# What is a real world example of negative forward interest rate?

As the title says, I am looking for a real world example where a forward interest rate is negative.

Theoretically this is not a problem at all, if I look for a 3M forward interest rate that starts in 3 months from now I just solve for $r_F$ in the equation $$\operatorname{df}(Date1,Date1+3M,r_{3M})\cdot \operatorname{df}(Date1+3M,Date1+6M,r_{F}) =$$ $$\qquad \operatorname{df}(Date1,Date1+6M,r_{6M})$$ where $r_{kM}$ is the $k$M-yield curve interest rate ($k=3,6$) and $\operatorname{df}$ is the discount factor.

It would also be interesting to see a reference to a negative yield curve interest rate.

A well known example of negative deposit rate is given on Wikipedia (Swedish Riksbank had an interest of -0,25% in July 2009).

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Forward interest rates are negative whenever the yield curve is negatively sloped. The US term structure was inverted most recently around 2007. Hard to find bank deposits that have negative yields (find countries experiencing deflation and you may find it), however, treasury bills during recent times of financial stress have yielded a negative rate. The Treasury is considering rules to allow for auctions that clear at negative rates.

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I agree that bonds with negative rates provide examples of this (although the forward bit is not necessary), but I don't think it is sufficient for the Yield curve to be decreasing - in general, the Yield curve will be decreasing at t if the (instantaneous) forward rate at t is below the yield at t. It could still be positive. An example of German bonds with negative yields is here. –  amgc May 30 '12 at 12:53
Thanks, but do you also have a concrete example of negative FWD rate? –  AD. May 30 '12 at 12:54
@amgc Right, it depends a lot on the days involved too. –  AD. May 30 '12 at 12:55

I have come across 2 markets where rates can be negative:

1. Inflation protected bonds. These bonds are pricd with real interest rates. You can think of them as (this is the Fisher equation: $$r = n - i$$ where $r$ is the real interest rate and $n$ is then nominal interest rate (the normal one) and $i$ is the (estimated or priced) inflation. Real rates for short maturities are often negative.

2. NDF implied yields. Some currencies can not be exchanged freely for off-shore investors (see NDF for examples). In these markets forward FX rates are traded and you can calculated implied interest rates from the traded forward (input: the forward FX rate, the domestic interest rate, the FX spot rate; output: an implied yield of the foreign currency that fits the inputs). Again for short maturities I have seen negative yields there.

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Thanks, in particular for the Fisher eq. link I have adopted it without knowing that it was called that or the background. :) –  AD. Jun 23 '12 at 21:12

A concrete example of negative forward rates is provided by the 3M CHF LIBOR futures. They're all trading above a price of 100, which implies negative forward rates.

See the prices here. Despite the prices of the forwards, CHF libor hasn't actually fixed negative yet. But the forwards are certainly all below zero.

Also, your formula for the forward rate doesn't strictly hold in today's interest rate world as the 3v6 basis spread can't be ignored. If you simply took 3M Libor, a 3Mx6M FRA, and the 6M Libor rate, that relationship would be violated.

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What you mean is that one has to take the spreads into account in the calculation? I work in a system where deals are valued, I guess the spread should be added to the interest rate - but that may of course differ from bank to bank. –  AD. Jun 25 '12 at 18:53
Can you explain why: price > 100 ensures the FWD interest rate to be negative? –  AD. Jun 26 '12 at 7:45
The interest future price is just 100 - forward rate (apart from a convexity adjustment). So if the price is greater than 100, the forward rate is negative. –  ldnquant Jun 29 '12 at 19:44
As for the spreads. The point is you have to treat 3M and 6M forward rates separately, this is typically done when bootstrapping your interest rate curves. –  ldnquant Jun 29 '12 at 19:45