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Additional Solved Problems

Solve various financial problems such as credit card interest rates, future value of annuity, and annuity rates. Understand compounding frequencies, real value of investments, and how different factors affect savings. Find detailed solutions and explanations.

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Additional Solved Problems

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  1. Additional Solved Problems Rate Conversions

  2. The Problem • Your credit card states that you are to pay 18% per year interest compounded monthly. • What is the annual rate compounded annually?

  3. Data Extraction • Macro Period = 1 year • Micro Period = 1 month • Micro Periods per Micro Period = 12 • Rate = 18% per year compounded monthly • Find Rate in terms of compounding annually

  4. Solution by Equation

  5. Continuous Compounding • If the bank was able to claim a lower rate than the true rate by quoting a nominal rate, can it do even better by compounding even more frequently? • The answer is yes. Let look at 19.56% in terms of compounding continuously

  6. Solution

  7. Comments • The three rates: 18% p.a. compounded monthly 19.56 p.a. compounded annually 17.86 p.a. compounded continuously are all equivalent • Note that there are 170 bases points between the highest and the lowest rates

  8. Additional Solved Problems Future Value of Annuity

  9. The Problem • How much will I have available in my retirement account if I deposit $2,000 per year into an IRA that pays on average 16% per year if I plan to retire in 40 years time? • If inflation is 3% per year, what is the real value of this amount? • Comment about this plan, given that I expect to live 20-years beyond retirement

  10. Categorization • Here we are talking about an a sequence of annual payments, so it is some kind of annuity • The problem is written loosely, and there is no indication of when the first cash flow occurs, right now at year-0, or at the end of this “year” in year-1. Assume at the end of the year at the beginning of year 1

  11. Categorization (Continued) • The evaluation point is not completely clear, but appears to be exactly 40-years from now • We are then looking at the future value of a regular annuity • We are searching for a future cash flow • The problem is to find the FV of a regular annuity

  12. Data Extraction • Step 1: • Pmt = $2,000 • I = 16% • n = 40 • FV = ? • Step 2: • PV of FV @ 3%

  13. Solution by Equation

  14. Calculator Solution

  15. Solution using Excel

  16. Solution using Excel

  17. Comments • Normally, you are not permitted to take a sum of money and distribute it over a number of years by simple division • e.g, you are not permitted to divide the FV by 20-years, your remaining life • The present value in this case is in real terms, and division is permitted. This gives $72,370.63 /year @ today’s spending power

  18. Additional Solved Problems Future Value Annuity Rate

  19. The Problem • Guaranteed! • Invest just $1000 per year Oddmann’s and we guarantee you to quadruple your investment in just 20 years! • Is this a good deal?

  20. Categorization • This is an annuity, because money is invested every year for a period of years • It is a future value of annuity, because the promise is to give money back to you at the end of the investment • In this case we solve for interest rate

  21. Data Extraction • FV = $1,000*20*4 = $80,000 • n = 20 • pmt = $1,000 • i = ?

  22. Solution by Equation Using the standard functions, and given general n, it is not possible to get consolidate both ‘i’s on the l.h.s.

  23. Solution by Calculator

  24. Comment • The prevailing interest rates will determine whether this return is high or low for its risk level • Remember that a guarantee is only as good as the guarantor, and a background check is necessary to determine whether Oddmann’s has the ability to fulfill their guarantee, given all future conditions

  25. Comments (Continued) • How liquid is your investment? The term indicates that the investment is for 20-years and can not be liquidated • If you stop paying after, say 5 years, as the contract may permit, and you still quadruple your money in the plan, is this a good deal? • This problem can’t be solved using n/I/PV/pmt/FV but is solved using your calculator’s IRR function, or Excel’s “Solve”

  26. IRR to solve Problem • The sequence is calculator dependent, but on the BAII+ is: • CF, 2nd CE/C, (CF0=0), down arrow, (C01 0), 1000, Enter, down arrow, (F01 1), 5, Enter, down arrow, (C01 0), down arrow, (F02 0), 14, enter, down arrow, (C03 0), 20000=/-, enter, down arrow, (F03 1) • Now hit IRR key, and then CPT • The solution is 8.45%

  27. Final Comment • The rate for investing over the whole period is 12.90%, but if you stop investing after 5-years the rate is 8.45% • Oddmann’s appears to have found a way to incorporate an implicit penalty for early withdrawal into their financial product

  28. Additional Solved Problems FV of an Annuity Solve for Periods

  29. Technical Issue • Some problems (other) textbooks are not realistic. They ask “How long it will take to save for …”, and assume that there is no final price inflation. An example is • How long will it take to save the $30,000 deposit on a new home if I can save $500 per month at a return of 0.7% per month?

  30. The Problem • Assume house inflation and salary inflation are about equal over time. How long will it take to save for the (real) $30,000 deposit for a home, given I can save a (real) $500 per month? A real return of 0.5% per month is available. (Inflation is irrelevant here)

  31. Categorization • This is an annuity evaluated at a future time • It is not clear whether it is regular or due, assume regular • We solve for the number of periods

  32. Data Extraction • Everything is is in terms of real spending power. • FV = 30000 • pmt = -500 • i = 0.5% (Watch this one, the interest factor is 1.005, not 1.05 nor 1.5. This is an easy error to make.) • n = ?

  33. Solution by Equation

  34. Calculator Solution

  35. Comments • The mathematics of finance works both on nominal cash flows and real cash flows, using nominal and real rates, respectively (Mixing nominal and real is not permitted) • Consider whether a nominal or real model matches the reality of the situation better in each case. The nominal case will occur more frequently, but isn’t always better

  36. Additional Solved Problems Future Value of an Annuity Find Payment

  37. The Problem • How much must you save each year in order to have saved $150,000 towards your retirement in 30 years? You believe that you can obtain a return of 9% on your investment over the whole period at an acceptable level of risk • Comment on your solution

  38. Categorization • Routine saving is an annuity, and we know what its future value is. We are not told of the starting date of the investment. Assume a regular annuity • We solve for annual payment

  39. Data Extraction • FV = 150,000 • i = 9% • n = 30 • pmt = ?

  40. Solution by Equation

  41. Calculator Solution

  42. Comments • A financial plan must be tailored to the needs of the individual. We need to take existing wealth, health, financial needs, current income, et cetera, into account • Given inflation, this plan is almost certainly inadequate for most individuals today

  43. Additional Solved Problems Special Category

  44. Why Growing Annuities? • I have all the tools necessary with regular annuities to solve growing annuity problems, so why the additional theory? • You had all that was required to solve annuities when you learned about lump-sums. Annuities gave you a tool that was faster to use, and corresponded with the real world • Growing annuities are computationally fast and safe to use, and reflect the real world

  45. The Problem • The last year’s cash-flow (just received) of the Diamond division of A&A Co was $1,000,000. A&A wishes to sell the division for strategic reasons • You are the CFO of the Diamond division, and you are planning a leveraged buyout

  46. The Problem (Continued) • You project that Diamond’s cash-flow will grow as follows: 14% from now to year 5, 10% from year 5+ until year 10, and 0% from year 10+ • The required rate of return for this risk is 11% • What is the value of Diamond?

  47. Notational Key • Denote • Starting period of i-th cash flow = ai • Ending period of i-th cash flow = bi • Growth in nominal cash flows between periods ai and ((bi +1) = ai+1) = gi • Cash flow at period s = Cs • Required rate of return on the project = r

  48. Table of Cash Flows

  49. Equation

  50. Excel Solution Cash Flows in Thousands of Dollars Yellow fields contain input data, and blue fields contain equations See Excel Notebook for Equations

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