Suppose you are paying a premium of one unit (say, $1) per year to another party and the discount rate is 5% per annum then what is the present value of all your payments for the next ten years. Easy, simply calculate the discount factors-say, we take the simple exponential discount factors as and multiply them by $1 and then add all the PVs. You will get $7.674.

In the above formula is the constant discount factor, which is equal to 5%. All of us do this kind of calculations day in and day out and it is trivial. But while doing a training session recently for a couple of graduate trainees in a bank, one of them asked us, just as an aside, if there is a closed form solution to this formula. That is, rather than taking the discount factors for each year using the value of time period and the discount rate in the exponential formula and then multiplying that DF by the premium and then summing it up over all the years, is there a way how in just one formula we perform the calculation and get the same result.

Though the question got us stumped initially and we had to defer it to the next session, it is actually a very simple and yet an intelligent question, and something quite fundamental. The problem is that most of the time we treat life a discrete, as an approximation of a continuous problem. If time is continuous-just as Merton demonstrated that Finance should be treated in continuous time-then there is indeed a closed form solution for most problems in finance.

Let's look at the above problem in continuous time. In continuous time, the time interval is divided into infinitesimally small segments and rather than summing up-any quantity-over discrete intervals, we integrate the area over all the infinitesimally small segments put together. Therefore, in continuous time the above problem reduces to integrating the discount factors over the time interval:

However, we know from basic integral calculus that:

Therefore, the above expression for the discount factor integration reduces to:

If we now use the upper limit of 10 years then the above formula give us:

Note that the closed form solution-integrating the discount factors-gives a PV of $7.869 which is quite close to $7.674. Thus rather than do the DF calculation year by year we can simply use the above formula in one stroke to calculate the premium paid for the entire ten year period. Most of the time we will ignore this kind of closed form treatment, because time for us is truly discrete (one year, one month, one day, etc.) and cannot be divided into infinitesimally small segments.

However, bear in mind that the above approach sometimes becomes totally indispensable in our calculations, especially when we are dealing with premium calculations of basket default swaps and some other credit derivative instruments

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