Respuesta :
Answer:
Ans. The best choice is:
A) An annuity that pays $1, 000 at the beginning of each year (PV=$6,759.02 with a discount rate of 10% annual)
Explanation:
Hi, well, I think that the best way of explaing is by discussing each option. In order to be alot more clear in the emplanation, let´s consider a 10% annual discount rate for all cases:
(A) $1, 000 at the beginning of each year, for 10 years
This is pretty straight forward, but please consider this, one of the payments is made in the present, therefore, when calculating the present value of the annuity, do not use 10 periods, use 9 and add 1000 to this calculation, everything should look like this.
[tex]PV=1000+\frac{1000((1+0.1)^{9}-1) }{0.1(1+0.1)^{9} } =6759.02[/tex]
(B) $1, 000 at the end of each year, for 10 years
We do the same as we did in A), but this time, we count 10 annuities and we don´t add 1000 at the beginning. Everything should look like this.
[tex]PV=\frac{1000((1+0.1)^{10}-1) }{0.1(1+0.1)^{10} } =6144.57[/tex]
(C) $500 at the end of every six months, for 20 semesters
This option requires that we transform the rate (10% Effective annual) into semi-annual terms. That is as follows.
[tex]r(Semi-annual)=(1+r(Annual))^{\frac{1}{2} } -1[/tex]
Therefore
[tex]r(Semi-annual)=(1+0.1))^{\frac{1}{2} } -1=0.0488[/tex]
Now, this is our discount rate, the one we need to use if the payments are made every six months. Let´s see how the math to this should look like.
[tex]PV=\frac{500((1+0.0488)^{20}-1) }{0.0488(1+0.0488)^{20} } =6294.52[/tex]
(D) $500 at the beginning of every six months, for 20 semesters (10 years)
We need to use the discount rate of C) (4.88% semi-annual), but the process is just like in A)
[tex]PV=500+\frac{500((1+0.0488)^{19}-1) }{0.0488(1+0.0488)^{19} } =6601.75[/tex]
Best of Luck.
