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Additional Solved Problems

Additional Solved Problems. Lump Sum Future Value. The Problem. You've received a $40,000 legal settlement. Your great-uncle recommends investing it for retirement in 27-years by “rolling over” one-year certificates of deposit (CDs) Your local bank has 3% 1-year CDs

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Additional Solved Problems

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  1. Additional Solved Problems Lump Sum Future Value

  2. The Problem • You've received a $40,000 legal settlement. Your great-uncle recommends investing it for retirement in 27-years by “rolling over” one-year certificates of deposit (CDs) • Your local bank has 3% 1-year CDs • How much will your investment be worth? • Comment.

  3. Categorization • Your capital gains will be reinvested. There is no cash-flow from the settlement for 27 years, so this is a lump sum problem. • There is some uncertainty in the cash flows because interest rate are static for just the first year, but we assume that it will be 3% until you retire • If you are unable to shelter your earnings, the IRS will want their cut

  4. Data Extraction • PV = $40,000 • i = 3% (or 3% * (1- marginal tax rate)?) • n = 27-years • FV = ?

  5. Solution by Equation

  6. Calculator Solution

  7. Comments • Your great uncle's a financial idiot • Given a 27-year investment, you should either • Invest the money more aggressively to accumulate the money you need to survive, or • Live! Blow the money on that red convertible!

  8. 3 Additional Solved Problems Lump Sum Interest Rate

  9. Problem 1 • If you have five years to increase your money from $3,287 to $4,583, at what interest rate should you invest?

  10. Algebraic Solution

  11. Problem 2 • An investment you made 12-years ago is today worth its purchase price. It has never paid a dividend. • Closer inspection reveals that the share price has been highly periodic, moving from $150 when purchased, to $300 in the next year, to $75 in the next, back to $150, before repeating

  12. 12-Year and Average Returns Compare with Average HPR

  13. Comments • Here we have the average holding period return being 41.67% per year, while the security has returned you nothing over the whole period! • Averages seduce us with their intuitiveness • The correct average to have used was the geometric average of return factors, not the arithmetic average of return rates

  14. Averages Must be Meaningful 1 • You walk 1 mile at 2 mph and another at 3 mph. What was your average speed? (2+3)/2 = 2.5 mph. • NO! • The first leg lasts 1/2 hour, and the second leg lasts 1/3 hours, total 5/6 hours. • So average speed is 2/(5/6) = 2.4 mph.

  15. Averages Must be Meaningful 2 • A little analysis shows that the correct mean for the walker is the harmonic mean • The correct mean for the return problem may be shown to be the geometric mean of the (1+return)’s • The appropriate mean requires thought

  16. Problem 3 • In 1066 the First Duke of Oxbridge was awarded a square mile of London for his services in assisting the conquest the England. The 30th Duke wished to live a faster paced life, and sold his holding in 1966 for £5,000,000,000. Examination of original project’s cost showed only the entry “1066 a.d.: to repair armor, £5” • What was rate of capital appreciation ?

  17. Categorization • We may assume that the Dukes lived quite well from leasing land to their tenants, but we are not interested in the revenue cash flows here, just the capital cash flows • There is a present cash flow, a future cash flow, and no annuity payments, so the problem is the return on a lump-sum invested for a number of periods

  18. Data Extraction • PV = 10 • FV = 5,000,000,000 • n = (1966 - 1066) = 1900 • i = ?

  19. Solution by Equation

  20. Solution by Calculator

  21. Comments • Note that a capital gain of only 1.1% per year results in a huge value over time • Time plus return is very potent • The real issue here is what is missing, namely the revenue streams

  22. Additional Solved Problems Lump Sum Number of periods

  23. The Problem • How many years would it take for an investment of $9,284 to grow to $22,450 if the interest rate is 7% p.a. ? • p.a. = per annum = per year

  24. Categorization • This is a lump sum problem asking for a solution in terms of time. Most of these problems are useful models of reality if expressed in real terms, not nominal terms • In any nominal situation, the terminal $22,450 will not be a constant, but will depend on the unknown time • We will assume that the numbers and rates are in real terms

  25. Data Extraction • PV = $9,284 • FV = $22,450 • i = 7% p.a. • n = ?

  26. Solution by Equation

  27. Additional Solved Problems Lump Sum Present Value

  28. The Problem • If investment rates are 1% per month, and you have an investment that will produce $6,000 one hundred months from now, how much is your investment worth today?

  29. Categorization • This is the most basic of financial situations, and involves finding the present value of a future payment given no periodic payments • The issue of risk is a little fuzzy. It is assumed that the rate given is for the project’s risk category

  30. Data Extraction • FV = $6000 • PV = ? • n = 100 months • i = 1%

  31. Solution by Equation

  32. Calculator Solution

  33. Additional Solved Problems Lump Sum Special Case: Doubling Rule of 72

  34. The Problem • Consider the following simple example: • Sol Cooper Investments have offered you a deal. Invest with them and they will double your investment in 10 years. What interest rate are they offering you? • We could solve this using • but this is over-kill

  35. Data Extraction • Doubling • n = 10 • i = ?

  36. Some Algebra

  37. Solution by Equation

  38. The Secret Reveled • Now you have seen the derivation of the rule of 72, you are now able to produce your own personal rules. Example: • “The Rule of a Magnitude” To increase your wealth by 10 times, the product of interest and time is 240, that is about (2.08/2)*ln(10) Example, how long will it take to increase your money ten times, given interest rates of 10%? N = 240/10 = 24 years, real answer is 24.16 years

  39. How good is the Rule of 72? • We have derived a rule using approximation methods, but have no idea how accurate it is • There are two tests we could apply • we could take some range, and determine the absolute maximum error of the rule in that range • we could simply graph the error • Graphs are fun:

  40. Graph of Rule of 72 Error • The high error in a part of the graph that does not interest us is hiding the error in the part that does. We have two choices • plot absolute error on a log scale • truncate the graph and re-scale • Truncation is fun

  41. Another Example • You are a stockbroker wishing to persuade a young client to reconsider her $50,000 invested in 3%-CDs. • Your client believes that stock mutual funds will return about 12% for the foreseeable future, but is averse to the volatility risks. Her money will remain fully invested for the next 48 years.

  42. Step 1 • The first step requires the calculation of how long is required to obtain a single doubling • CDs: 72/3 = 24 years to double • Mutual fund: 72/12 = 6 years to double

  43. Step 2 • The second step requires the calculation of how many doublings will occur during the lives of the investments • CDs: 48/24 = 2 doublings • Mutual fund: 48/6 = 8 doublings

  44. Step 3 • The third step calculates the value of the investment in 48 years • CDs: 2 doublings of $50,000 • = $200,000 • Mutual fund: 8 doubling of $50,000 • =256 * $50,000 • =$12,800,000 in 48 years

  45. Conclusion • We shall discover that her risk is smaller than she imagines, but she will be about 64 times more wealthy if she accepts that risk • Using the accurate method, her respective wealths are $206,613 and $11,519,539, • The lesson is to start to invest early, and accept some risk

  46. Growth at 3 and 12 % • The following graph shows her wealth increases over 10 years at a 3% and 12% • The graph was cut at 10 years because the 12% rate of growth is so large that it dwarfs the 3% growth, making the graph meaningless

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