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Homework answers / question archive / What is the equal payment series for 12 years that is equivalent to a payment series of $15,000 at the end of the first year, decreasing by $1,000 each year over 12 years? Interest is 8% compounded annually
What is the equal payment series for 12 years that is equivalent to a payment series of $15,000 at the end of the first year, decreasing by $1,000 each year over 12 years? Interest is 8% compounded annually.
The NPV of the variable-payment stream over the 12 year period can be calculated using the following standard equation 1.
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(1)15,000(1+.08)1+14,000(1+.08)2+13,000(1+.08)3+...+4,000(1+.08)12=78,407.3(1)15,000(1+.08)1+14,000(1+.08)2+13,000(1+.08)3+...+4,000(1+.08)12=78,407.3
Now, the equation for calculating the NPV under the fixed-payment option is specified in the following equation 2, where X is the fixed-payment amount that must be determined. Note that the stream of payments is forced to be equal to the solution found for the variable-payment stream specified in equation 1.
(2)X(1+.08)1+X(1+.08)2+X(1+.08)3+...+X(1+.08)12=78,407.3(2)X(1+.08)1+X(1+.08)2+X(1+.08)3+...+X(1+.08)12=78,407.3
Using either a spreadsheet or a calculator, the value of X can be determined to equal 10,404.26.
That is,
{MathJax fullWidth='false' \frac{15,000}{{(1+.08)}^1}+\frac{14,000}{{(1+.08)}^2}+\frac{13,000}{{(1+.08)}^3}+...+\frac{4,000}{{(1+.08)}^{12}}=\frac{10,404.26}{{(1+.08)}^1}+\frac{10,404.26}{{(1+.08)}^2}+\frac{10,404.26}{{(1+.08)}^3}+...+\frac{10,404.26}{{(1+.08)}^{12}}=78,407.3
