to-maturity curve will be flat. If we expect growth acceleration (growth of the growth rate), the rate structure will grow with the maturity. However, if we expect growth to slow down, the structure of the rates will decrease.

We thus perceive **the informative function of the yield-to-maturity curve**, which makes it possible to inform the observer of the expectations of financial market operators with regard to expectations of the growth rate of the economy.

WE ALSO SEE THAT THE PENALISATION OF THE LONG-TERM CASH FLOWS BY THE DISCOUNTING PROCESS IS NOT INEVITABLE.

When the economic outlook is trending downwards, the rate structure should be decreasing. But **we must not necessarily deduce that this form of the yield curve is synonymous with disaster.** It can very easily correspond to a return to normal after a period of over-excitation. For example, coming back to the present, if the growth rate of the economy is particularly high because of catch-up effects, marking a significant gap compared with the sustainable growth rate in the long term, the rate structure should be decreasing and the short-term discount rate higher than the discount rate applicable for a longer time frame.

It is only the action of the central banks, which is particularly noticeable on short maturities, **that is preventing such a statistical observation today.**