MATH 373 Fall 2016 Test 1 September27, 2016

Transcription

1 MATH 373 Fall 2016 Test 1 September27, Ellie lends Aakish 10,000 to be repaid over the next three years with level annual payments of Ellie takes each payment and reinvests it at an annual effective interest rate of 6.2%. Determine the yield rate earned by Ellie on this loan when reinvestment is taken into account , 000(1 i) 4000(1.062) 4000(1.062) , 000(1 i) 12, ((1 i) ) ( ) 1i i %

2 2. Jeremy invested 100,000 in a fund three years ago. Nine months after that initial investment, the fund was worth 120,000 and he invested an additional 40,000 in the fund. One year ago, the fund had a value of 200,000 and Jeremy withdrew 60,000. Today the fund is worth 140,000. Estimate Jeremy s annual dollar weighted return over the three year period using simple interest. A 100, 000 B 140, 000 C 40, , , 000 I B A C 140, , , , 000 j I A C (1 t ) C (1 t ) , , 000 j , ,000 40, , / (1 ) (1 ) T 1/3 i j (1 i) ( ) 1 i i %

3 3. Anam borrows 10,000 to be repaid with two payments. The first payment is P at the end of two years and the second payment is 2P at the end of five years. The annual effective interest rate on the loan is 7.3%. Calculate the Outstanding Loan Balance at the end of the third year. 10, 000 P(1.073) 2 P(1.073) , 000 P[(1.073) 2(1.073) ] 10, 000 P (1.073) 2(1.073) OLB 3 3 Accumulated Value of Past Payments 10,000(1.073) (1.073) Or OLB 3 2 Present Value of Future Payments (2)( )(1.073)

4 4. Agarwal Automotive is building a new factory. Agarwal will invest 100 million at time 0 to build the factory. Additionally, Agarwal expects to receive the following profits from the factory over the next four years: Time Amount 1 20 million 2 X million 3 40 million 4 35 million After 4 years, the factory will be obsolete and will no longer be used. This factory will result in an Internal Rate of Return of 10% based on the above cash flows. Calculate the Net Present Value at an interest rate of 12% To find X (in millions): (1.1) 20(1.1) X (1.1) 40(1.1) 35 X (1.21) 100(1.4641) 20(1.331) 40(1.1) 35 X NPV (in millions): NPV (1.12) (1.12) 40(1.12) 35(1.12) NPV

5 5. JT is the beneficiary of an annuity due that makes quarterly payments for 13 years. The first payment is The second payment is The third payment is Each payment is 25 larger than the prior payment. JT takes each payment and deposits it into an account earning a nominal rate of 10% compounded quarterly. Calculate the amount that JT will have at the end of 13 years. P 1000 Q 25 n 4(13) 52 (4) i Q n n AV Pa ( a nv ) (1 i) (1 i) n n i ( ) (( ) 52(1.025) ) (1.025) ) ( ) 1000( ) ( ) 43, ( ) 160, Note that we multiply by (1+i) to get the accumulated value and by (1+i) to get change the value from an annuity immediate to an annuity due.

6 6. Shannon has a loan of 100,000 that is being repaid with level monthly payments of 1000 followed by a drop payment of D. Madison has a loan of 100,000 that is being repaid with level monthly payments of 1000 followed by a balloon payment of B. The interest rate for both loans is 9% compounded monthly. Calculate B D. (12) i We can use our calculator: I / Y 0.75; PV 100, 000; PMT 1000; CPT N Or n n ln(.25) 100, 000 1, n n ln(1.0075) After 185 payments: Using our calculator, 2nd Amort P1 185; P2 185; BAL Or OLB , 000(1.0075) , , Then: B (1.0075) D D B

7 7. John has just won the lottery! John has the choice of the following two options to receive his winnings: a. A perpetuity that pays 1000 at the end of the first year, 2000 at the end of the second year, 3000 at the end of the third year, etc. At an annual effective interest rate of i, this perpetuity has a present value of 159,045. b. An annuity due with 20 annual payments. The first payment is 10,000. The second payment is 10,000(1.05). The third payment is 10,000(1.05) 2. Each payment thereafter is 1.05 times the prior payment. Calculate the present value of Option b. using an interest rate of i. For a: ,045 Set x=1/i i i x 1000x159, x (1000)( 159, 045) x i i x For b at i=0.0825: PV 10, 000 (10, 000)(1.05) v... (10, 000)(1.05) v , , 000(1.05) v 10, , 000(1.05) (1.0825) 1 v(1.05) 1 (1.05) / (1.0825) ,

8 8. A bag of M&Ms costs Jeff has enough money to purchase 75 bags of M&Ms. However, Jeff instead decides to invest his money at an interest rate of 10%. At the end of 3 years, Jeff can purchase 80 bags of M&Ms. Calculate the annual rate of inflation over the three year period. At time 0: 75(150) Money that Jeff has. After 3 years: (1.1) Money that Jeff has. $ Cost per bag. 80bags Inflation: (1 j ) 1 j j j %

9 9. You are given that: a. vt () 1 t b Calculate the effective interest rate in the 10 th year. 1 1 v( t) a( t) t a( t) v( t) a'( t) a(0) 1 (0) 1 a'( t) 2 10 at ( ) 1 (10) i 10 a(10) a(9) (1 10(.08)) (1 9(.08)) a(9) (1 9(.08)) 1.72

10 10. Yuhu invests 10,000 in Fund A at the beginning of each year for 10 years. The fund earns an annual effective interest rate of 7.5%. At the end of each year, the interest earned in Fund A is removed from Fund A and invested in Fund B. Fund B earns an annual effective interest rate of 5%. Determine the amount that Yuhu will have at the end of 10 years. Fund A: 10,000(10) 100,000 Fund B: Year 1= 10,000(0.075) 750 Year 2= (10,000 10,000)(0.075) 1500 and so on until Year 10= 10,000(10)(0.075) 7500 P 750 Q 750 n 10 i 0.05 Q n AV Pa ( a nv ) (1 i) n n i n (10)(1.05) (1.05) ) ( ) 15000( ) ( ) 48, Total: A B100,000 48, ,101.81

11 11. Tanner borrows 25,000 to buy a new car. The loan has a nominal interest rate of 12% compounded monthly. The loan will be repaid with 48 level monthly payments of Q, but the payments will be deferred with the first payment at the end of 4 months from the date of the loan. Tanner makes all payments as scheduled. Right after the 24 th payment, Tanner sells her car and pays off the loan by paying the outstanding loan balance. Determine the amount that Tanner must pay to pay off the loan. i(12) , 000 Qa (1 i) , 000 Q (1.01) Q OLB24 Qa Qa OLB 24 14,

12 12. Matt invests 10,000 in a fund for nine years. During the first two years, Matt earns a force of interest of 8%. For the next three years, Matt earns an annual effective discount rate of 5%. For the last four years, Matt earns a nominal interest rate of 4% compounded quarterly. Determine the annual effective interest rate that Matt earned over the nine year period (2) 3 10, 000(1 i) 10, 000( e )(1 0.05) 1 4(4) (1 i) ( e )(0.95) (1.01) (1 i) ( )( )( ) 9 (1 i) (1 i) ( ) i %

13 13. Emily invests 25,000 in a Fund earning an effective annual interest rate of i. Based on the Rule of 72, Emily expects to have 50,000 at the end of 8 years. Stephanie borrows money to purchase a car. The loan requires Stephanie to make monthly payments of 600 for five years. The annual effective interest rate on Stephanie s loan is i. Calculate the amount of Stephanie s loan i 9% i 8 (12) 12 (12) (12) 1/12 i i i (12) 12( n) i ( ) L P L i (12) 600 (12) L 600( ) 29,