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FINC 404 Fall 2011 Assignment 1 Solution 1. I need your group names! Groups of 4 people please. 2. You can deposit $10,000 into an account paying 9% annual interest either today or exactly 10 years from today. How much better off will you be at the end of 40 years if you decide to make the initial deposit today rather than 10 years from today? Solution Deposit now:Deposit in 10 years: FV40=PV (1+k)40FV30=PV10 x (1+k)30 FV40=$10,000 x (1. 09)40FV30=PV10 x (1. 09)30 FV40=$314,090. 00 (approx)FV30=$132,680. 00 (approx)
You would be better off by $181,410 ($314,090 – $132,680) by investing the $10,000 now instead of waiting for 10 years to make the investment. 3. Assume that you just won the state lottery. Your prize can be taken either in the form of $40,000 at the end of each year for the next 25 years (i. e. $1,000,000 over 25 years) or as a single amount of $500,000 paid immediately. a. If you expect to be able to earn 5% annually on your investments over the next 25 years, ignoring taxes and other considerations, which alternative should you take?
Why? b. Would your decision in part (a) change if you could earn 7% rather than 5% on your investments over the next 25 years? Why? Solution a. PVAn = PMT/k * (1-1/(1+k)^n) PVA25 = $563,760 At 5%, taking the award as an annuity is better; the present value is $563,760, compared to receiving $500,000 as a lump sum. b. PVA25 = $466,160 At 7%, taking the award as a lump sum is better; the present value of the annuity is only $466,160, compared to the $500,000 lump sum payment. 4.
For each of the mixed streams of cash flows shown in the following table, determine the future value at the end of the final year if deposits are made into an account paying annual interest of 12% assuming no withdrawals are made during the period and that the deposits are made: a. At the end of each year. b. At the beginning of each year. | |Cash Flow Stream | |Year |A |B |C | |1 |$900 |30,000 |1,200 | |2 |1,000 |25,000 |1,200 | 3 |1,200 |20,000 |1,000 | |4 | |10,000 |1,900 | |5 | |5,000 | | a. | |Cash Flow Stream | |Year |A | FV |B | FV |C |FV | |1 | $ 900. 0 | $ 1,129. 0 | $ 30,000. 0 | $ 47,205. 6 | $ 1,200. 0 | $ 1,685. 9 | |2 | $ 1,000. | $ 1,120. 0 | $ 25,000. 0 | $ 35,123. 2 | $ 1,200. 0 | $ 1,505. 3 | |3 | $ 1,200. 0 | $ 1,200. 0 | $ 20,000. 0 | $ 25,088. 0 | $ 1,000. 0 | $ 1,120. 0 | |4 | | | $ 10,000. 0 | $ 11,200. 0 | $ 1,900. 0 | $ 1,900. 0 | |5 | | | $ 5,000. 0 | $ 5,000. 0 | | | |FV Sum | | $ 3,449. | | $ 123,616. 8 | | $ 6,211. 2 | b. If payments are made at the beginning of each period the present value of each of the end-of-period cash flow streams will be multiplied by (1 + i) to get the present value of the beginning-of-period cash flows. A$3,449. 0 (1 + . 12) = $3,862. 50 B$123,616. 8 (1 + . 12) = $138,450. 00 C$6,211. 2 (1 + . 12) = $$6,956. 80 5. Janet Boyle intends to deposit $300 per year in a credit union for the next 10 years and the credit union pays an annual interest rate of 8%. a.
Determine the future value that Janet will have at the end of 10 years given that end of year deposits are made and no interest is withdrawn if: 1) $300 is deposited annually and the credit union pays interest annually 2) $150 is deposited semiannually and the credit union pays interest semiannually 3) $75 is deposited quarterly and the credit union pays interest quarterly b. Use your finding in part (a) to discuss the effect of more frequent deposits and compounding of interest on the future value of the annuity. a. (1)Annual(2)Semiannual FVA10=$4,346. 0FVA10=$4,466. 70 (3)Quarterly FVA10=$4,530. 15 b. The sooner a deposit is made the sooner the funds will be available to earn interest and contribute to compounding. Thus, the sooner the deposit and the more frequent the compounding, the larger the future sum will be. 6. Harte Systems, Inc. , a maker of electronic surveillance equipment, is considering selling to a well known hardware chain the rights to make its home security system. The proposal deal calls for payments of $30,000 and $ 25,000 at the end of years 1 and 2 and for annual year-end payments of $15,000 in years 3 through 9.
A final payment of $10,000 would be due at the end of year 10. a. Lay out the cash flows involved in the offer on a time line. b. If Harte applies a required rate of return of 12% to them, what is the present value of this series of payments? c. A second company has offered Harte a one-time payment of $100,000 for the rights to market the home security system. Which offer should Harte accept? Solution a. Cash Flows $30,000$25,000$15,000$15,000$15,000$10,000 |—————|—————|—————|—————|—-(((-——|—————|—> 01234910 End of Year b. Cash Flow StreamYearCFx(1+k)n=Present Value A1$30,000x. 893=$ 26,790 25,000x. 797=19,925 3-9* 15000 [(1. 12)7-1] x 1=54,585 0. 12 (1. 12)2 1010,000x. 322=3,220 $ 104,520 Calculator solution$ 104,508. 28 *The PV for this 7-year annuity is obtained by first getting the present value of 15000 annuity for 7 years then discount for another 2 years. 7. You have decided to endow your favorite university with a scholarship. It is expected to cost $6,000 per year to attend the university into perpetuity. You expect to give the university the endowment in 10 years and will accumulate it by making annual end of year deposits into an account.
The rate of interest is expected to be 10% for all future time periods. a. How large must the endowment be? b. How much must you deposit at the end of each of the next 10 years to accumulate the required amount? Solution a. Present value of a perpetuity=PMT x (1 ( i) =$6,000 x (1 ( . 10) =$6,000 x 10 =$60,000 b. PMT = $ 3,764. 82 (rearrange FVA equation) 8. Marian Kirk wishes to select the better of two 10-year annuities, C and D. Annuity C is an ordinary annuity of $2,500 per year for 10 years. Annuity D is an annuity due of $2,200 per year for 10 years. a.
Find the Future value of both annuities at the end of year 10, assuming that Marian can earn (1) 10% annual interest (2) 20% annual interest. b. Use your findings to indicate which annuity has the greater future value at then end of the end of year 10 for both (1) 10% and (2) 20% interest rates. c. Find the present value of both annuities, assuming that Marian can earn (1) 10% and (2) 20% annual rates d. Use your findings to indicate which annuity has the greater present value at then end of the end of year 10 for both (1) 10% and (2) 20% interest rates.
Solution a. Annuity C (Ordinary)Annuity D (Due) (1) FVA10%,10=$39,842. 50FVAdue=$38,567. 54 (2) FVA20%,10=$64,897. 50FVAdue=$68,531. 76 b. (1)At the end of year 10, at a rate of 10%, Annuity C has a greater value ($39,842. 50 vs. $38,567. 54). (2)At the end of year 10, at a rate of 20%, Annuity D has a greater value ($68,531. 76 vs. $64,896. 71). c. (1) Annuity C (Ordinary)Annuity D (Due) PVA10%,10=$15,362. 50PVAdue=$14,870. 90 (2) PVA20%,10=$10,480PVAdue=$11,066. 88 d. (1)At the beginning of the 10 years, at a rate of 10%, Annuity C has a greater value ($15,362. 0 vs. $14,870. 90). (2)At the beginning of the 10 years, at a rate of 20%, Annuity D has a greater value ($11,066. 88 vs. $10,480. 00). 9. Joan Messineo borrowed $15,000 at a 14% annual rate of interest to be repaid over 3 years. The loan is amortized into three equal annual end of year payments. a. Calculate the annual end of year loan payment. b. Prepare the loan amortization schedule |PV |15000 | | | | | |k |0. 4 | | | | | |n |3 | | | | | |A? |($6,460. 97) |> or from PVA equation | | | | | | | | | |Loan Amortization | | | | | | |a |b |c |d=b-c e=a-d | |Year |Beg Balance |Payment |Interest |Principal |End Balance | |1 |15000 |$6,460. 97 |2100 |$4,360. 97 |$10,639. 03 | |2 |$10,639. 03 |$6,460. 97 |1489. 464 |$4,971. 51 |$5,667. 52 | |3 |$5,667. 52 |$6,460. 97 |793. 4527 |$5,667. 52 |$0. 00 |