Asked • 03/27/19

Negative probabilities - Can we have negative payments in bonds?

In [Half of a Coin: Negative Probabilities](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.592.2043&rep=rep1&type=pdf), 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?

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