a. $500(1.06) = $530.00.
b. $500(1.06)2 = $561.80
c. $500(1/1.06) = $471.70
d. $500(1/1.06)2 = $445.00
$1,552.90(0.3220) = $500.03; i = 6%: $1,552.90(0.5584) = $867.14.
The present value is the value today of a sum of money to be received in the future. For example, the value today of $1,552.90 to be received 10 years in the future is about $500 at an interest rate of 12 percent, but it is approximately $867 if the interest rate is 6 percent. Therefore, if you had $500 today and invested it at 12 percent, you would end up with $1,552.90 in 10 years. The present value depends on the interest rate because the interest rate determines the amount of interest you forgo by not having the money today.
The general formula is FVAn = PMT(FVIFAi,n).
FVA10 = ($400)15.9374 = $6,374.96.
($200)5.5256 = $1,105.12.
($400)5 = $2,000.00.
FVAn(Annuity due) = PMT(FVIFAi,n)(1 + i). Therefore,
FVA10 = $400(15.9374)(1.10) = $7,012.46.
FVA5 = $200(5.5256)(1.05) = $1,160.38.
FVA5 = $400(5)(1.00) = $2,000.00.
The general formula is PVAn = PMT(PVIFAi,n).
PVA10 = $400(6.1446) = $2,457.84.
$200(4.3295) = $865.90
$400(5) = $2,000.00.
PVAn (Annuity due) = PMT(PVIFAi,n)(1 + i).
Therefore, $400(6.1446)(1.10) = $2,703.62.
PVAn (Annuity due) = $200(4.3295)(1.05) = $909.20.
PVAn (Annuity due) = $400(5)(1.00) = $2,000.00.
$100(0.9259) = $ 92.59 $300(0.9259) = $ 277.77
$400(0.8573) = 342.92 $400(0.8573) = 342.92
$400(0.7938) = 317.52 $400(0.7938) = 317.52
$400(0.7350) = 294.00 $400(0.7350) = 294.00
$300(0.6806) = 204.18 $100(0.6806) = 68.06
PVA = $1,251.21 PVB = $1,300.27
a. 7 percent: $700 = $749(PVIFi,1); PVIFi,1 = 0.9346.
b. 7 percent.
c. $201,229/$85,000 = 2.3674 = FVIFi,10; i = 9%.
d. $9,000/$2,684.80 = 3.3522 = PVIFAi,5; i = 15%.
Using the tables, the payment can be calculated as $25,000 = PMT(PVIFA10%,5) and PMT = $25,000/3.7908 = $6,594.91. Notice that rounding errors occur when using the tables in Appendix A.
Repayment Remaining
Year Payment Interest of Principal Balance
1 $ 6,594.94 $2,500.00 $ 4,094.94 $20,905.06
2 6,594.94 2,090.51 4,504.43 16,400.63
3 6,594.94 1,640.06 4,954.88 11,445.75
4 6,594.94 1,144.58 5,450.36 5,995.39
5 6,594.93* 599.54 5,995.39 0
$32,974.69 $7,974.69 $25,000.00
*The last payment must be smaller to force the ending balance to zero.
b. Here the loan size is doubled, so the payments also double in size to $13,189.87.
c. The annual payment on a $50,000, 10-year loan at 10 percent interest would be $8,137.27. Because the payments are spread out over a longer time period, more interest must be paid on the loan, which raises the amount of each payment. The total interest paid on the 10-year loan is $31,372.70 versus interest of $15,949.37 on the 5-year loan.
With tables, proceed as follows:
Security Z: PV = FV(PVIFi,10)
$422.41 = $1,000(PVIFi,10)
PVIFi,10 = 0.4224.
Look this value up in Table A-1, across the 10-year row. The 0.4224 appears in the 9% column, so Security Z has a 9% rate of return.
Security B: Here the task is a bit more difficult with tables, because two variables are involved. Ordinarily, a lot of trial-and-error calculations would be involved, but in this case we make a lucky guess of 8%:
PV = $80(PVIFAi,10) + $1,000(PVIFi,10)
$1,000 = $80(6.7101) + $1,000(0.4632) = $536.81 + $463.20 = $1,000.01
b. With a calculator, for the "zero coupon bond," enter N = 10, I = 6, PMT = 0, FV = 1000, and press PV to get the value of the security today, $558.39. The profit would be $558.39 - $422.41 = $135.98, and the percentage profit would be $135.98/$422.41 = 32.2%.
For the "coupon bond," enter N = 10, I = 6, PMT = 80, FV = 1000, and then press PV to get PV = $1,147.20. The profit is $147.20, and the percentage profit is 14.72%.
With the tables, look up PVIFA6%,10 and PVIF6%,10 and proceed normally to get the same results as with a calculator.
c. Here we compound cash flows to obtain a "terminal value" at Year 10, and then find the interest rate which equates the TV to the cost of the security.
There are no intermediate cash flows with Security Z, so its TV is $1,000, and, as we saw in Part a, 9% causes the PV of $1,000 to equal the cost, $422.41. For Security B, we must compound the cash flows over 10 years at 6%. Enter N = 10, I = 6, PV = 0, PMT = 80, and then press FV to get the FV of the 10 year annuity of $80 per year: FV = $1,054.46. Then add the $1,000 to be received at Year 10 to get TVB = $2,054.46. Then enter N = 10, PV = -1000, PMT = 0, FV = 2054.46, and press I to get I = 7.47%.
So, if the firm buys Security Z, its actual return will be 9% regardless of what happens to interest rates--this security is a zero coupon bond which has zero reinvestment rate risk. However, if the firm buys the 8% coupon bond, and rates then fall, its "true" return over the 10 years will be only 7.47%, which is an average of the old 8% and the new 6%.
A similar process could be employed using the tables, but it would not be possible to determine the exact actual rate on the coupon bond.
d. The value of Security Z would fall from $422.41 to $321.97, so a loss of $100.44, or 23.8%, would be incurred. The value of Security B would fall to $773.99, so the loss here would be $226.01, or 22.6% of the $1,000 original investment. The percentage losses for the two bonds is close, but only because the zero's original return was 9% versus 8% for the coupon bond.
The "actual" or "true" return on the zero would remain at 9%, but the "actual" return on the coupon bond would rise from 8% to 9.17% due to reinvestment of the $80 coupons at 12%.
a. First, determine the annual cost of college. The current cost is $12,500 per year, but that is escalating at a 5 percent inflation rate:
College Current Years Inflation Cash
Year Cost from Now Adjustment Required
1 $12,500 5 (1.05)5 $15,954
2 12,500 6 (1.05)6 16,751
3 12,500 7 (1.05)7 17,589
4 12,500 8 (1.05)8 18,468
Now put these costs on a time line:
18 19 20 21
-15,954 -16,751 -17,589 -18,468
How much must be accumulated by age 18 to provide these payments at ages 18 through 21 if the funds are invested in an account paying 8 percent, compounded annually?
$15,954(PVIF8%,0) = $15,954
$16,751(PVIF8%,1) = 15,510
$17,589(PVIF8%,2) = 15,079
$18,468(PVIF8%,3) = 14,660
$61,203
Thus, the father must accumulate $61,203 by the time his daughter reaches age 18.
b. She has $7,500 now (age 13) to help achieve that goal. Five years hence that $7,500, when invested at 8 percent, will be worth $11,020:
$7,500(1.08)5 = $11,020.
c. The father needs to accumulate only $61,203 - $11,020 = $50,183. The key to completing the problem at this point is to realize the series of deposits represent an ordinary annuity rather than an annuity due, despite the fact the first payment is made at the beginning of the first year. The reason it is not an annuity due is there is no interest paid on the last payment which occurs when the daughter is 18. Thus,
$50,183 = PMT(FVIFA8%,6).
PMT = $6,841.
Another way to approach the problem is to treat the series of payments as a five-year annuity due with a lump sum deposit at the end of Year 5: $50,183 = FVADUE,5 + PMT = PMT[(FVIFA8%,5)(1.08)] + PMT = PMT[5.8666(1.08) + 1]. Therefore, PMT = $50,183/7.3353 = $6,841.
$2,000(1/1.14)6 = $2,000(0.4556) = $911.20.
$1,000 today is worth more. The present value of $2,000 at 14 percent over six years is $911.20, which is less than $1,000.00. Alternatively, the future value of $1,000 is: FV6 = $1,000(1.14)6 = $1,000(2.1950) = $2,195.
Solve for PMT = $802.43. Set up amortization table:
Pmt of
Period Beg Bal Payment Interest Principal End Bal
1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57
2 9,697.57 802.43 484.88
$984.88
PMT = $298,315.55. Then set up the amortization table:
Year Balance Payment Interest Principal Balance
1 $1,000,000.00 $298,315.55 $150,000.00 $148,315.55 $851,684.45
2 851,684.45 298,315.55 127,752.67 170,562.88 681,121.57
Fraction that is principal = $170,562.88/$298,315.55 = 0.5718 = 57.18%.
Using a tabular method: Information given:
1. Will save for 10 years, then receive payments for 25 years.
2. Wants payments of $40,000 per year in today's dollars for first payment only. Real income will decline; inflation is 5 percent. Therefore, to find the inflated fixed payment use the following equation:
FV10 = PV(FVIF5%,10) = $40,000(1.6289) = $65,156.
3. He now has $100,000 in an account which pays 8 percent, annual compounding. We need to find the FV of the $100,000 after 10 years:
FV10 = PV(FVIF8%,10) = $100,000(2.1589) = $215,890.
4. He wants to withdraw $65,156 per year for 25 years, with the first payment made at the beginning of the first retirement year. So we have a 25-year annuity due with payments of $65,156 at an interest rate of 8 percent annually.
PVADUE,25 = $65,156(PVIFA8%,25)(1.08)
= $65,156(10.6748)(1.08) = $751,169.45.
5. Since the original $100,000 which grows to $215,890 will be available, he must save enough to accumulate $751,169.45 - $215,890 = $535,279.45.
6. The $535,279.45 is the FV of a 10-year ordinary annuity. The interest earned is 8 percent. Therefore, use the following equation to calculate the yearly payments:
FVA10 = PMT(FVIFA8%,10)
$535,279.45 = PMT(14.487)
PMT = $36,948.95.