# Solved problems in financial mathematics

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1. The bank charges 50 rubles of ordinary simple interest for the use of 3,000 rubles within 60 days. What is the rate of simple interest in such a transaction? 7

2. A promissory note with a maturity of 100,000 rubles was sold at a simple discount rate of 3.5% 72 days before the maturity date. Find discount and revenue. 7

3. At what annual rate of compound interest does money double in 12 years? 7

4. What amount, when paid after 3 years, is equivalent to 10 thousand rubles paid after 10 years from now, if the rate of interest is 5% per year? eight

5. What quarterly contributions should be made to a bank that accrues 1.5% per quarter in order to save 500 thousand rubles in 5 years? eight

6. Ivanov deposits 500 rubles in a savings bank at the end of each quarter. At the end of each year, the bank charges 4% compound interest. What amount will be in Ivanov’s account in 5 years? nine

7. How much money do you need to have in your account to ensure an eternal annuity of 1,500 rubles per month if the bank charges 3% per quarter? nine

8. A bond for 100 thousand rubles, which pays 5% per annum, will be redeemed in 15 years at face value. At what price should it be bought to provide the buyer with a rate of return of 3% per annum?. nine

9. Calculate what is more profitable for the investor: to receive 20,000 rubles today or to receive 35,000 rubles in 3 years if the interest rate is 17%. ten

10. How many years will it take to get 20,000 rubles out of 1000 rubles deposited in the bank if the interest rate is 14% per annum?. ten

11. What should be the loan interest rate in order to increase 10,000 rubles to 30,000 rubles for a deposit term of 5 years? ten

12. Capital amounting to 4000 monetary units (CU) is invested in the bank for 80 days at 5% per annum. What will be its final value. eleven

13. For how many years you need to invest capital at 9% per annum, so that the interest payment is equal to its double amount. eleven

15. A promissory note with a par value of CU20,000. with a maturity date of 03.11.05. taken into account on 03.08.05 at 8% per annum. Find the discount and discount the value of the bill. 12

16. Let CU 20,000 be invested in the bank. at 10% (d) per annum. Find the final amount of capital if the billing period is: a) 3 months; b) 1 month. thirteen

17. Nominal annual rate – 30%. Find the equalizing interest rate when calculating compound interest every 3 months. thirteen

18. Every three months, CU 500 is invested in the bank. What will be the total amount of these deposits at the end of the 10th year with an interest rate of 8% and annual capitalization. thirteen

19. How much will the annual contributions of CU 2,000 increase? within 4 years at 8% per annum, if capitalization is made every three months and the first contribution is made at the end of the first year. fourteen

20. One of the bank deposits accumulated CU 200,000 over 20 years. Find the amount initially deposited in the account if the annual interest rate (d) is 8%. fourteen

21. Let the first deposit in the bank be CU 2000, and each subsequent deposit is reduced by CU 100. in relation to the previous one. Find the amount of deposits at the end of the 10th year, if they are made annually, postnumerando, the interest rate is 4% per annum, the capitalization is annual. fifteen

22. Find the present value of the sum of 10 postnumerando deposits of CU 5000. at 8% per annum, if capitalization is carried out every six months. fifteen

23. Let the amount of the loan be CU20,000. Depreciation is carried out by the same annuities for 10 years at 2% per annum. Find the amount of debt payment for the second and third years, if interest is capitalized annually. fifteen

24. Determine the simple interest rate at which the initial capital in the amount of 10,000 rubles. will reach in 180 days the amount of 19,000 rubles. sixteen

25. Credit in the amount of 15,000 rubles. issued from 26.03 to 18.10 at a simple 24% per annum. Determine the amount of debt for various options for calculating interest. 17

26. The Bank announced the following conditions for issuing a loan for a year: for the first quarter, the loan interest is 24%, and in each subsequent quarter, the interest rate on the loan increases by 3%. Determine the amount to be returned to the bank if the loan is issued for a year and amounts to 15,000 rubles (simple interest) 17

27. The deposit agreement is concluded for a period of 2 years and provides for the accrual and capitalization of interest for six months. The deposit amount is 15,000 rubles, the annual rate is 16%. Calculate the amount on the client’s account by the end of the term. eighteen

28. The owner of a bill with a nominal value of 19,000 rubles. and with a circulation period of 1 year presented it to the issuing bank for accounting 60 days before payment. The bank accounted for it at a rate of 60% per annum. Determine the discounted value, that is, the amount received by the owner of the bill, and the amount of the discount. nineteen

29. Determine the value of the bank’s annual discount rate, which is equivalent to the simple interest rate of 24% per annum (n = 1 year). nineteen

30. Interest is accrued quarterly on deposits at a nominal annual rate of 16%. Determine the amount of the deposit for accumulation in 1.5 years of the amount of 19,000 rubles. 20

31. The Bank offers long-term loans at 24% per annum with quarterly interest, 26% per annum with semi-annual interest and 20% per annum with monthly interest. Determine the most profitable loan option for the bank. 20

32. The bank issues a loan at 24% per annum. The semi-annual inflation rate was 3%. Determine the real annual interest rate adjusted for inflation. 21

33. What interest rate on deposits should be assigned so that the real return on the deposit, taking into account inflation of 3%, is 10% per annum? 21

34. Calculate the inflation rate for the year with a monthly inflation rate of 3%. 22

35. Deposit 15,000 rubles. deposited in the bank for six months with monthly compound interest at a nominal rate of 72% per annum. Determine the real income of the depositor if the expected monthly inflation rate is 3%. 22

36. The property lease agreement is concluded for 5 years. Rent is paid in amounts S 1 u003d 19,000 rubles, S 2 u003d 20,000 rubles, S 3 u003d 21,000 rubles. at the end of the 1st, 3rd and 5th years. According to the new payment schedule, two amounts are paid: S 4 u003d 22,000 rubles. at the end of year 2 and S 5 at the end of year 4. The bank interest rate is 5%. Determine S 5 . 23

37. Determine the amount of annual payments at a compound rate of 5% per annum to create a fund in 6 years in the amount of 19,000,000 rubles. 24

38. Calculate the amount of the fund that can be formed in 2 years by depositing 19,000 rubles at the end of each year. Interest on deposits is charged at a rate of 5%. 24

39. The monthly rent for an apartment is 1,800 rubles. The due date is the beginning of the month. Calculate the amount of the equivalent payment charged for the year ahead. The bank deposit rate is 48% per annum. 25

40. A two-year bond with a face value of 1,000 rubles. has 4 semi-annual coupons with a yield of 20% per annum each. Calculate the price of its initial placement, assuming a comparison rate of 16%. 25

41. A zero coupon bond was bought at an auction at a rate of 40 and sold at a rate of 58 after 90 days. Calculate the return on investment using compound and simple interest. 26

42. Provide an amortization plan for a 5-year loan of 1,500,000 rubles, repaid: in equal amounts; equal fixed payments. The interest rate on the loan is 5%. 26

43. A capital of CU 4,000 is invested in a bank for 80 days at 5% per annum. What will be its final value. 28

44. For how many years is it necessary to invest capital at 9% per annum so that the interest payment is equal to its double amount. 28

45. Find the present value of the sum of 10 postnumerando contributions of 5000 CU. at 8% per annum, if capitalization is carried out every six months. 28

46. Let the loan amount be CU20,000. Depreciation is carried out by the same annuities for 10 years at 2% per annum. Find the amount of debt payment for the second and third years, if interest is capitalized annually. 29

1. The bank charges 50 rubles of ordinary simple interest for the use of 3,000 rubles within 60 days. What is the rate of simple interest in such a transaction?

Decision:

Simple interest is calculated using the formula:

R = iP * (t/T);

50 = i 3000* (60/365);

i = 365*50 /(3000*60) = 0.1014 (10.14%)

or:

S = P(1+i); (50+ 3000) = 3000 (1+i); 3050 = 3000 + 3000i; 50/3000 = i; i = 0.0167 (1.67%) – for 60 days (two months); for the year: i = 0.0167*365/60 = 0.101388 (10.14%);

2. A promissory note with a maturity of 100,000 rubles was sold at a simple discount rate of 3.5% 72 days before the maturity date. Find discount and revenue.

Decision:

In the case of a simple discount:

P = S (1 – nd);

Revenue:

P u003d 100000 (1 – 0.035 * 72/365) u003d 100000 * 0.993 u003d 99300 rubles.

The discount will be:

100000 – 99300 u003d 700 rubles.