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1 University of Virginia - math1140: Financial Mathematics Fall 2011 Exam 1 7:00-9:00 pm, 26 Sep 2011 Honor Policy. For this exam, you must work alone. No resources may be used during the quiz. The only things you can use are this handout, your brain, a calculator, and a writing tool. Cellphones are not allowed. You may not aid or accept aid from other students. Directions. Answer all questions including all sub-parts. You may use the backs of pages for your scratch work, but I will only grade answers that are written in the answer spaces, or that are found following clearly marked arrows from these spaces. The space provided for each answer is designed to be big enough to easily fit a full credit, correct answer. If you feel like you need more space to write your answer, then either you have extremely large handwriting, or your answer is incorrect, inelegant, or you are providing more detail than is needed for full credit. Your Name: UVa Id: E1-1

2 Problem 1: Simple Interest. (10) a. (3) Define simple interest and give the formula for calculating it. Include definitions of all the symbols used in the formula. The interest is the charge for the use of the money. The basic simple interest formula is I = Pit where I is the interest, P is the principal (the borrowed or invested money), i is the interest rate per time period, t is the term (the length of the loan in time units). b. (3) Bob takes out a $5, 000, 15% simple interest, loan for nine months. How much does Bob have to pay at the due date? The principal is P = $5, 000, i = 15% per year, and the term is t = 9 months. We apply the future value formula S = P(1 + it) = $5, 000( ) c. (4) If you desire to triple your money at 8% simple interest, how long it will take you? Let x denote the principal. Since we are looking to triple the money, then the future value is 3x. The rate of interest is i = 8% per year. We apply the future value formula S = P(1 + it). 3x = x( t) We divide both sides by x and we have 3 = t. Then t = 2/0.08. E1-2

3 Problem 2: Discounting a note. (10) a. (5) Explain the discounting of a note. Note: I not asking you to discuss discount interest. You should give the three parties involved and explain how the sell price of the note is calculated. The discounting of a note is the sell of a note by the owner of the note to a third party before the maturity value. The interest rate can be different than the interest rate of the original note, and the sell price will be calculated as the present value of an investment with term equal to the time interval from the sell date to the maturity date and the maturity value equal to the maturity value of the note. The three parties involved are the borrower, the entity taking out the loan, the lender, the entity lending the money to the borrower, and the third party, the entity buying the note from the lender. b. (5) A $2, 000 corporate note paying 10% interest and maturing in six months is bought today. After three months this note is sold to a third party requiring 15%. Find the amount the third party paid for the note. Find the actual interest the original note holder made on his investment. The maturity value of the note is S = $2, 000( ) = $2, 100. S The third party will pay P = = $2, = $2, The original holder invested $2, 000 and, in 3 months, sold the note for $2, The interest made on the investment is $2, $2, 000 = $24.1. E1-3

4 Problem 3: NPV and IRR. (10) a. (2) Define the net present value. The net present value at the interest rate i is the sum of investments and expenses (negative) and returns (positive), all brought to the present using the interest rate i. b. (2) Define the internal rate of return. The internal rate of return is the interest rate that makes the net present value zero. c. (3) Today, Nora invests $1, 000 in her garden. She will buy $100 worth of gardening supplies each month for the next four months. In five months she will sell her tomatoes for $5, 000 at the farmer s market. What is the net present value of Nora s investment at 5% simple interest? = $1, $ /12 + $ /12 $ /12 + $ /12 $5, 000 = $3, /12 d. (3) Set-up the equation used to calculate the internal rate of return of the investment. 0 = = $1, $ IRR 1/12 + $ IRR 2/12 $ IRR 3/12 + $ IRR 4/12 $5, IRR 5/12 IRR = 790% E1-4

5 Problem 4: Partial Payments. (10) a. (5) Kathrine purchased a used car on July 1 for $8, 500 and paid $1, 500 down. The dealer agreed to a 7% interest rate and two partial payments of $2, 000 on Sep 1 and $2, 000 on Dec 1. The balance is due on Feb 1. Use the Merchant s Rule to calculate the final payment. Calculate the term in months (aka use approximate time and ordinary interest). The balance calculated using the Merchant s Rule is $8, 500( ) + $1, 500( ) +$2, 000( ) + $2, 000( ) = $3, b. (5) Kathrine purchased a used car on July 1 for $8, 500 and paid $1, 500 down. The dealer agreed to a 7% interest rate and two partial payments of $2, 000 on Sep 1 and $2, 000 on Dec 1. The balance is due on Feb 1. Use the US Rule to calculate the final payment. Calculate the term in months (aka use approximate time and ordinary interest). The balance at the first payment is $8, 500( ) + $1, 500( ) + $2, 000 = $5, The balance at the second payment is $5, ( ) + $2, 000 = $3, The final balance is $3, 170.6( /12) = $3, E1-5

6 Problem 5: Equations of Value and Add-on Loans. (10) a. (5) A woman owes $1, 000 due in one year and $2, 000 due in 3 years. What is the cash equivalent of these two amounts today, at 10%? The cash equivalent of the two amounts is $1, 000 $2, = $2, b. (5) You purchase a new refrigerator, and finance it by borrowing $975 from the store, which charges add-on interest at an annual rate of 10%. If you are to make monthly payments for 18 months, how much is the payment? The monthly payment for an add-on loan is the maturity value divided by the number of months $975( /12) 18 = $62.29 E1-6

7 Problem 6: Discount Interest. (10) a. (5) Give the basic discount interest formula and define all the symbols. The basic discount interest formula calculates the discount D and it is D = Sdt. The discount is the interest charge, paid upfront, S is the amount (the money paid back), d is the discount rate (the percentage of the amount for a given time interval), and t is the term in time units. b. (5) Find the discount of $1, 000 in two years at a bank discount rate of 8%. Note: I am asking for the discount, D, not the discount rate d. D = Sdt = $1, = $160 E1-7

8 Problem 7: Discount Interest. (10) a. (5) On March 1, 2003, Alice bought a 3-month, $5, 000 CD earning 8% simple interest. Later, on April 1, 2003, she bought a six months $4, 000 note earning 6% simple interest. Find Alice s proceeds for both securities on May 1,2003 at a discount rate of 8%. First we need to calculate the maturity value of the two financial instruments using simple interest formulas and then we use discount interest to move them back to May 1. The proceeds for both securities on May 1, 2003 is $5, 000( )( ) + $4, 000( )( ) = $ b. (5) John bids 96.4 on a 28-day $500, 000 T-bill. Find the purchase price, face value, discount yield, and investment yield. The purchase price is PP = (B/100)FV = $500, 000 = $482, 000. The face value is FV = $500, 000. The discount yield is the discount rate when using ordinary interest. We calculate it using the formula D = S P = Sdt, more specifically FV PP = FVdm/360. Thus d = FV PP FV m 360 = $500, 000 $482, 000 $500, = 46.29% The investment yield is the simple interest rate when using exact interest. We calculate it using the formula I = S P = Pit, more specifically FV PP = PPim/365. d = FV PP PP m 365 = $500, 000 $482, 000 $482, = 48.68% E1-8

9 Problem 8: Compound Interest. (10) a. (5) Give the compound amount formula and define all the symbols used. S = P(1 + i) n where S is the amount, P is the principal, i is the interest rate per period and n in the total number of conversion periods. b. (5) Suppose that a savings account pays 5% interest per quarter compunded quarterly. If $2, 000 is deposited, how much is in the account 2 years later? S = $2, 000( ) 8 = $2, E1-9

10 Problem 9: Decisions, decisions. (20) Alice needs $10, 000 today. Alice has $10, 000 invested in a certificate of deposit paying 8% in six months. She considers three otions for obtaining the $10, 000: Option 1) take out a loan with nominal rate 10% compounded monthly, for six months. Option 2) take out a discount interest loan with a discounted rate of 6%, for six months. Option 3) withdraw $10, 000 from the CD, forfeiting the interest on the amount withdrawn. Which option should Alice choose? Justify your answer. If Alice uses the first option, then she earns interest from the certificate of deposit and pays interest for the loan. The certificate of deposit after six months has value The loan has a maturity value of $10, 000( ) = $10, ( $10, ) 6 = $10, If Alice uses option 1, then she loses $10, $10, 400 = $ If Alice uses the second option, then she earns interest from the certificate of deposit and pays interest for the loan. The certificate of deposit after six months has the same value as in the first option. The loan has a maturity value of S = P 1 dt = $10, = $10, I Alice uses option 2, then she makes $10, 400 $10, = $90.72 If Alice uses option 3, then she does not earn interest and she does not have to pay interest. She makes no profit. We conclude, the best option for Alice is 2. E1-10

11 Problem 10: Extra Credit.(10) You can borrow money from Bank A at 6% simple interest, where time is computed using the number of days that have passed. You can also invest money at Bank B for 6% simple interest, and in this case time is computed using the number of months that have passed. You hit upon the idea of borrowing from Bank A and investing that money with Bank B during the month of February. How much do you need to borrow and invest in order to realize a profit of $10, 000 at the end of the month? Use exact interest for Bank A and use months for Bank B. ( ) If I borrow x dollars from bank B, then I have to pay back x If I invest x dollars with bank A, the the maturity value is x ( ) 28. To make a profit of $10, 000, the maturity value of the x dollars invested with bank A has to be $10, 000 more than the maturity value of a loan of x dollars with bank B. The equation we need to solve is ( x ) ( + $10, 000 = x ) x + $10, 000 = 1.005x ( )x = $10, x = $10, 000x = $ E1-11

12 Problem 11: Extra Credit.(10) Doris can invest $20, 000 in an account paying 0.65% interest per month compounded monthly, or she can invest the same amount in an account paying simple interest with the time calculated in days. If her investment will be from January 1 to May 1, what annual interest rate must the second account pay for the two investments to yield the same return? The serial number of January 1 is 1 and the serial number of May 1 is 121. Use exact interest. The investment of $20, 000 in an account paying 0.65% interest per month compounded monthly from January 1 to May 1 has a maturity value of $20, 000( ) 4 = $20, Let i denote the interest rate of the account paying simple interest with the time calculated in days. The maturity value is ( $20, i 120 ) 365 To determine the interest rate i we need to solve the equation ( $20, = $20, i 120 ) 365 and we get i = 7.98%. E1-12

13 Problem 12: Extra Credit.(10) Krusty is negotiating his next contract for his show. He would like to insure that the residuals (payments for later repeat broadcasts) will have a present value of $500, 000. There will be three annual payments, one now, one in one year and another payment in two years. The payments will increase by 5% each year. If money is worth 7% simple interest, how large is the first payment? Let x be the first payment. The second payment is 5% larger, thus 1.05x. The third payment is 5% larger than the second, thus (1.05) 2 x. Since the present value of the three payments is $500, 000, then the sum of the three payments moved to the present must be $500, 000. x x (1.05)2 x = $500, x x x = $500, x = $500, 000 x = $180, E1-13

14 Problem 13: Extra Credit.(10) On January 1, 1995, Maggie received a payment of $2, 500. On January 1, 2005, she will receive another payment of $5, 000. On January 1, 2015, she will receive a final payment of $10, 000. If we assume a nominal rate of 7% convertible semiannually, what is the present value of all of these payments on January 1, 2002? The value of $2, 500 from January 1, 1995 on January 1, 2002 is $2, 500( /2) 1 4 = $4, The value of $5, 000 from January 1, 2005 on January 1, 2002 is $5, 000( /2) 6 = $4, The value of $10, 000 from January 1, 2015 on January 1, 2002 is $10, 000( /2) 26 = $4, The present value of all of these payments on January 1, 2002 is $4, $4, $4, E1-14

15 Problem 14: Extra Credit.(10) John won $10, 000 at the state lottery. He has two options to receive his money: Option A) Four equal payments of $2, 500 every six months for two years Option B) Two equal payments of $5, 000 every year for two years For both options the first payment is today. If money is worth 5%(12), which option should John choose? John should choose the option with the biggest net present value. The present value of option A is $5, $5, 000( /12) 12 = $9, The present value of option B is $2, $2, 500( /12) 6 + $2, 500( /12) 12 + $2, 500( /12) 18 = $9, John should use option A. E1-15

16 Problem 15: Extra Credit.(10) Explain why is an add-on loan more expensive for the borrower than a simple interest loan. The add-on loan requires monthly payments but those payments do not help reduce the interest charges. The present value of the monthly payments at the loan s interest rate is bigger than the principal. E1-16

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