In Half of a Coin: Negative Probabilities, the author mentions bond duration.

Suppose we have payments at times $t = 1,2,...,n$ denoted respectively by $R_1, R_2, ..., R_n$ and the discount factor is $v = \frac{1}{1+i}$ where $i$ is effective interest rate. Then the bond value today is given by

$$B = \sum_{t=1}^{n} R_tv^t$$

The bond duration is

$$D = \frac{\sum_{t=1}^{n} tR_tv^t}{\sum_{t=1}^{n} R_tv^t}$$

It can be seen that $$D = E[T]$$


$T$ is a random variable with range $t = 1,2,...,n$ each having probability $\frac{R_t v^t}{B}$

The author says something like we can have negative probabilities if we have negative $R_t$'s. So this is a kind of bond where instead of making a payment we get a certain amount of money? Is there such a thing? Or is that only in theory?

Is there another way we can have negative probabilities when it comes to bonds?

  • 1
    $\begingroup$ Could you provide a working link to the document? $\endgroup$ Commented Jul 14, 2016 at 9:26
  • $\begingroup$ @muffin1974 done! $\endgroup$
    – BCLC
    Commented Jul 15, 2016 at 22:56

2 Answers 2


There is no need to resort to negative-coupon bonds. A negative $R_t$ is simply a negative payment.

For a simple example, build a portfolio consisting of being long a $n$ maturity bond paying a coupon C on $t \in \left\lbrace 1, 2, ..., n \right\rbrace$ and short a zero-coupon bond with face value $V > C$ maturing at $t^*$. Then, $R_t > 0$ for $t \ne t^*$, but $R_{t^*} < 0$. In other words, a net positive cashflow at all $t \in \left\lbrace 1, 2, ..., n \right\rbrace \setminus \left\lbrace t^* \right\rbrace$, with a net negative cashflow at time $t^*$. Assume also that interest rates are strictly positive.

Clearly, $D = E \left[ T \right]$ still. Assuming that $B = \sum_{t=1}^n R_t \nu^t> 0$ (the net actualized value of the portfolio is positive), $p_{t^*} = \frac{\nu^{t^*} R_{t^*}}{B} < 0$. Also, $\sum_{\tau \ne t^*} p_\tau > 1$.

If instead $B < 0$, then $p_{t^*} > 0$, but $p_\tau < 0$ for $\tau \ne t^*$ (all the other "probabilities" are negative).

  • $\begingroup$ is it correct to say that you did not mix up coupon and payment? $\endgroup$
    – BCLC
    Commented Jul 20, 2016 at 1:32
  • $\begingroup$ A coupon is a positive cashflow, since one is long the bond. The payoff of a ZC bond (short position) is a negative cashflow. Made some edits to make it clearer. $\endgroup$
    – ocstl
    Commented Jul 20, 2016 at 7:55
  • $\begingroup$ Ah thanks, ocstl. ^-^ 1 do we need $-R_{t^{*}} > R_t$? 2 do we need $t^{*} \in \{1,2,...,n\}$? 3 So $T$ then represents duration of a portfolio of those two bonds? $\endgroup$
    – BCLC
    Commented Jul 20, 2016 at 12:06
  • $\begingroup$ Edit so $T$'s expected value then is the following $$D = \frac{\sum_{t=1}^{n} t(R_t-f(t))v^t}{\sum_{t=1}^{n} (R_t-f(t))v^t}$$ where $f(t) = R_{t^{*}}1_{t^{*}}(t)$ ? $\endgroup$
    – BCLC
    Commented Jul 20, 2016 at 12:12
  • 1
    $\begingroup$ (1) No. One only needs to have cashflows of both signs to generate a negative probability. (2) No. Ideally, $t^* \in \mathbb{N}$. But only signed cashflows (with both signs). (3) $E \left[ T \right]$ is the duration. As for $D$, since $R_{t^*} < 0$ (and assuming $R_t > 0$ for $t \in \left\lbrace 1, 2, ..., n \right\rbrace$, there should be a plus sign instead of a minus. Otherwise, looks good. $\endgroup$
    – ocstl
    Commented Jul 20, 2016 at 21:15

The answer is NO, with very few exceptions

There might be bonds with negative coupon(s), and the Bloomberg search even finds some, but there are plenty of reasons why negative coupons are impractical. Instead of having negative coupons on the issue, there are bonds with low or 0 coupons, issued at a premium and having a negative yield.

Here are some of the reasons I can think of why negative coupons are problematic:

-) Default risk: For debt instruments an important thing is the credit rating of the issuer. Ultimately, this translates into default risk, meaning any outstanding coupon payments and the nominal are at risk. A negative coupon would make the bond holder pay, exposing the bond issuer to credit risk. In general terms, this credit risk profile would be very inhomogeneous as the issuer typically has no control over the entity that holds the bond. Almost like a portfolio of bank loans. Even worse, because the issuer has little knowledge about the bond holders. In addition, bond holders can change if security is traded on the secondary market.

-) Settlement/Clearing: As far as I know, the large houses (such as Euroclear, Clearstream) refuse to take care of negative coupons. Additionally, for floating rate coupons, they comment the following:

"Securities with a floating rate coupon and an interest rate calculation linked to a negative benchmark interest rate may result in a theoretical negative coupon. However, such coupons would usually be announced as ‘NIL payment’ or ‘coupon pays zero’. The possible reasons are the following:

  • a ‘floor level rate’ (total or benchmark interest rate) has been defined in the terms and conditions of such securities
  • the issuer does not exercise the right to collect negative interest due to the administrative burden and cost of such an operation
  • the terms and conditions of the security define that the interest payment is due to be paid from the issuer to the investor and there is no reverse covenant in place for the investor to pay the issuer."

-) Taxation In a lot of countries there are taxes on interest income such as coupons for some or all investors. While things are clear for positive coupons, what happens if there are negative coupons? Does the issuer have to pay taxes on interest income for that? Can a bond holder use these as offsetting losses? I'm far from being a tax expert but I think there are plenty more questions on this.

-) Trading will also be an issue. With negative accrued interests, things could get a little messy. But while the issuer needs to collect coupons from many different entities, this is usually a 1:1 thing meaning that there simply could be a cash payment the size of the accrued interest (which is negative) attached to the trade. In fact, this already happens when a bond is traded between the "ex-date" of the coupon and the factual payment date. So the secondary market should be of little concern.

For the sake of completeness: you can find bonds with negative coupons:

Bloomberg search: If you search Bloomberg if there are bonds with negative coupons, the search results are almost 0. There are, as a matter of fact, some bonds with negative coupons in the system. For those bonds, I can only imagine that there is a close connection between the issuer and the bond holder(s), making a rigorous collection of coupon payments possible. See also this article.

  • $\begingroup$ I'm sorry for being unclear. I was looking something more to do with the motivation of the question rather than my guessed answer $\endgroup$
    – BCLC
    Commented Jul 20, 2016 at 12:08
  • 1
    $\begingroup$ @BCLC No problem. I tried to answer your question as it was stated in its original form, and I think I did a fair job. As for the whole "negative probability" thing, I have no idea what this is all about so I am not the right person to answer this. $\endgroup$
    – vanguard2k
    Commented Jul 20, 2016 at 14:51

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