1 1. LS7a An account was established 7 years ago with an initial deposit. Today the account is credited with annual interest of $860. The interest rate is 7.7% compounded annually. No other deposits or withdrawals have been made. How much is the end-of-day balance? o o **Note** For some reason the test answers to this question exhibit very large rounding errors. For example, the correct test answer is 12,034. Even the answer that Dr. Downs comes to on his example is about $10 off from the test answer. I have no explanation for why he does this, but be aware.
2 2. MC5c Here are two future expenses that you want to save for today: $5,200 payable in 6 years, and $7,700 payable in 9 years. You make an investment today that perfectly finances the future expenses if the investment earns a target 12.8% average annual rate of return (compounded annually). The investment indeed grows sufficiently to finance your first expense. Unfortunately, for the entire investment horizon your actual annual rate of return falls short of the target by 240 basis points per year. When it is time to pay the second expense, how much money do you lack? First, discount back your two future payments to find out how much you deposited in the account. N = 6 N = 9 I/Y = 12.8 I/Y = 12.8 PV = CPT = -2, PV = CPT = -2, PMT = 0 PMT = 0 FV = 5,200 FV = 7,700 Total Deposit = 2, , = 5, Now you need to see how much your initial deposit actually grew during the first 6 years. N = 6 I/Y = = 10.4 PV = -5, PMT = 0 FV = CPT = 9, Now subtract your first payment and see how much your remaining balance grows over the next 3 years. N = 9 6 = 3 I/Y = = 10.4 PV = (9, ,200) = 4, PMT = 0 FV = CPT = 5, Since you owe 7,700 and you only have 5, your shortfall is 7,700 5, = 2,202.04
3 3.1 ROR1 A venture capitalists provides a company equity financing of $11.0 million. After 7 years the company repurchases the equity for $47.4 million. There are no other cash flows between the two. Find the average annual geometric rate of return, and also find the cumulative rate of return. N = 7 I/Y = CPT = = Geometric Return PV = -11 PMT = 0 FV = 47.4 Cumulative Rate of Return 3.2 ROR3c Two years ago you purchased a stock for $37. One year ago the price had moved to $10. Today it is at $49. Which one statement about the annual average rate of return is correct? a) The geometric average return is 17.3% and the arithmetic average return is 158.5%. b) The geometric average return is 17.3% and the arithmetic average return is 137.8%. c) The geometric average return is 13.1% and the arithmetic average return is 158.5%. d) The geometric average return is 15.1% and the arithmetic average return is 137.8%. e) The geometric average return is 15.1% and the arithmetic average return is 158.5%. N = 2 I/Y = CPT = = Geometric Return PV = -37 PMT = 0 FV = 49 Arithmetic Average Return 3.3 LS4c A deposit exactly 14 years ago of $2,300 earns 9.5% annual interest compounded annually. There have been no other deposits or withdrawals. As of today, how much total interest-on-principal has accumulated?
4 3.4 LS6a A savings account was established with $36,000 exactly 9 years ago. The account earns 5.3% compounded annually. Otherwise, the account has been left alone. When the annual interest is credited to the account today, how much interest is credited? o N = 9 1 = 8 o I/Y = 5.3 o PV = -36,000 o PMT = 0 o FV = CPT = 54, =
5 4.1 BS38 The Federal Reserve Board of Directors uses three significant tools to influence market activity. Which of these statements is the most accurate description of one of these Fed tools? Accurate Statements The official government "discount rate" is the interest rate charged by Federal Reserve District banks to member public and private banks. The reserve requirement on member bank accounts regulates the amount of loans that banks may lend to business and individual borrowers. Buying and selling currencies and government securities in the global financial marketplace affects supply and demand conditions for capital. 4.2 LS24 You wish to purchase in 10 years an item that today costs $1,400. The cost is expected to inflate at an annual rate of 4.4%. You make a deposit today that perfectly finances the future purchase. The observed interest rate that your savings earns is 7.6%. Describe the relation between your deposit, inflation, and the discount rate. a) the deposit equals the real cost of $1400 discounted at the nominal rate 7.6% b) the deposit equals the real cost of $1400 discounted at the real rate 3.07% c) the real interest rate is 7.6% d) Two choices, A and B, are correct e) None of the A-B-C choices are correct Correct Answer = B The real interest rate is the rate that your purchasing power increases at and can be found by
6 5.1 LS14 Today your account was credited with its annual interest of $5,210. The account was established some time ago with a $17,150 initial deposit. No other deposits or withdrawals have been made. The account earns 8.7% annual interest. How many years ago was the account established? o o N = CPT = 16 I/Y = 8.7 PV = -17,150 PMT = 0 FV = 65, LS15 Some time ago a $136,000 initial deposit opened an account. No other deposits or withdrawals have been made. Today the annual interest was credited to the account. Total lifetime interest now equals $71,210. The account earns 6.2% annual interest. How many years ago was the account established? N = CPT = 7 I/Y = 6.2 PV = -136,000 PMT = 0 FV = 136, ,210 = 207,210
7 6. LS10a In exactly 20 months a bill of $15,120 is due. Today you deposit money such that if the account earns a target rate of return of 0.84% per month, the bill is perfectly financed. Unfortunately, your account earns 15 basis points less than your target. When the bill is due, how much money do you lack? N = 20 = 20 I/Y = 0.84 = = 0.69 PV = CPT = -12, = -12, PMT = 0 = 0 FV = 15,120 = CPT = 14, Since you owe 15,120 and you only have 14,676.48, you lack 15,120 14, =
8 7. CY10a In exactly 14 years a bill of $26,040 is due. Today you deposit money such that if the account earns a target rate of return of 6.00% per annum, compounded quarterly, the bill is perfectly financed. No other deposits or withdrawals have been made. Your account actually accumulates $22,154. What was the actual average annual percentage rate? N = 14 =14 I/Y = 6 = CPT = 4.78 PV = CPT = -11, = -11, PMT = 0 = 0 FV = 26,040 = 22,154
9 8. CY21 Suppliers X and Z are competing to sell your company supplies. The full price of supplies from supplier X is $1,300 and they offer these payment plans: 4.9% discount if you pay within 10 days, otherwise pay full price within 230 days. The full price with supplier Z is $1,390 and they offer these payment plans: 5.8% discount if you pay within 25 days, otherwise pay full price within 290 days. Your company financing rate is 7.7% compounded daily. Find the supplier and payment plan that represent the lowest present value of cost. Supplier X Plan 1 Supplier X Plan 2 N = 10 N = 230 I/Y = 7.7/365 = I/Y = 7.7/365 = PV = CPT = 1, PV = CPT = 1, PMT = 0 PMT = 0 FV = 1,300 (1,300 x 0.049) = 1, FV = 1,300 Supplier Z Plan 1 Supplier Z Plan 2 N = 25 N = 290 I/Y = 7.7/365 = I/Y = 7.7/365 = PV = CPT = -1, PV = CPT = 1, PMT = 0 PMT = 0 FV = 1,390 (1,390 x 0.058) =1, FV = 1,390 Supplier X plan 1 has the lowest present value, but if you buy from supplier Z the lowest present value plan is plan 1
10 9.1 FF6 By how many basis points does 4.4% differ from 8.5%? = TR15 What are proper definitions for PVIFA, FVIFA, and the discount rate? Proper Definitions the present value interest factor for an annuity, PVIFA(r,N), equals the initial deposit into an account earning the periodic rate r that perfectly finances a series of N one-dollar withdrawals the future value interest factor for an annuity, FVIFA(r,N), equals the total accumulation in an account earning the periodic rate r that results from a series of N one-dollar deposits the discount rate is the periodic percentage return subtracted from the future cash flow for computing present value 9.3 TR33 From a series of periodic rates of return there are two procedures for computing the average rate of return per period: the arithmetic average and the geometric average. Is the following statement about these two statistics TRUE or FALSE: The geometric average periodic rate of return always is greater than or equal to the arithmetic average periodic rate of return. False TRUE or FALSE: The arithmetic average periodic rate of return always is greater than or equal to the geometric average periodic rate of return. True
11 10.1 FV7 Your parents contribute $135 monthly to a college savings plan for you that earns 8.00% compounded monthly. The first deposit was exactly 9 years ago. Find the account balance after today's monthly deposit and crediting of monthly interest. N = 9 x 12 = 108 I/Y = 8/12 = PV = -135 PMT = -135 FV = CPT = 21, FV9 An account is today credited with its monthly interest thereby bringing the account balance to $5,450. The interest rate is 10.10% compounded monthly. You plan to make monthly withdrawals of $55 each. The first withdrawal is in exactly one month and the last in exactly 12 years. Find the account balance immediately after the last withdrawal. N = 12 x 12 = 144 I/Y = 10.10/12 = PV = -5,450 PMT = 55 FV = CPT = 2,908.44
12 11. FV5 An account is today credited with its annual interest thereby bringing the account balance to $7,490. The interest rate is 9.10% compounded annually. You plan to make annual withdrawals of $700 each. The first withdrawal is in exactly one year and the last in exactly 18 years. Find the account balance immediately after the last withdrawal. N = 18 I/Y = 9.10 PV = -7,490 PMT = 700 FV = CPT = 6,722.12
13 12. FV6 Today you inherit an account with a balance of $7,400. For a while you don't do anything with the account but it continues to accrue interest. Exactly 20 months from today you start an ambitious savings plan and deposit $200 into the account. You plan to deposit that much each month. Exactly 40 months from today you reconsider your plan, make your last deposit, and make no additional deposits. You nonetheless leave the account alone and it continues to accrue interest at a rate of 8.6% compounded monthly. You finally close the account exactly 7 years from today. How much is the total accumulation? N = 19 = = 21 = (7 x 12) 40 = 44 I/Y = 8.6/12 = = 8.6/12 = = 8.6/12 = PV = -7,400 = -8, = -14, PMT = 0 = -200 = 0 FV = CPT = 8, = CPT = 14, = CPT = 19,663.68
14 13.1 TR3 Suppose two alternative investments promise cash flow streams that possess equal lives. Further, suppose the simple sum of the cash flows for each investment is the same amount. Given a positive interest rate, which investment has the smallest present value? a) an investment that is being discounted by a small discount rate. b) an investment which generates equal cash flows each period. c) an investment which generates most cash flows at the beginning of its life. d) there is no reliable relationship between the distribution of cash flows and present value. e) an investment which generates most cash flows at the end of its life. Answer = E 13.2 TR4 Suppose two alternative investments promise cash flow streams that possess equal lives. Further, suppose the simple sum of the cash flows for each investment is the same amount. Given a positive interest rate, which investment has the biggest present value? a) an investment that is being discounted by a large discount rate. b) an investment which generates equal cash flows each period. c) there is no reliable relationship between the distribution of cash flows and present value. d) an investment which generates most cash flows at the end of its life. e) an investment which generates most cash flows at the beginning of its life. Answer = E 13.3 TR5 Suppose two equal-cost alternative investments promise cash flow streams that possess equal lives. Further, suppose the simple sum of the after-tax cash flows for each investment is the same amount. Given a positive interest rate, which investment has the smallest present value? a) an investment which generates most cash flows at the end of its service life. b) an investment which generates equal cash flows each period. c) an investment that is being discounted by a small discount rate. d) there is no reliable relationship between the distribution of cash flows and present value. e) an investment that has relatively high sales revenues. Answer = A
15 14.1 PV7 A friend received an inheritance 5 years ago and put all funds into an account earning 8.00% compounded quarterly. Exactly one quarter after establishing the account the friend started withdrawing $1,300 per quarter. Today she'll make another quarterly withdrawal, and quarterly interest will be credited to the account, and then the balance will be $18,711. How much was the friend's inheritance? N = 5 x 4 = 20 I/Y = 8/4 = 2 PV = CPT = -33, PMT = 1,300 FV = 18, PV9 You might invest in a security that will return after-tax cash flow to you of $1,100 per year for 7 years (first cash flow one year from now), after which the security likely can be sold immediately for $7,100. You make an offer to buy the security so that you'll get a 10.30% rate of return (compounded annually). Find the offer price. N = 7 I/Y = 10.3 PV = CPT = -8, PMT = 1,100 FV = 7,100
16 15. PV10b You might invest in an asset that will return after-tax cash flow to you of $2,200 per month for 40 months (first cash flow one month from now), and after receiving the last cash flow you'll immediately receive after-tax net proceeds from liquidation equal to $82,500. You make an offer to buy the asset so that you'll get your "target" annual rate of return of 16.80% (compounded monthly). The seller makes a counteroffer that is $8,300 higher than your offer. Find your annual rate of return if you buy at the counteroffer price and receive the expected cash flows. N = 40 N = 40 I/Y = 16.8/12 = 1.4 I/Y = CPT = x 12 = PV = CPT = -114, PV = -114, ,300 = -122, PMT = 2,200 PMT = 2,200 FV = 82,500 FV = 82,500
17 16. FV17 Today you open an account with a $8,800 deposit that earns 8.70% compounded annually. You've set a target for the account so that in exactly 6 years its balance will be $11,500. To reach the target you'll adjust the balance annually; each year's adjustment will be exactly the same amount and the first adjustment occurs exactly one year from now. After the last annual adjustment in exactly 6 years, and crediting of that year's interest, the account balance exactly equals the target. Describe the annual adjustment that you make each year. N = 6 I/Y = 8.7 PV = -8,800 PMT = CPT = FV = 11,500 Each year you make a withdrawal of
18 17.1 TR1 Which statement describes the "rule of 72"? a) The approximate number of years required for a deposit to double equals 72 divided by the percentage interest rate. b) The number of months required for a deposit to double equals the decimal interest rate divided by 72. c) The simple sum of cash flows required for an investment to earn a positive rate of return equals the investment cost divided by 72. d) The number of months required for a deposit to double equals the decimal interest rate times 72. e) The simple sum of cash flows required for an investment to earn a positive rate of return equals the investment cost times 72. Answer = A 17.2 TR2 For the simple time value lump-sum relation with monthly compounding find the most accurate statement. For all cases, hold everything constant except the stated variables. Accurate Statements When the interest rate doubles then the total interest more than doubles. When the term doubles then the total interest more than doubles. When the beginning wealth doubles then the total interest exactly doubles TR27 Banks advertise loans so that you'll borrow money from them and pay them interest. They also advertise deposits so that you'll open an account with them and they'll pay you interest. Which statement below most likely describes the relation between the effective annual rate ("EAR") and annual percentage rate ("APR"). Accurate Statements The EAR generally is bigger than the APR. The APR generally is smaller than the EAR. When banks advertise for loans they likely quote the APR. When banks advertise for deposits they likely quote the EAR.
19 18. TS1b You wish to establish an endowment fund that will provide student financial aid awards every semiannum, perpetually. To finance the scholarships you will make a series of equal deposits into a savings account. The deposits will be made semiannually equal to $2,200 each, with the first one today and the final one in 7 years. The first award is to be granted one semiannum after the last deposit. The savings rate is 6.30% compounded semiannually. How much is each award? N = 7 x 2 = 14 I/Y = 6.3/2 = 3.15 PV = -2,200 PMT = -2,200 FV = CPT = 41,370.46
20 19. AM3d The Company borrowed $174,000 at 9.60% to be repaid monthly over 15 years. They just remitted payment number 83. How much interest-to date has been paid? N = 15 x 12 = 180 = 83 I/Y = 9.6/12 = 0.8 = 9.6/12 = 0.8 PV = 174,000 = 174,000 PMT = CPT = -1, = -1, FV = 0 = CPT = -122, ( ) ( ) ( )
21 20.1 AM5a Your friend is taking out a mortgage for $127,000 at 8.70% repayable with monthly payments over 25 years. She respects your financial expertise and asks "how many payments will I have to make before I reduce the principal balance by half its original amount." You pull out your calculator, and tell her the number of payments she'll make to reduce the balance by half is: N = 25 x 12 = 300 N = CPT = 219 I/Y = 8.7/12 = I/Y = 8.7/12 = PV = 127,000 PV = 127,000 PMT = CPT = -1, PMT = -1, FV = 0 FV = -127,000/2 = -63, AM9c You have just bought a house by borrowing $220,000 at a 8.70% annual interest rate (compounded monthly) repayable with fixed payments over 25 years. When finally in the far-off future you make your last payment, how much of that last payment will be principal? N = 25 x 12 = 300 N = (25 x 12) 1 = 299 I/Y = 8.7/12 = I/Y = 8.7/12 = PV = 220,000 PV = 220,000 PMT = CPT = -1, PMT = -1, FV = 0 FV = CPT = 1,788.28
22 21. CB2a Consider the following cash flows for two mutually exclusive investments: at time 0: CF A = ($780) and CF B = ($1,050) at time 1: CF A = $547 and CF B = $117 at time 2: CF A = $319 and CF B = $327 at time 3: CF A = $124 and CF B = $950 Which statement is true? a) if the financing rate is 16.8% then project A is the better of the two b) if the financing rate is 5.8% then project A is the better of the two c) if the financing rate is 8.52% then projects A and B create the same amount of wealth d) if the financing rate is 11.6% then project B is the better of the two e) if the financing rate is 14.2% then project B is the better of the two Use CF Function on calculator Project A Project B CF ,050 C C C Project A NPV Project B NPV 16.8% % % % % Answer = A
23 22. CB8 The bank issued a $138, year mortgage (monthly payments) with an annual interest rate of 8.20%. They just received payment number 110 and have decided to sell the loan. The buyer of the loan expects to receive an annual rate of return equal to 10.20%. For the original bank that issued the loan, what was the internal rate of return? N = 25 x 12 = 300 = (25 x 12) 110 = 190 = 110 I/Y = 8.2/12 = = 10.2/12 = 0.85 = CPT = x 12 = PV = -138,000 = CPT = -101, = -138,000 PMT = CPT = 1, = 1, = 1, FV = 0 = 0 = 101,939.95
24 23.1 FF23 The payback period assesses the quality of a cash flow stream. What are some general advantages and/or disadvantages of this assessment measure? the payback period ignores the time value of money a short payback period is better than a long payback period the payback period ignores cash flows occurring after the payback point is reached 23.2 TR32 What factors are relevant for a household choosing between borrowing with a 15-year versus a 30-year loan (assume that fees, interest rates, and all else are equal). Correct Answers Total lifetime interest definitely is less with a 15-year loan but that doesn't mean it is a better choice. Total lifetime interest definitely is more with a 30-year loan but that doesn't mean it is a worse choice. When the household expects short-run liquidity problems as their careers commence but for the long-run they expect high income growth then borrowing with a 30-year loan may be advantageous. When the household expects that in the short-run their uses for money will have low utility but that in the long-run their uses for money will have higher utility then borrowing with a 15-year loan may be advantageous. When the household expects that in the short-run their uses for money will have high utility but that in the long-run their uses for money will have lower utility then borrowing with a 30-year loan may be advantageous TR36 Which statement is most consistent with the Net present value (NPV) and Internal rate of return concepts? Consistent Statements The IRR for a project is the financing rate at which the project's NPV is zero. The NPV for a project is the amount of capitalized economic profit that a project creates. When the project's actual financing rate is less than the IRR then the NPV is positive. When the project's actual financing rate exceeds the IRR then the NPV is negative. When a project's NPV is positive then the actual financing rate is less than the IRR.
25 24. CB10a Your company is analyzing purchase of a machine costing $8,700 today. The investment promises to add $9,000 to sales one year from today, $12,500 two years from today, and $15,000 three years from today. Incremental cash costs should consume 50% of the incremental sales. The tax rate is 35% and the company's financing rate is 15.8%. The investment cost is depreciated to zero over a 3-year straight-line schedule. Find the project's net present value and internal rate of return. What is important is not the increase in sales but the increase in profits, so for cash flows you have to remove costs and taxes, but you also have to add in the tax benefit from depreciation. [ ( ) ( )] ( ) [ ( ) ( )] ( ) [ ( ) ( )] ( ) NPV = 2, IRR = 30.12%
26 25. CB3c You took out a 30-year mortgage (monthly payments) for $125,000 at 8.90% and payment number 31 is due today. You are deciding whether you should refinance the outstanding principal by borrowing at today's lower rate of 5.90% an amount that pays off the old loan. The new loan is for 30 years as of today. The total fees for getting the new loan equal 3.6% of the original loan's outstanding principal. The first payment for the new loan would be due one month from today. Suppose you pay the fees today with funds from your savings account. What is the net present value of the refinancing venture if your "personal discount rate" is 15%? N = 30 x 12 = 360 = 31 = 360 I/Y = 8.9/12 = = 8.9/12 = = 5.9/12 = PV = 125,000 = 125,000 = 122, PMT = CPT = = = CPT = FV = 0 = CPT = -122, = 0 Now use the Cash Flow Function CF0 = -122, x = -4, C01 = = F01 = = 329 C02 = F02 = 31 NPV = 16,491.10