Total Present Value of Multiple Cash Flows I understand how to calculate the total accumulated and present values of multiple cash flows over n years, but I don't quite understand how this works when m of those n years aren't included.
For example, say I have the following cash flows:


*

*$50 due in 1 year from today

*$100 due in 2 years from today

*$200 due yearly at 3, 4, 5, ..., 20 years from today
How would I go about calculating the total present value today of these cash flows with an interest rate of 8% per annum?
 A: You just discount the first two and add them up. The last you can either do the same way-a spreadsheet makes it easy-or you can recognize that you have a geometric series.  The value of the cash flow is $200(1+0.08)^{-3}+200(1+0.08)^{-4}+\dots 200(1+0.08)^{-20}$ and sum the finite geometric series.  The ratio is $1/1.08$ and the starting term is ????
A: Under the time value of money, due to the constant interest rate, your cash flow stream is equivalent to $50$ for $20$ years, $50$ from years $2$ through $20$, and $100$ from years $3$ through $20$. (Draw a timeline of both scenarios to convince yourself that this is true.) 
The $50$ for $20$ years has present value $$50\left(\dfrac{1-(1/1.08)^{20}}{0.08}\right) = 50a_{\overline{20}|8\%}$$
(I hope you know this formula if you're doing a finance class!). The $50$ from years $2$ through $20$ consists of $19$ payments, with present value $$50\left(\dfrac{1-(1/1.08)^{19}}{0.08}\right) = 50a_{\overline{19}|8\%}$$
but this values the annuity at year $1$ and not at the present time, so we need to bring this back one year by multiplying by $\dfrac{1}{1.08}$:
$$\dfrac{50}{1.08}\left(\dfrac{1-(1/1.08)^{19}}{0.08}\right) = 50a_{\overline{19}|8\%}v_{8\%}\text{.}$$
The $100$ from years $3$ through $20$ has present value (using a similar method as above):
$$\dfrac{100}{1.08^2}\left(\dfrac{1-(1/1.08)^{18}}{0.08}\right) = 100a_{\overline{18}|8\%}v^2_{8\%}\text{.}$$
Hence, the present value is 
$$50a_{\overline{20}|8\%} + 50a_{\overline{19}|8\%}v_{8\%} +  100a_{\overline{18}|8\%}v^2_{8\%}\text{.}$$
