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?
Interest rates have been negative in Japan for some time. The Japanese government borrows in the bond market to finance deficits and instead of making interest payment they actually receive them
