Examples and Causes of Negative Forward Interest Rates
Summary
The document gives examples of negative interest rates and explains how they can appear in forward markets. A negatively sloped yield curve can imply a negative forward rate; the responses mention an inverted US term structure and Treasury bills trading at negative yields during financial stress. They also point to CHF LIBOR futures priced above par as evidence of negative implied forward rates, even though CHF LIBOR had not yet fixed below zero at the time described.
Other examples include negative real rates on inflation-protected bonds and negative foreign-currency yields implied by non-deliverable forward prices. The document cautions that simple forward-rate calculations using LIBOR tenors can fail when the 3-month versus 6-month basis spread matters. These are illustrative market examples rather than a current survey, and the quoted market conditions are historical; the discussion does not provide data or a worked calculation for any example.
Key ideas
- A negatively sloped yield curve can imply negative forward rates.
- CHF LIBOR futures priced above par were cited as evidence of negative implied forward rates.
- Treasury bills may trade at negative yields during periods of financial stress.
- Inflation-protected bonds can have negative real yields, and NDF prices can imply negative foreign-currency yields.
- Basis spreads can invalidate a simple forward-rate calculation based on a single LIBOR curve.
Tags
Full text
# What is a real world example of negative forward interest rate?
# 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).
## Answer by Ram Ahluwalia (score 7)
https://quant.stackexchange.com/a/3559
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.
## Answer by ldnquant (score 6)
https://quant.stackexchange.com/a/3674
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.
## Answer by Richi Wa (score 4)
https://quant.stackexchange.com/a/3657
I have come across 2 markets where rates can be negative:
- 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.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.