# Negative probabilities - Can we have negative payments in bonds?

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 $$D = E[T]$$

where

$T$ 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?

• Also, in a traditional bond, you make a lump-sum payment and receive smaller payments over time. Not sure what you mean by making a payment. – Kontorus Jul 14 '16 at 16:15
• You cannot have a negative probability. – 123 Jul 15 '16 at 3:34
• @123 You can if it's in a quasiprobability space. Edited – BCLC Jul 16 '16 at 2:00
• No. One cannot reasonably discuss negative probabilities. The negative probability of an event is ridiculous. Perhaps you mean negative expected values? That is certainly possible. Negative values. Sure. – 123 Jul 16 '16 at 3:56
• A quasiprobability distribution is something entirely different than a probability. This is definitely true since the former violates the axioms of the later. If you want to discuss quasiprobability distributions as used in quantum mechanics or mathematical finance then that's fine but perhaps be more clear. – 123 Jul 16 '16 at 4:11