Financial mathematics problem. ( Deferred annuities ).

Question

We need to calculate present value(value at t=0) of the payments of amount $1$ made at $t = m+1 , m+2 , ...... , m+n$ and no payments are made between $t=0$ to $t=m$ , effective rate of interest is $i$.

We calculate the above as follows :

$a_{m}^{n} = v^{m+1} + v^{m+2} + v^{m+3}$ . . . . . . $v^{m+n}$ , { $v^{t} = (1+i)^{-t}$ }

NOTE : We don't consider the payment made at $t = m$.

Now the payments are made continuously between the same time period with force of interest $\delta$ , so the present value is given by : $$ \int_{m}^{m+n} e^{-\delta t} dt$$

My question is , why we are considering the payment made at $t = m$ here ?, whereas when payments were made yearly we didn't..

What's the difference ? Can anyone help ?

Answer 1

The payments are made continuously in the second case. So you really aren't evaluating "at $t=m$", they are just your integral bounds. You can think of the continuous payments beginning instantaneously after time $t=m$ and ending instantaneously before time $t=m+n$ if that helps.

Answer 2

The notation is a bit confusing for annuities (and not something I've seen before).

But to understand this, you need to recall that $a_{\overline{n}|}$, the present value of an annuity-immediate, finds the present value of $n$ payments of $1$ starting at time $1$ to time $n$. The payment at time $0$ is excluded. This is true no matter at what time you compute $a_{\overline{n}|}$: the payment at the time of valuation is excluded.

As for your second question, there is no "present value" at time $m$. This is because the so-called present value at time $m$ would be $$\int_{m}^{m}e^{-\delta t}\text{ d}t = 0$$ but we still need to include continuous payments near time $m$, so we integrate from $m$ anyway.