Question

The answer given from the book is $3,099.00 Please do not copy another users incorrect solution to this problem and only solve if you can get to this answer. Thank you!

(4) It is reported that the present value of an annuity paying $1,000 at the end of each year for Y years is $12,692.58, and the present value of an annuity paying $300 each year for four years is $1,069.12. Find the present value of an annuity paying $500 at the end of each two years for Y years.

Answer 1

To solve this problem, we first need to determine the Interest Rate (i.e Rate of Return) offered by the Investment.

As per the problem, "Present Value of the annuity paying $300 each year for 4 years is $ 1,069.12"

Using the RATE function in Excel, let us determine the Interest rate as shown below :-

8 PMT 300 9 PV -1069.12 10 Years 4 11 4.7850% 12 RATE 13

PMT 300 PV -1069.12 10 Years 4 11 12 =RATE(C10,C8,C9) RATE 13

Note:- The first snapshot is the final value obtained and the second one is formula used.

Now let us try to determine the "Number of Years" for which the Investment will be done

According to the problem "Present Value of the annuity paying $1,000 each year for Y years is $ 12,692.58"

We know that Rate of Interest = 4.785%

15 PV -12692.58 16 PMT 1,000 4.7850% 17 RATE 18 19 Years 20

15 PV -12692.58 16 РMT 1000 =RATE(C10,C8,C9) 17 RATE 18 =ROUND(NPER(C17,C16,C15) ,0) 19 Years 20 21 22 Years 20

Now we know that total no of years = 20

Let us plot the Cash Flow for the final annuity

Years Cash Flows 24 25 1 26 2 500 27 3 28 4 500 29 5 30 6 500 31 7 8 500 33 9 34 10 500 35 11 36 12 500 37 13 38 14 500 39 15 16 40 500 41 17 42 18 500 43 19 44 20 500 45 32

Let us discount the cash flows using the PV Factors for the respective periods

23 Years Cash Flows PV Factors Discounted Cash Flows 24 25 1 0.954335067 0 26 2 500 0.91075542 455.3777101 27 0.869165835 0 28 4 500 0.829475435 414.7377177 29 5 0.791597495 0 30 6 500 0.755449249 377.7246244 31. 7 0.720951709 0 0.688029498 344.014749 32 8 500 33 9 0.656610677 0 34 10 500 0.626626594 313.3132972 35 11 0.598011733 0 36 12 500 0.570703567 285.3517837 37 13 0.544642427 0 38 14 500 0.519771367 259.8856836 39 15 0.496036043 16 40 500 0.47338459 236.692295 41 17 0.451767514 42 18 500 0.431137581 215.5687906 43 19 0.411449712 44 20 500 0.392660889 196.3304445 45 SUM 3098.997096 46

Cash Flows Discounted Cash Flows 24 Years PV Factors =1/ (1+4.785%)^A25 =1/ (1+4.785%)^A26 =1/ (1+4.785%)^A27 -1/(1+4.785% )^A28 =1/ (1+4.785%)^A29 -1/(1+4.785% )^A30 =1/(1+4.785%)^A31 1/(1+4.785%)^A32 =1/(1+4.785%)^A33 1/(1+4.785%)^A34 1/(1+4.785%)^A35 =1/(1+4.785%)^A36 | =1/(1+4.785%)^A37 =1/(1+4.785%)^A38 | =1/(1+4.785%)^A39 =1/(1+4.785%)^A40 | =1/(1+4.785%)^A41 =1/(1+4.785%)^A42 1/(1+4.785%)^A43 =1/(1+4.785%)^A44 C25*B25 25 1 C26*B26 26 2 500 C27*B27 27 3 -C28*B28 28 4 500 C29*B29 29 5 30 6 -С30*В30 500 31 7 C31*B31 -С32*В32 32 8 500 -C33*B33 33 9 34 10 -С34*В34 500 -С35*В3: -С36*В36 36 12 500 -С37*В37 37 13 38 14 500 C38*B38 C39*B39 39 15 500 C40*B40 40 16 41 17 C41*B41 42 18 -C42*B42 500 C43*B43 43 19 44 20 C44*B44 500 -SUM(D25:D44) 45 SUM

Hence, Present Value = $3,099