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Time Value of Money
I have brought up the time value of money before and because of current discussions I would like to explore this issue. Standard engineering economics tells us that money today is worth more then money tomorrow. The percentage increase in dollar value of money in a year compared to the value of money today is known as the expected rate of return. This number reflects how much money a business plans to earn on each dollar invested. Without knowledge of this rate of return the interest rate or effective interest rate can be used as an effective rate of return. Often people take 10% as the expected rate of return because that is what they think they should be able to make on the stock market.
In another thread a solar sail was mentioned and I used the ratio of the mass of the solar sail to the mass of the cargo it deceived as a means to compare a solar sail to a conventional rocket. I was told that I cannot do this because a solar sail is reusable. I argue that if we consider the expected rate of return the comparison is totally valid.
Let x be the cost of building a solar sail now and y be the cost of building an equally valuable solar as the one that returns to earth in four years. Keep in mind the sail that returns to earth in four years will not be as good as the original. It could have a lower thrust due to micrometeorite punctures and the control systems could be worn out. Thus in four years perhaps a sail half the size of the original could be equally as viable as the sail that returned form the four year trip. But we shell assume for simplicity that solar sail does not depreciate in value over the four years either by wearing out or becoming obsolete.
Thus the cost to build the solar sail in for years is x. This is equivalent in dollars at the time it was built to:
x*(100/(100 + P))^4
and the total value today of a sail that never whereas out is:
y=sum_{n=0}^{oo} x*(100/(100 + P))^(4*n) =1/(1-(100/(100 + P))^4 )
Taking the expected rate of return to be 10% the multiplicity factor in value due to the sail being reusable is:
y=1/(1-(100/(100 + 10))^4 )= 1/(1-(0.909)^4 )=3.1520
Thus if the solar sail takes 4 years to return to earth and has the same mass fraction as chemical propulsion then it economically it is only 3 times as good. If the solar sail has a mass fraction 4 times as good and takes 2 years for a return flight then economically it is 12 times as good. Thus once conceivable image the cost of transportation from earth obit to mars being reduced by a factor of 12 as a consequence of a solar sail.
If simillar const reduction can be made in getting mass to LEO then the total cost of a mars mission could be brought from 40 billion dollars to under 10 billion dollars, except that solar sails are too slow for human travel and that price reduction assumes that the solar sail costs as much to build as a chemical rocket. Should an equivalent mass solar sail be more expensive the savings will be reduced or disappear completely.
Now lets consider the equation more generally and say we have some super fast reusable vehicle that can make the flight there and back in 2 months. The multiplicity factor due to the vehicle being reusable becomes:
y=1/(1-(0.909)^(2/12))=63.3877
Thus a reusable vehicle that can make the trip there and back in a month (perhaps a fusion powered GCN) can cost 63 times as much as chemical propulsion to build and be economically as good as chemical propulsion. However, if it can be built for the same price then the cost of the interplanetary travel will be greatly reduced and if similar const reductions can be made to deliver goods to LEO then the cost of a mars mission will fall to under a billion dollars.
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