 we can view the present value as follows if the present value is positive the investor is getting out more than they're putting in But if the present value is negative the investor is putting in more than they're taking out Since we want to provide an objective measure of value We often assume the net present value to be zero and if the net present value is zero Then the investment has a fair value for the investor and the financial institution So again, there's many variables in this equation So let's see if we can solve for each one given the others For example suppose you deposit 1,000 now and 1,000 next year to get a payout of 2,500 in two years Let's find the effective interest rate and we'll assume compound interest so we'll assume a net present value of zero and our discount function 1 divided by 1 plus i to the t and We'll compute so our deposit of 1,000. That's a negative our deposit of 1,000 at time one that's negative 1,000 v1 and Our withdrawal payout of 2,500 at time two. That's 2,500 v of two To simplify the solution process. We note that our discount function 1 over 1 plus i to the t and We note that v of 1 1 over 1 plus i and v of 2 1 over 1 plus i squared Now while we could solve this equation as is for i We'll make the following substitution because it will simplify our computations. We'll let v equal 1 divided by 1 plus i then v of 1 is v and v of 2 is v squared and our equation becomes quadratic and We can solve this using the quadratic formula. So again not Computing until we reach the end of the problem gives us Since i and 1 plus i must be positive. We can ignore the negative square root then solve for the interest rate i and So our interest rate works out to be 15.83 percent effective Let's consider another problem. Suppose you deposit $1,000 at the end of the next three years at 5% annual interest When will you be able to withdraw $10,000? So there are transactions of minus 1,000 at t equals 1, 2 and 3 because they're deposits and they represent money going away from you and 10,000 at time t which we don't know so the equation for net present value is We'll isolate the expression containing the unknown quantity Now to solve it note that the left-hand side is a product so we'll divide both sides by 10,000 and If we take the reciprocal of both sides that will illuminate one fraction It's an exponential equation. So we'll hit both sides with a log and solve and Again, we save our rounding until the end of the problem and we find that t is about 26.66 years So let's see if this is a plausible answer So we can simplify the scenario suppose we make all of our deposits 1,000 by 3 at time Zero and let this amount accrue a 5% interest for 26.66 years and a total amount will be or A little over $11,000 which is consistent with the withdrawal of 10,000 Now since the scenario assumed all deposits were made at the start it overestimated the balance So if we assume all deposits are made at t equals 2 then the interest would accumulate for 24.66 years and the amount would be or a little over $99.99 which is again consistent with the $10,000 withdrawal So that 26.66 years seems to be a reasonable solution