Chapter 1 and 2 Study Quanatative
The Parks plan to take three cruises, one each year. They will take their first cruise 9 years from today, the second cruise one year after that, and the third cruise 11 years from today. The type of cruise they will take currently costs $5,000, but they expect inflation will increase this cost by 3.5% per year on average. They will contribute to an account to save for these cruises that will earn 8% per year. What equal contributions must they make today and every year until their first cruise (ten contributions) in order to have saved enough at that time for all three cruises? They pay for cruises when taken.
Cost of first cruise = 5,000 x 1.035^9 = 6,814.49 PV of first cruise cost = 6,814.49/(1.08^9) = 3,408.94 PV of second cruise cost = [(1.035^10)/(1.08^10)] x 5,000 = 3,266.90 PV of third cruise cost = [(1.035/1.08)^11] x 5,000 = 3,266.90 PV of all three = 3,408.94 + 3,266.90 + 3,130.78 = $9,806.62. This is the amount needed in the account today, so it's the PV of a 10-payment annuity due. Solve for payment at 8% = $1,353.22
A bank quotes certificate of deposit (CD) yields both as annual percentage rates (APR) without compounding and as annual percentage yields (APY) that include the effects of monthly compounding. A $100,000 CD will pay $110,471.31 at the end of the year. Calculate the APR and APY the bank is quoting.
For APR, PV = 100,000; FV = —110,471.31; PMT = 0; N = 12; CPT In' 0.8333, which is the monthly rate. The APR = 12 x 0.8333, or 10%. APY = (110,471.31/100,000) — 1 = 0.10471 = 10.471% (equivalent to a compound monthly rate of 0.8333%) Note that the APY, which includes the effects of compounding, must logically be larger than the APR, which does not.
A client has $202,971.39 in an account that earns 8% per year, compounded monthly. The client's 35th birthday was yesterday and she will retire when the account value is $1 million. At what age can she retire if she puts no more money in the account? At what age can she retire if she puts $250 per month into the account every month, beginning one month from today?
PV = —202,971.39; 1/Y = 8 / 12 = 0.6667; PMT = 0;. FV = 1,000,000; CPT → N = 240. 240 months is 20 years; she will be 55 years old. Don't clear TVM functions. PMT = —250; CPT → N = 220, which is 18.335 years, so she will be 53.
Terry Corporation preferred stocks are expected to pay a $9 annual dividend forever. If the required rate of return on equivalent investments is 11%, a share of Terry preferred should be worth: a) $81.82 b) $99.00 c) $122.22
a) 9/0.11 = $81.82
A company's dividend in 1995 was $0.88. Over the next eight years, the dividends were $0.91, $0.99, $1.12, $0.95, $1.09, $1.25, $1.42, and $1.26. Calculate the annually compounded growth rate of the dividend over the whole period.
This problem is simpler than it may appear. The dividend grew from $0.88 to $1.26 in eight years. We know, then, that 0.88(1 + i)^8 = 1.26, where i is the compound growth rate. Solving for i we get [(1.26/0.88)^1/8] - 1= 4.589%. You could also just enter 1.26/0.88, press [√] three times, get 1.045890, and see that the answer is 4.589%. This technique for solving for a compound growth rate is a very useful one, and you will see it often. Calculator solution: PV = 0.88; N = 8; FV = —1.26; PMT = 0; CPT → I/Y = 4.589%
An investment (a bond) will pay $1,500 at the end of each year for 25 years and on the date of the last payment will also make a separate payment of $40,000. If your required rate of return on this investment is 4%, how much would you be willing to pay for the bond today?
We can take the present value of the payments (a regular annuity) and the present value of the $40,000 (lump sum) and add them together. N = 25; PMT = —1,500; i = 4; CPT → PV = 23,433.12 and 40,000 x (1/1.04)^25 = 15,004.67, for a total value of $38,437.79 Alternatively, N = 25; PMT = —1,500; i = 4; FV = —40,000; CPT → PV = 38,437.79
An investor has just won the lottery and will receive $50,000 per year at the end of each of the next 20 years. At a 10% interest rate, the present value of the winnings is closest to: a) $425,678 b) $637,241 c) $2,863,750
a) N = 20; I/Y = 10; PMT = -50,000; FV = 0; CPT → PV = $425,678.19
How much must be invested today, at 8% interest, to accumulate enough to retire a $10,000 debt due seven years from today? The amount that must be invested today is closest to: a) $5,835 b) $6,123 c) $8,794
a) N = 7; I/Y = 8; FV = -10,000; PMT = 0; CPT → PV = $5,834.90
If $10,000 is borrowed at 10% interest to be paid back over ten years, how much of the second year's payment is interest (assume annual loan payments)? a) $937.26 b) $954.25 c) $1,037.26
a) To get the annual payment, enter PV = —10,000; FV = 0; I/Y = 10; N = 10; CPT → PMT = 1,627.45 The first year's interest is $1,000 = 10,000 x 0.10, so the principal balance going into year 2 is 10,000 — 627.45 = $9,372.55. Year 2 interest = $9,372.55 x 0.10 = $937.26
Given an 8.5% discount rate, an asset that generates cash flows of $10 in Year 1, $20 in Year 2, $10 in Year 3, and is then sold for $150 at the end of Year 4, has a present value of: a) $108.29 b) $135.58 c) $163.42
a) Using your cash flow keys, CF0 = 0; CFI = 10; CF, = -20; CF3 = 10; CF = 150; I/Y = 8.5; NPV = $108.29
An investor will receive an annuity of $4,000 a year for ten years. The first payment is to be received five years from today. At a 9% discount rate, this annuity's worth today is closest to: a) $16,684 b) $18,186 c) $25,671
b) 1. Find the PV of the 10-year annuity: N = 10; IN = 9; PMT = -4,000; FV = 0; CPT → PV = 25,670.63 This is the present value as of the end of Year 4; 2. Discount PV of the annuity back four years: N = 4; PMT = 0; FV = -25,670.63; I/Y = 9; CPT → PV = 18,185.72
A share of George Co. preferred stock is selling for $65. It pays a dividend of $4.50 per year and has a perpetual life. The rate of return it is offering its investors is closest to: a) 4.5% b) 16.9% c) 14.4%
b) 4.5/65 = 0.0692 or 6.92%
Given an 11% rate of return, the amount that must be put into an investment account at the end of each of the next ten years in order to accumulate $60,000 to pay for a child's education is closest to: a) $2,500 b) $3,588 c) $4,432
b) N = 10; I/Y = 11; FV = -60,000; PV = 0; CPT → PMT = $3,588.08
An analyst estimates that XYZ's earnings will grow from $3.00 a share to $4.50 per share over the next eight years. The rate of growth in XYZ's earnings is closest to: a) 4.9% b) 5.2% c) 7%
b) N = 8; PV = -3; FV = 4.50; PMT = 0; CPT → I/Y = 5.1989
If $10,000 is invested today in an account that earns interest at a rate of 9.5%, what is the value of the equal withdrawals that can be taken out of the account at the end of each of the next five years if the investor plans to deplete the account at the end of the time period? a) $2,453 b) $2,604 c) $2,750
b) PV = -10,000; I/Y = 9.5; N = 5; FV = 0; CPT → PMT = $2,604.36
How much must be invested today at 0% to have $100 in three years? a) $77.75. b) $100.00. c) $126.30.
b) Since no interest is earned, $100 is needed today to have $100 in three years
If $1,000 is invested today and $1,000 is invested at the beginning of each of the next three years at 12% interest (compounded annually), the amount an investor will have at the end of the fourth year will be closest to: a) $4,779 b) $5,353 c) $6,792
b) The key to this problem is to recognize that it is a 4-year annuity due, so switch to BGN mode. N = 4; PMT = -1,000; PV = 0; I/Y = 12; CPT → FV = 5,352.84 Switch back to END mode.
An investment is expected to produce the cash flows of $500, $200, and $800 at the end of the next three years. If the required rate of return is 12%, the present value of this investment is closest to: a) $835 b) $1,175 c) $1,235
b) Using your cash flow keys, CF0 = 0; CFI = 500;. CF, = 200; CF3 = 800; I/Y = 12; NPV = $1,175.29 Or you can add up the present values of each single cash flow: PV1 = N = 1; FV = -500; I/Y = 12; CPT → PV = 446.43 PV2 = N = 2; FV = -200; I/Y = 12; CPT → PV = 159.44 PV3 = N = 3; FV = -800; I/Y = 12; CPT → PV = 569.42 Hence, 446.43 + 159.44 + 569.42 = $1,175.29
What is the effective annual rate for a credit card that charges 18% compounded monthly? a) 15.38% b) 18.81% c) 19.56%
c) EAR = [(1 + (0.18/12)]^12 - 1 = 19.56%
The amount an investor will have in 15 years if $1,000 is invested today at an annual interest rate of 9% will be closest to: a) $1,350 b) $3,518 c) $3,642
c) N = 15; I/Y = 9; PV = -1,000; PMT = 0; CPT → FV = $3,642.48
An investor is looking at a $150,000 home. If 20% must be put down and the balance is financed at 9% over the next 30 years, what is the monthly mortgage payment? a) $799.33 b) $895.21 c) $965.55
c) N = 30 x 12 = 360; I/Y = 9/12 = 0.75; PV = —150,000(1 — 0.2) = —120,000; FV = 0; CPT → PMT = $965.55
Fifty years ago, an investor bought a share of stock for $10. The stock has paid no dividends during this period, yet it has returned 20%, compounded annually, over the past 50 years. If this is true, the share price is now closest to: a) $4,550. b) $45,502. c) $91,004
c) N = 50; I/Y = 20; PV = -10; PMT = 0; CPT → FV = $91,004.38
Given daily compounding, the growth of $5,000 invested for one year at 12% interest will be closest to: a) $5,600 b) $5,628 c) $5,637
c) N= 1 x 365 = 365; I/Y = 12/365 = 0.0328767; PMT = 0; PV = —5,000; CPT → FV = $5,637.37
If $5,000 is invested in a fund offering a rate of return of 12% per year, approximately how many years will it take for the investment to reach $10,000? a) 4 years b) 5 years c) 6 years
c) PV = -5,000; I/Y = 12; FV = 10,000; PMT = 0; CPT → N = 6.12 Rule of 72 → 72/12 = six years
An investor is to receive a 15-year, $8,000 annuity, with the first payment to be received today. At an 11% discount rate, this annuity's worth today is closest to: a) $55,855 b) $57,527 c) $63,855
c) This is an annuity due. Switch to BGN mode. N = 15; PMT = -8,000; 1/Y = 11; FV = 0; CPT → PV = 63,854.92 Switch back to END mode.
