Risk Free Rate of Return

The risk-free rate of return is a fundamental concept when it comes to understanding economic behaviour across time. It is deeply intertwined with our perception of time and impatience, when it comes to making decision of how we spend our income across time.

Our analysis begins with the intertemporal consumer problem. We have some consumer/household representing all consumers in the economy, that aims to maximise their utility (happiness) across their lifetime through purchasing units of an ‘aggregate consumption good’, in different amounts across their lifetime. Their consumption at each time period is restricted by the total number of units of the good they can afford – which is their income for the period. The restriction of consumption at each period is called a budget constraint. However, consumers can transport units of consumption across time – that is, between budget constraints – through an object referred to as an asset. A consumer can choose to swap a unit of the good for as many units of the asset as that buys them, and then swap the number of assets they have for as many units of the good it gets them in the future. In effect, they are exchanging units of consumption today for units in the future (exchanging across periods). The consumer who lives for two-periods has the following problem:

The consumer must choose what amount of goods they purchase across at each time period, with the ability to exchange consumption across time, such that they maximise their utility in their lifetime – in other words, they choose the optimal combination of consumption now and in the future (a consumption ‘bundle’). Now, when it comes to the consumer choosing which bundle is best for them, there is a key piece of information they must infer: when I exchange out one unit of a good today, how many units do I get in the future? Naturally, this is determined by the how the price of the asset changes. Combining the two budget constraints, by eliminating the asset, and we get the following:

This is called the intertemporal budget constraint (IBC). It expresses our consumption and income across time, all in terms of units of consumption today. The ratio of the asset prices, gives us how many units of the good today are required to be exchanged for a unit of the good in the future. We can consider this as the price of a unit of future consumption, in terms of units of current consumption. The inverse of this defines what is called the (gross) ‘risk-free rate of return’ - the price of a unit of present consumption, in terms of consumption tomorrow.

When the risk-free rate is greater than 1, we require more than one unit of the good today to get one unit of the good tomorrow. If this is the case, then we crucially have the Time Value of Money (TVM) principle: a fixed unit of income (remember - that is a unit of consumption here), is worth more today than tomorrow, as if it was collected today and then put into an asset, you would have a larger amount of income tomorrow.

With optimisation techniques (calculus) and some algebra, for the consumer to maximise their utility, given the risk-free rate, they must choose a consumption bundle consistent with the following equation:

This is called the Euler Equation. Effectively, it says, at the optimum, the utility gained from an additional unit of consumption today, is equal to the utility from the additional units, acquired through intertemporal exchange, of consumption tomorrow - they can’t use arbitrage to increase their utility. The risk-free rate gives the rate of return on exchanging between present and future consumption (the ratio of asset prices above), such that the additional utility to the consumer at their optimum is the same.

Take note of the function u. Typically in economics, we assume this function is increasing, and its derivative is decreasing. The former ensures something we have assumed implicitly throughout – that purchasing more of a good never harms the consumer’s utility. The latter ensures that as a consumer purchases more of a good, their utility increases gradually less and less. The parameter beta, is fixed between 0 and 1, and is called the subjective discount factor. It represents how consumers are innately impatient (they have time preference), and it has important implications alongside the decreasing derivative of u. Looking at the Euler equation, we can see that if the subjective discount factor equals the inverse of the risk-free rate, the optimising consumer will purchase equals amounts of the good today and tomorrow (they perfectly smooth their consumption). The subjective discount factor here acts as enough of a counterweight, that despite the returns on saving today to consume tomorrow, consumers choose to split their total consumption equally. Furthermore, if TVM principle breaks (a risk-free rate of less than 1), the risk-free rate and subjective discount factor together mean there will be much higher consumption today than tomorrow (so the derivative of u evaluated at tomorrow’s consumption, is much higher than the derivative of u evaluated at today’s consumption). The key implication here is, in the context of equilibrium analysis (where there are other optimising agents in the economy), it would be unlikely to get a consumption plan, and hence risk-free rate, like this - and so with standard time preference, we expect TVM to usually hold in equilibrium. Note that this discussion also highlights how TVM is a concept not directly concerned with inflation – it is to do with real (risk-free) returns, rather than nominal returns (from which real returns can be recovered via the Fisher Equation).

This is called the intertemporal budget constraint (IBC). It expresses our consumption and income across time, all in terms of units of consumption today. The ratio of the asset prices, gives us how many units of the good today are required to be exchanged for a unit of the good in the future. We can consider this as the price of a unit of future consumption, in terms of units of current consumption. The inverse of this defines, for the consumer at its optimal bundle, what is called the ‘risk-free rate of return’ - the price of a unit of present consumption, in terms of consumption tomorrow.

Looking back at the IBC, we see that the consumer’s entire future consumption and future income have both been expressed in the equivalent number of units of present consumption, through their price - the inverse of the risk-free rate. This is referred to as discounting, or finding the present value (PV) of these variables. More generally, the PV of a single unit of future consumption, is the number of units of present consumption that the single unit in the future costs. In order to discount that unit of future consumption, it simply needs to be divided by the risk-free rate. We could have equivalently denominated the IBC in terms of units of consumption tomorrow. If we did this, then the consumer’s entire present consumption and present income would have been expressed in the equivalent number of units of future consumption. The price converting the units of consumption would be the risk-free rate, and as a result, both entire present consumption and income, would have been expressed in their future value (FV).