From 2011Q4 to 2012Q3, Paypal's total payment volume - the sum USD value of all transfers through its system - was $136 billion.

Let us postulate that in the future, Bitcoin/BitPay is able to absorb 10% of Paypal's market share in online money transfers - this is ambitious, but not delusionally so. In that case, Bitcoin would have an annual TPV of $13.6 billion. The bitcoin network seems to take slightly over an hour - let's say 1.2 hours - to finalize a transaction (according to the "six confirmations" rule used by most clients). This means that assuming the network's transfers are uniformly distributed (the most conservative estimate possible - generally speaking there will be spikes and the biggest spike is the most important), Bitcoin must be able to represent $13.6/365/20 = $1.8 million in value at any given time. Folding in the data that the peak day for PayPal transactions - Black Friday - has a volume equal to about 248% that of an average Friday, we can conservatively estimate the peak requirement of "how much value the Bitcoin network must represent" as about $4.46 million.

The most recent estimate that I have read - a paper analyzing the Bitcoin network - seems to believe that about 78% of Bitcoins in existence are either dead or in long-term storage. Assuming this ratio remains roughly accurate, the total number of coins available to represent transactions is (21 Million * 0.22) = 4.4 million BTC.

Thus: if Bitcoin is able to absorb 10% of Paypal's market share in online money transfers, the lower bound for the value of 1BTC is about $1.01.

Nice work. I like your exercise. Here is another interesting exercise for you.

Suppose we use your math, but introduce one small tweak. Rather than relying on proof-of-work, bitcoin is redesigned as a proof-of-stake system. All txn fees go to bitcoin owners rather than hardware owners. Assume that the average txn fee of 0.1% of a sent balance. Fees then contribute to bitcoin's market value. The interest rate becomes very important here because it determines the conversion of expected fee revenue streams into present market value.

0.1% implies that $13.6 billion in txns generates $13.6 million in fees per annum. Assume an annual interest rate of 10% (this is high because of risk). $13.6 million in fees per annum then has a present value of $136 million.

Lower Bound on Market Cap = ($4.46 million + $136 million) = 140.46 million

Coins = 21 million [the hoarded coins are sitting around earning fees in this case]

Lower Bound on Price = $6.70

Suppose that bitcoin is extremely low risk and has a 1% annual interest rate. With such low-risk, $13.6 milloin in fees per annum equates to a present value of $1.36 billion.

Lower Bound on Market Cap = ($4.46 million + $1.36 billion) = $1.36446 billion

Coins = 21 million [the hoarded coins are sitting around earning fees in this case]

Lower Bound on Price = $64.97

As you can see,

**the valuation of bitcoin is crippled by the PoW design.**[You could increase market cap arbitrarily by increasing the fee %, but this needs to be kept low to allow bitcoin to capture paypal's market share. I think 0.1% is reasonable.]