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Most financial models include a risk-free rate or risk-free asset.
Why should there be such thing as a positive risk-free rate?
I dont see why an asset would provide a positive (real) return if it hadn't any risk to it: If there was an asset with positive return without risk, everyone would buy it until its price reflects no return.
Or: Why would there be a positive riskfree rate if there is no free lunch?
Risk-free rate is that you get for letting someone else use your money in a riskless manner. Suppose we live in a world where there is no risk whatsoever. In particular, if you lend someone \$100 there is 100% certainty that he will pay you back in a year. Before the pay date, he can do whatever he wants with your $100, while you have no access to it. Even though the risk is zero, you still have to be compensated for the inconvenience, and that's what the risk-free rate is for.
The risk free rate is important and the reason for the inclusion and consideration of the risk free rate is that investors do not get compensated for not taking on risk. Now, we can argue whether the risk free rate truly provides risk free returns (we all should know that it does not, but ...) but it is important in the context of pricing risky assets that the market does not compensate for not taking risk.
Couple other points:
Another reason why we essentially "change measure" and "price, using risk neutral probabilities" is because such apparatus makes pricing the hedge, and hence the risky asset itself much easier. The hardest part in pricing risky assets is that we do not know at which rate a risky asset's future cash flows should be discounted at.
I provided reasons why the notion of a risk free rate of return should be included in most asset pricing models. But that does not mean that such risk free rate of return cannot be negative. It seems illogical and counter intuitive at first, but investors are willing to pay a price even to lend money if the "expected return" on other competing investments is inferior. Unfortunately many academicians and academic research has not caught up with the fact of such investor behavior. Investors always choose the relatively most attractive investment not the absolutely best investment.
The reason why not everyone invests in a "risk free asset"? Simple. Because investors compare investments on the basis of risk-adjusted returns and because many investors are rational to their utility. If an investor prefers to get rich quick then he/she will invest in an asset that provides a potential return of 50% even if the risk of ruin is elevated relative to a risk free asset that pays 2%-3% p.a., for example.
I would disagree in that any "fundamental theorem of asset pricing" plays a role in answering the question why we need to consider the risk free rate when pricing risky assets. The theorems help in answering the question whether we can price derivatives in a risk free context, hence, whether we are allowed to change measure without completely messing up the accuracy of pricing.
My non-rigorous answer:
The future is uncertain. Even if there is no financial risk to investing in the "risk free" asset there is personal risk. For example, I could get hit by a car and die. Even if I survive till the moment that I liquidate my investment I will have less time left in my life to enjoy it. I need to be compensated for giving up this immediate compensation, which is what the risk free rate does.
In my opinion, risk free rate is not necessarily positive and not so important to pricing theory.
It happened to be positive in most cases, but imagine a planet using Uranium-235 instead of gold as the money and unknowingly suffers from a shrinking population, likely the risk free rate is negative.
Below are what I regard as important in pricing theory, starting from "risk free measure". "Risk free rate" is implied by the discount rate $D(0; t)$ but not required to be positive.
Definition: risk-neutral measure $P$ is equivalent to $P^0$ and under which, the discounted stock price $D(0; t)S_i(t)$ is a martingale, $\forall i$.
Lemma: the discounted portfolio value $D(0; t)X(t)$ is a martingale.
Definition: An arbitrage is a portfolio value process $X(t)$ satisfying $X(0) = 0$ and also satisfying for some time $t > 0$, $P[X(T) 0] = 1$, $P[X(T) > 0] > 0$.
First fundamental theorem of asset pricing: If a market model has a risk-neutral probability measure, then it does not admit arbitrage.
Definition: A market model is complete if every derivative security can be hedged.
Second fundamental theorem of asset pricing: Consider a market model that has a risk-neutral probability measure. The model is complete if and only if the risk-neutral probability measure is unique.
I suggest to distinguish three things: Why risk-free rate? What defines its magnitude and sign? And what is its role in specific economic models?
"Why risk-free rate" is very simple to answer: Because you want to compare cash at different points in time. One assumes that the difference between cash at time $t_1$ i.e. $C(t_1)$ and at time $t_2$ is $C(t_1) - D(t_1,t_2)C(t_2)$, where $D(t_1,t_2)$ is the/a discount factor and $r(t_1, t_2) = 1 / D(t_1, t_2)$ the according risk-free rate. This is pure convention and has nothing to do with risk, pricing models or the real world. Even if you believed this were utter nonsense and money at different times can be directly compared just by subtracting the amounts, you do not get around this, as you are just saying $D=1$ or -equivalently- that risk free rate is zero.
"What defines its magnitude and sign" Often theories or models assume positive risk free rates and user9403 gave good reasons why this may be a reasonable assumption. There are also situations possible where the risk-free rates might be negative. One example being storage cost for the risk-free asset, e.g. the cost of secure vaults to hoard your cash. A standard textbook on macroeconomics should give you more details.
Note that contrary to your comment positive risk-free rate does not necessarily create arbitrage. To buy the risk-free asset you have to offer something in exchange unless you borrow what you offer. But your cost of borrowing will always be higher than the risk free rate.
It is a decision problem, it is always a decision problem...
The most basic decision problem is a lottery that gives you X (X > 0) for a chance of %Y and nothing for %(100-Y). Someone comes and offers you to buy your ticket for Z (0 < Z < X, otherwise it is trivial). What would you do? If Z is close to X, you need to be more risk-willing (or seeking) to refuse the deal. If Z is close to 0 you need to be more risk-averse to accept the deal. Usually if Z is somewhere close to X*Y/100, you are called risk-neutral. The function that "regulates" your responses to such situations is called your utility function (curve). In the most basic terms it is between risky asset (i.e. a stock) and a risk-free asset (bond).
The problem is then modeled as an equilibrium to find the "fair price". There is usually an implicit assumption on your utility function (some diminishing returns curve, less and less marginal utility for marginal dollar gains). Fair price can be defined as "a point where no party gets a better deal than the other or by making a deal with a 3rd party". Because if it would make me worse on a deal, I would not agree. And if I can make a better deal with a 3rd party I would take that deal. If the deals are not correctly priced, they can be constantly abused (it is not arbitrage, it is the law of large numbers). The fair price means you are indifferent to taking any side of the bet or investing in risk-free rate.
In the case of positive risk-free rate, a third option arises; keep your money to yourself. Though, negative risk-free rate occurs in real life. So, there is more to life than trading (probably keeping large amounts of money safe might be more expensive than the risk-free rate provided).
There is a positive risk free rate because there exists bodies that are willing to take the risk and believe they will make more money than they offered to you.
The question "is there really a risk-free rate?" notwithstanding, why would you choose to hand over your money for a fixed return if you can access to the same deal the person will make with your money? Suppose there are infinite deals and all probabilities and payoffs are well defined. Would utility curves matter? The only thing that would prevent utility curves to be obsolete is the absorbing state (losing all your money). You cannot return from that. Assuming not playing is not an option, eventually there will be a single winner with all the money and all others broke.
In real life, time is also a resource so the economy expands, rules are more complex and probabilities not clear. As a result, utility curves are various and changing. There is also consumption. So positive interest rate is a belief for the offering party they can do better performance with that money of yours and an incentive for you to avoid spending that money or investing in something else.
No free lunch is something associated with infinite abuse of the contracts.
And a technical reason for options pricing, keeping risk free rate positive helps some models' properties to stay same. For example if your risk free rate is negative it might be justified to exercise a non-dividend paying stock's american option early and not justified to exercise the put.
Ashwath Damodaran explains risk free rate in his 2008 paper, "What is risk free rate? A Search for the Basic Building Block". He has explained risk free rate from the perspective of investment.
If we invest in an risk free asset then we expect guaranteed return. "An investment that delivers the same return, no matter what the scenario, should be uncorrelated with risky investments with returns that vary across scenarios." The investment is termed as risk free because there is no variance around the expected return. And the actual return is always equal to the expected return. In the real world, we define risk free return as government bond rate (which is the closest we can get, assuming no default rate for the country).
Why positive risk-free rate? It happens to be positive in our case because we have positive expected return for a risk free asset.
"No free lunch": Yes you are right. We do not have any risk free asset in this world but in models and while solving problems we define least risky asset as risk free.
In models, US T-bill is termed as risk free but during mortgage crisis, was it really risk free?
A risk free rate is the return rate from investing in an asset that has the lowest risk found in the market. It is a naming convention. The least risky of all returns is labelled as 'risk free' for the purpose of various models and resulting discussions.
Another parallel answer is that you must understand what financial risk is in the first place. It is, most of the time (in investments), the risk of a return not exactly matching the expected value. Often represented by concepts of standard deviation of expected (or historical) returns.
Once you understand this concept of risk and the naming convention, you understand the meaning and use of 'risk free'.
Edited: The risk free rate is positive because the factors of production, and the perception of time (from an individual's perspective) are limited. Limited supply of desirable goods such as houses (or limited capacity to make them from land, labour and capital), gives them a positive value in society (versus say air, which is in almost unlimited supply, though that may change at some point). The second part of the argument is that because of our human perception of time being something limited (either because see our own lives as ending at some point, or anyway can't recall the last one), we have an innate preference to consume now rather than later (not withstanding separate effects such as diminishing marginal utility such as what happens when one eats too many chocolates at once). This 'time preference' means that we put a value on immediacy, such that more distant utility flows are worth less (= discount rate or risk free rate).
Note, what I think are, the necessary conditions to the above. 1) Limited supply, 2) Positive utility from consumption, and 3) Perception of limited time.
Thus, a positive risk free rate, probably wouldn't apply in the case of hydrogen (too abundant), nor to ebola (no one wants it), nor from the perspective of the Highlander (since he can afford to wait).
There are some useful corollaries of the above argument that help explain some structural trends in the global modern world economy, in the way that demographics play out in the demand and supply of capital, and the way in which this impacts the risk free rate.
