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In all the classic texts on equities derivatives, there is an assumption of the risk-free rate r. We can immediately dismiss the concept of a fixed rate; all interest rates are variable (and martingale?) in reality.

But also, as is more apparent in the current crisis, currency has a cost of carry, and no counterparty is risk-free (witness the lowering of sovereign credit ratings). On the other hand, there is almost always inflation; in growth periods the inflation is driven by high base rates, and in recession it is driven by quantitative easing.

Can we say, then, that the risk-free rate is in fact only ever lower than inflation, and that the rate is never fixed?

If you disagree, in what circumstances is the risk-free rate higher than inflation?

Edit: answers appreciated, but the crux of the issue is this: how can a rate be actually risk-free if it is above inflation? That implies there is a resource which yields increasing value with no risk; but no commodity permits this? Where is the disconnect?

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up vote 5 down vote accepted

Take a look at historical short-term risk-free rate proxies such as Fed Funds, LIBOR, short Treasuries, and you will find plenty of periods where rates have been significantly above or below inflation (as measured by any CPI series) in the same period. In fact, controlling this difference, known as the real interest rate, is the primary tool of modern central bank monetary policy. The real rate is typically positive when inflation is considered too high, as Volcker thought in the early 80s. Back then, Volcker raised Fed Funds to 20% while inflation peaked at 13.5%. Thus this period is a clear example of when the "risk-free" rate is much higher than inflation.

Setting a real risk-free rate above inflation is really quite simple for the government, which has near direct control over short term rates. I can understand your confusion for long term rates, though. The key idea is that in times of strong economic expansion, the market is actually quite willing to "yield an increase in value with no risk," as you put it. The "commodity" that permits this is the overall economy. In fact, it is the present situation which is more unusual, where the economy is so weak that the market is unwilling to offer a return to investors for delaying consumption.

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Inflation usually has little to do with the "risk-free" nature of a rate. Also, interest rates follow inflation rather than the other way round i.e. QE does not "cause" inflation. QE was performed because inflation was too low in the first place and the interest rates had to be artificially lowered "below" the 0-0.25% range as the real interest rate (ie nominal rate-inflation rate) was too high making it harder to borrow money. Similarly, in a growing economy, it is the narrowing output gap that causes high inflation - not the "base rate".

The most recent example of a risk free rate being higher than inflation is of the early period in the current recession. The inflation rate was negative while the Federal Funds target was 0-0.25% http://www4d.wolframalpha.com/Calculate/MSP/MSP277819h32f8138ca0h2000001d27hg6g82060fe2?MSPStoreType=image/gif&s=15&w=496&h=202&cdf=Coordinates&cdf=Tooltips

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Risk-free interest should never go negative, so in a period of deflation the risk-free interest should be higher than the negative inflation rate.

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If the risk free rate is lower than inflation, basically you are loosing money and then the rate used is not risk free. Best solution for risk free rate is government bonds adjusted for inflation + inflation rate

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