Time Value of Money Concepts and Calculations

6. Sam wants to accumulate $75,000 in 7.5 years to purchase a boat. He expects to earn an annual rate of return on invested funds of 12% compounded quarterly. How much does Sam need to invest today to meet his goal? A. $30,899 B. $31,489 C. $32,057 D. $66,00

A The answer is $30,899. Sam needs to set aside $30,899 now to achieve his financial goal. Make sure to use a quarterly discount rate in deriving this amount. Using 1 P/YR, the keystrokes on the HP 10bII/HP 10bII+ are 75,000 FV; 12 ÷ 4 = 3 I/YR; 7.5 × 4 = 30 N; solve for PV −30,899.007, or $30,889.

8. Jerry makes a contribution to his retirement account at the end of each quarter. He has $60,000 in current savings and wants to reach an account balance of $100,000 in the next six years. Jerry's investments will earn an annual rate of return of 7% compounded quarterly. Calculate the amount that he needs to contribute each quarter. A. $300.17 B. $305.43 C. $491.91 D. $555.5

B The answer is $305.43. Jerry needs to contribute $305.43 each quarter, with keystrokes on the HP 10bII/HP 10bII+ as follows: END mode 100,000 FV 60,000 +/− PV 7 ÷ 4 = 1.75 I/YR 6 × 4 = 24 N Solve for PMT = −305.4260, or $305.43

21. Zack invested $15,000 in a mutual fund six years ago, and it has had the following end-of-year distributions: End of Year Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Amount $175 $225 $225 $225 $250 $215 Zack redeemed $4,500 of the fund in year 5, and sold the remainder of his shares for $16,500 at the end of year 6. What is Zack's internal rate of return (IRR)? A. 6.80% B. 7.30% C. 6.96% D. 7.12%

B The answer is 7.30%. END mode, 1 P/YR 15,000, +/−, CFj 175, CFj 225, CFj 3, SHIFT, Nj (three $225 cash flows—could also just enter each separately) 4,750, CFj ($250 distribution plus $4,500 redemption) 16,715, CFj ($215 distribution plus $16,500 redemption) SHIFT, IRR/YR = 7.3030, or 7.30%

Ordinary Annuity

Payments are made at the end of each period Ex: Mortgage payments

Serial payments

Payments increase each year by the amount of inflation (to maintain a constant or real dollar amount).

Annuity Due

Payments made at the beginning of each period Ex: Lease payments

You've determined the net present value (NPV) of your client's investment to be $32,500. If your client's required rate of return is 8%, which of the following is most likely to be the investment's internal rate of return (IRR)? A) 3% B) 0% C) 7% D) 9%

The correct answer is 9%. If the net present value (NPV) of the investment is a positive number, the investment's IRR must be greater than or equal to the investor's required rate of return. Here, the investor's required rate of return is 8%, so the IRR must be greater than or equal to 8%. 4.3.1

Terry wants to save $1 million (in today's dollars) for his retirement, which is 20 years away, by depositing money in his brokerage account at the end of every year using the serial payment method. He assumes he can earn 7% on his investments and that inflation will average 3% over the 20-year savings period. How much does Terry need to deposit at the end of the first year to meet his goal?

The answer is $35,009.39. The required payment at the end of the first year is $35,009.39. The keystrokes on the HP 10bII/HP 10bII+ are as follows: 1,000,000, FV; [(1.07 ÷ 1.03) − 1 × 100] = 3.8835, I/YR; 20, N; Solve for PMT = -33,989.7035; PMT of -33,989.7035 × 1.03 = -35,009.3946, or $35,009.39. 4.2.3

Gilbert purchased several gold coins for $30,000. Today, he sold the coins for $55,045.91. Gilbert estimated the average annual rate of return, compounded monthly, on the coins was 9%. Approximately how many years did Gilbert own the coins (rounded to the nearest 0.00)?

The answer is 6.77. 9 ÷ 12 = 0.75, I/YR 30,000, +/‒, PV 55,045.91, FV Solve for N = 81.2325 ÷ 12 = 6.7694 years (6.77, rounded) 4.1.5

Rule of 72

To calculate the number of years for an investment to double in value, simply divide 72 by the annual interest rate.

20. Drake purchased rental property four years ago for $128,900. His cash flows for each year were as follows: Year End Year 1 Year 2 Year 3 Year 4 Inflows $6,000 $6,600 $7,400 $8,400 Outflows $3,200 $1,400 $400 $1,400 If the property is worth $145,000 at the end of the fourth year, what would Drake's IRR be? A. 5.80% B. 6.99% C. 6.69% D. 6.27%

B The answer is 6.99%. END mode, 1 P/YR 128,900, +/−, CFj 2,800, CFj (This is a positive cash flow: 6,000 inflow minus 3,200 outflow.) 5,200, CFj 7,000, CFj 152,000, CFj SHIFT, IRR/YR = 6.9854, or 6.99% Note that the purchase price is entered as the first cash flow (Cash is purchasing property, so it would be a negative number, an outflow). Also, the final cash flow at the end of the fourth year (net inflow of $7,000) is added to the value of the property ($145,000) to come up with an end of year 4 cash flow of $152,000.

18. Yvette wants to accumulate $80,000 for a future goal in nine years. She can deposit $18,000 today and wants to know what payment she would need to make at the beginning of each six-month period to reach her goal. She is conservative with her investments and wants to assume a 3% rate of return. A. $6,123.95 B. $2,715.23 C. $2,755.96 D. $6,215.81

B The answer is $2,715.23. BEG mode, 1 P/YR SHIFT, C 18,000, +/−, PV 3 ÷ 2 = 1.5, I/YR 9 × 2 = 18, N 80,000, FV Solve for PMT = −2,715.2300, or $2,715.23

2. James invested $20,000 in an account earning a 9% annual rate of interest compounded monthly. Calculate his account value at the end of eight years if all interest is reinvested at the 9% rate. A. $39,851 B. $39,980 C. $40,978 D. $174,400

C The answer is $40,978. James's account will be worth $40,978 at the end of the eight-year period. Make sure to use a monthly interest rate in deriving this amount. The keystrokes are 20,000 +/− PV; 9 ÷ 12 = 0.75 I/YR; 8 × 12 = 96 N; solve for FV= 40,978.4246, or $40,978.

4. Roxanne has already saved $3,000 for a down payment on a future house. She adds $500 at the end of every six months into an account earning an annual rate of 5.5% compounded semiannually. Calculate the amount Roxanne will have accumulated in four years. A. $7,406 B. $7,528 C. $8,134 D. $8,255

C The answer is $8,134. Roxanne will have accumulated $8,134 in four years. This is derived using the following keystrokes: END mode 3,000 +/− PV 500 +/− PMT 5.5 ÷ 2 = 2.75 I/YR 4 × 2 = 8 N Solve for FV= 8,134.0608, or $8,134 Not all FV calculations begin with a zero PV. Here, Roxanne has already saved $3,000, making the initial PV entry 3,000 +/− PV. In addition, you should note that the signs for PV and PMT on the HP 10bII/HP 10bII+ can both be either positive or negative, but one cannot be positive and the other negative. Note: if you came up with an answer of negative $679.78, you have mistakenly entered the payment of $500 as a positive number. The $500, along with the initial PV of $3,000, should be entered as a negative number.

Annuity

Equal payments or deposits on a regular basis

Bernie and Betty purchased their home eight years ago for $239,500. They made a 20% down payment, and financed the balance using a 30-year mortgage with a 5.15% interest rate. Taxes and insurance increase the payment by $300 per month. What is their outstanding principal balance?

The answer is $165,071. Set the calculator for 12 payments per year or 12 P/YR. Next be sure the calculator is in End Mode. A 20% down payment of $47,900 means that Bernie and Betty financed the balance of $191,600, and this is used as the PV in the calculation. Because we must first calculate the regular monthly payment, N = 360 (or 30 years x 12 months per year). The interest or I/YR = 5.15 and all that needs to be done is to calculate the payment or PMT = $1,046.19. In calculating eight years of payments, we are examining the results of 96 payment periods or 8 x 12 = 96; To accomplish this we must press the following keys: 1 [INPUT]; 96 [SHIFT], [AMORT] look under the FV key for AMORT. Once this has all been done the following should be on your screen 1 - 96. Then push the [=] key and the principal paid thus far in eight years will show up; Press the [=] key again and interest paid to date shows up; Press [=] key one more time and the remaining principal balance will be displayed. LO 4.2.1

John wants to start his own business in six years and will need $200,000. He assumes inflation will average 4% and that he can earn a 9% compound annual after-tax rate of return on his investments. What serial payment should John invest at the end of the first year to attain his goal?

The answer is $30,727.95. END mode 200,000, FV [(1.09 ÷ 1.04) − 1] × 100 = 4.8077, I/YR 6, N; Solve for PMT = -29,546.1090 × 1.04 = -30,727.9533, or $30,727.95 (change sign) 4.2.3

George and Chelsea want to make sure they will have enough funds available to send their son Oliver to college. Oliver is eight years old and will begin a four-year college program at age 20 after working full time for two years following high school graduation. The annual tuition today is $10,000, and it is expected to increase annually by 5%. George and Chelsea estimate that they can get a 7% after-tax return on their money. If George or Chelsea were to die, what would be the amount of insurance needed today to provide for Oliver's education?

The answer is $31,012. The answer is calculated by inflating the $10,000 at 5% for 12 years = $17,959. Next, enter $17,959 as the first PMT, and calculate the PVAD (BEGIN) for four years using the inflation-adjusted interest rate (1.9048) = $69,845. Finally, enter $69,845 as a FV, discounted at the after-tax return of 7% for 12 years, and solve for the PV ($31,012). Keystrokes: Step 1 END Mode, 1 P_Yr 12, N 5, I/YR 10,000, +/-, PV Solve for FV = 17,958.5633 Step 2 BEG Mode, 1 P_Yr 4, N ([1.07 ÷ 1.05] - 1) x 100 = 1.9048, I/YR 17,958.5633, +/-, PMT 0, FV Solve for PV = 69,845.1886 Step 3 END Mode, 1 P_Yr 12, N 7, I/YR 69,845.1886, FV Solve for PV = 31,012.0990, or $31,012 LO 4.2.3

An investor makes an initial deposit of $20,000 into a mutual fund. Each subsequent year he deposits an additional $2,500 into the fund. What will be the value of the account in eight years if the fund earns 9% annually?

The answer is $67,422.44. Make sure the calculator is in the END mode, 1 P/YR. 8, N 9, I/YR 20,000, +/‒, PV 2,500, +/‒, PMT Solve for FV = 67,422.4373, or $67,422.44 The account value after eight years would be $67,422.44. 4.2.4

Clayton wishes to start saving for a lump-sum amount of $75,000 (in today's dollars) that is needed in seven years. He assumes an inflation rate of 2% and an investment rate of return of 9%. Assume Clayton wishes to save annually using the level payment method. What is his required deposit? (Round to the nearest dollar.)

The answer is $9,364. The required annual savings deposit under the level payment approach is $9,364, with keystrokes as follows on the HP: 10bII/HP 10bII+: Step 1: Inflate the lump-sum in today's dollars into the future need. 75,000, +/−, PV 7, N 2, I/YR Solve for FV = 86,151.4251 Solve for the required annual level payment using the inflated lump-sum value. (Be sure to clear your calculator) SHIFT, C ALL FV = 86,151.4251 7, N 9, I/YR Solve for PMT = -9,363.8429, or $9,364 (rounded) 4.1.6

When calculating the net present value (NPV) of a potential investment, the investor's desired rate can be called I. the required rate. II. the internal rate. III. the opportunity cost. IV. the cost of capital.

The answer is I, III, and IV. The investor's desired rate may be called the required rate, cost of capital, or opportunity cost. These terms may be used interchangeably in a net present value (NPV) calculation. Internal rate of return (IRR) is the discount rate that, when applied to the cash flows of an investment, equates the net cash inflows to the net cash outflows. 4.1.1

When a single sum is converted to a future value (FV) that represents the value to which the single sum grows over a given period at an assumed or actual rate of return, this process is called

The answer is compounding. Compounding is the process of interest being earned on increasing sums over time. 4.1.1

Phil secures a $350,000, 15-year mortgage with an annual interest rate of 5.5%. What will be the unpaid principal balance on Tommy's mortgage at the end of 10 years? A) $200,281.78 B) $168,128.00 C) $149,718.22 D) $145,238.22

The correct answer is $149,718.22. The unpaid principal balance on Tommy's mortgage at the end of 10 years will be $104,761.78. The keystrokes on the HP 10bII/HP 10bII+ are as follows: END mode; 350,000 PV; 15 × 12 = 180 N; 5.5 ÷ 12 = 0.4583 I/YR; solve for PMT = -2,859.7921; 1 INPUT 120 [Shift] AMORT; (pressing the = key toggles you through amortization totals for months 1 through 120); = - 200,281.7762 (total principal paid through 120 months) = -142,893.2758 (total interest paid through 120 months) = 149,718.2238 (remaining principal balance through 120 months of payments) 4.2.1

Bryan wants to open a photography studio in four years. To do so, he needs to accumulate $175,000 (in today's dollars) to adequately finance this venture. He assumes he can earn an 8% compound annual after-tax rate of return on investment and inflation will average 3.5%. What will be Bryan's serial payment at the end of the third year? A) $40,997.88 B) $45,455.08 C) $42,432.81 D) $43,917.96

The correct answer is $45,455.08. END mode FV= 175,000i= 4.3478 [(1.08 ÷ 1.035) − 1] × 100 n= 4 PV= 0 PMTOA= (40,997.8820) × 1.035 = 42,432.8078 (change sign) 42,432.8078 × 1.035 = 43,917.9561 43,917.9561 × 1.035 = $45,455.08 4.2.3