### Introduction

The risk-free interest rate is the interest rate that can be made without taking a significant risk. As such, it can be seen as a proxy for the return on capital of doing, essentially, nothing. It is classically considered to be the return of a short-dated government bond. In this node, we will first see how risk-free this risk-free rate actually is. We will then see how the risk-free rate is important in valuation of financial products.

### Remaining risks

Although I've called this return on a government bond a "risk-free rate", it is not actually risk-free. There are three theoretical risks remaining:

##### Default risk

A government could theoretically

default on its

bonds. Thing is, if a government issues

bonds in its own currency, it can just

print money until it can pay the bond. the holder of the

bond is still screwed, but so is everyone else holding

assets denominated in that currency. As such, the risk of

default for a

government is no "additional" risk, apart from the currency risk. Note that it is not clear that this is true for countries in the

Eurozone, as here, the currency is shared by several countries, neither of which has total control over it.

##### Liquidity risk

When buying these

bonds, you might not be able to

sell them because there is no

buyer. This is also a bit unlikely: short-dated government bonds are very liquid, certainly for first-world issuers. Secondly, the bond is short-dated, often 3 months: you'll get your money back anyway in 3 months, which is pretty short.

##### Market risk

Imagine you bought a bond for 4%, to mature in 3 months. You will get 101 dollar for every 100 dollar you put in. Now, imagine the government raises the interest rate to 5%. Someone buying this new bond will then make 25 cents more. This will depress the value of your bond by about 25 cents, as it is silly to get 101 when you can get 101.25. Note that this computation is a bit simplified, and for longer-dated bonds, durations and compounding have to be included to get an accurate answer.

The thing is, interest rates normally don't swing by this much. A .5% change in 3 months would already be a lot, and this means you lose about a dime on the bond - not spectacular compared to the $1 interest. Secondly, if this truly bothers you, you could pick 4-week bonds as well.

### Use of the risk-free rate

In mathematical finance, the risk-free rate is the equivalent of stuffing cash under the mattress. Cash you get in the future is to be discounted with this risk-free rate (often the risk-free rate appropriate for that duration). Cash you need to pay in the future accrues interest with this interest rate.

Secondly, it is a yardstick for how well an investment is doing. Having a return of 10 % is not impressive if the risk-free rate is 11 %. In principle, the risk-free rate needs to be subtracted from all returns before they are compared. This is, incidentally, also (one of) the reasons why a guaranteed product can be worth less than the guaranteed value: if this value is only paid out in the future, the risk-free interest rate is forfeited, and it is essentially a zero-coupon bond.

Another use for the risk-free rate is to see how it compares to your savings account. If a bank pays a lot less than this, you should probably find a different savings account.

### Negative risk-free rates

Conventional wisdom holds that the risk-free rate is always positive. After all, if you lend money, you want at least your

principal back. However, during the height of the financial crisis in 2008, this was no longer true: the effective risk-free rate on the American dollar became slightly negative. Now, you might think it is possible to just take your money, stuff it in an old mattress, and have a better deal. This is in principle true, and as such, there are limits to how far negative a rate can be: if it gets too low, someone will just

sell these

bonds, withdraw the cash and put it in a warehouse. Because there are practical issues with this - you need a warehouse, and some way to

guard it - this

arbitrage only becomes interesting at a significantly negative interest rate.

### Conclusion

The risk-free rate is a

yardstick for how

cash returns. It is normally taken to be the return on short-dated government bonds, which is probably the closest proxy. An investment should be compared to the risk-free interest rate to assess how much it returns.