Math review in financial case study course. Chapter 2 Quantitative/Analytical Problems

Kansas State University Module 2 - Week 2 - Comprehensive Math Review PFP 836 Case Studies in Personal Financial Planning Spring Semester 2016 The Case Approach to Financial Planning Chapter 2 Quantitative/Analytical Problems 1. Holding-Period Returns Last year Marla purchased 100 shares of stock for $8 per share. She paid a flat $75 to purchase the shares. Since making her purchase, she has received $200 in dividends. Marla is concerned that the stock price will fall below its current FMV of $7. Calculate her holding-period return if she sells today and pays a $75 commission. HINT: TWO transaction fees of $75 were paid. 2. Dollar-Weighted Returns Brad purchased 10 shares of stocks for $10 per share. He paid a $5 commission. One year later, he purchase another 10 shares at $9 per share. Again, he paid a $5 commission. In the second year, he got discouraged and sold 10 shares at $7 per share, paying another $5 commission. At the end of 3 years Brad liquidated his holdings, paying $5 in commissions, at $8 per share. What was his dollar-weighted rate of return? 3. Time-Weighted Returns Marybeth purchased one share of stock for $92. At the end of year 1, the stock was worth $95. At the end of year 2, the stock was worth $100; and at the end of year 3, the stock was worth $92. What was Marybeth's time-weighted return? How does this compare to her holding period rate of return? 4. Mean Returns Maurice purchased a coin collection several years ago. Each year he has had the collection appraised by a reputable coin dealer. Maurice has calculated the yearly percentage gain or loss based on the appraisal, as shown below: Year 1: 10% Year 2: 5% Year 3: 9% Year 4: -5% Year 5: 2% Year 6: -3% Year 7: 12% What is the average (mean) return of the coin collection over the seven-year period? Use the rate of return data provided by Maurice to calculate the geometric mean return. How does this compare to the mean return? 5. Weighted Average Returns a. Sherman is a novice investor. The following table shows his portfolio holdings and market values for each holding. Please calculate Sherman's weighted average return for his portfolio. Junk Bond Fund Gold Fund Small Cap Fund Index Fund CD TOTAL b. Market Value $ 13,000.00 $ 19,500.00 $ 36,000.00 $ 9,800.00 $ 41,000.00 Rate of Return 7.50% 12.00% 3.40% 6.20% 1.00% $ 119,300.00 Sonya and Josh own several pieces of expensive jewelry. The following table summarizes their most important gem assets: Asset Diamond Ring Sapphire Necklace Emerald Pendant Ruby Brooch TOTAL Market Value Purchase Price $ 12,000 $ 7,250 $ 23,000 $ 18,875 $ 19,500 $ 19,000 $ 21,200 $ 23,000 $ 75,700 $ 68,125 Use this information to calculate the holding-period return for all of their jewelry. 6. Real and Nominal Rates Consumer price increases have been averaging 3.50% per annum in South Korea. If SoHyun can earn a nominal rate of return on her investment portfolio of 10.25%, what is her real rate of return, assuming continuous compounding? 7. Tax-Exempt and Taxable Rates a. b. 8. Thomas Jones is in the 25% marginal federal tax bracket, the 4% marginal state tax bracket, and the 2% marginal city tax bracket. If he earns 9.25% on his portfolio, what is his after-tax rate of return? Stephanie is considering purchasing a fixed-income investment. She has narrowed her list of bond choice to two AAA-rated investments. The first is a corporate bond that matures in seven years. The bond yields 6.35%. The second bond also matures in seven years; however, this bond is a municipal bond issued by the state in which Stephanie resides. The bond has a current coupon rate at 4.79%. Stephanie is in the 25% marginal federal tax bracket and the 3.50% marginal state tax bracket. Which bond should she invest in to maximize her after-tax rate of return? Effective Rate Conversion Problems a. b. 9. A client was earning an EAR of 7%. The client was contributing to the account on a monthly basis. What would be the appropriate EPR? A client wants to compute the estimated future value of a bank account. The account pays a semiannual rate of 4% (EPR). The client is investing in the account on a biweekly basis. What is the appropriate EPR to be able to solve for an estimated future value for the client? Future Value Problems Lanisha currently earns $45,000 per year. She expects her salary to increase by 3.5% per year. She plans to work for another 20 years. How much will she earn in her final year of work? 10. Present Value Problems David and Iantha expect their 10-year-old daughter to get married someday. They estimate that her wedding would cost $30,000 today. Wedding costs will increase by 3% per year, and she will marry in 15 years. David and Iantha earn a pre-tax annual rate of 5%. They are in the 20% marginal tax bracket. What amount do they need to set aside today for their daughter's wedding? 11. Future Value of Annuity Problems Sue wants to save $1,000,000 in 20 years. She estimates she can earn 8.5% on savings. She intends to make deposits at the beginning of every year in a series of equal payments starting today. How much does she need to save each year to reach her goal? 12. Present Value of Annuity Problems Roshanna has just won the lottery. Her required rate of return is 6%. Should she take the annual annuity payment of $125,000 for the next 20 years or an immediate lumpsum payout of $1.5 million? 13. Future Value of Growing Annuity Problems Martha wants to start saving for college. She estimates that she will need $50,000 when she starts college four years from now. She plans to save $9,500 this year and increase deposits by 5% annually (payments at the end of each year). She can earn 7% on her savings. Will she meet her savings goal of $50,000 for college four years from now? 14. Present Value of Growing Annuity Problems Mike wants to receive $40,000 per year pre-tax from his investments for the next 30 years. He is concerned about inflation, which he expects to average 3%. He can earn a pre-tax rate of 6% on his money. How much does he need in an account to generate $40,000 in annual inflation-adjusted income (payments at the end of each year)? 15. Present Value of Delayed Annuity Problems Jonas desires fixed annual income of $85,000 beginning 20 years from now and lasting for 20 years. He plans to deplete the account. His annual required return is 9.5%. How much does he need to invest today to achieve his goal? 16. Present Value of Perpetuity Problems Jose Marie and his wife, Ayna, want to ensure that they do not outlive their money. They want to make end-of-year withdrawals that start at $80,000 after tax and increase by 4% per year. They can earn an annual effective rate of 7.2% after tax. How much do they need to have invested to make withdrawals that last forever? 17. Internal Rate of Return a. Ruth invested $16,000 in an exchange-traded fund six years ago. Dividends and earnings were automatically reinvested in new shares. Although Ruth considers herself to be a buy-and-hold investor, she nonetheless made a few trades during the time period, as follows: $3,025 redemption at the end of the second year $1,825 redemption at the end of the third year $4,200 additional investment into the fund at the end of the fifth year $19,885 received at the end of the sixth year, when Ruth redeemed the shares. What was Ruth's internal rate of return on the investment? 18. Net Present Value Kristy purchased shares of a mutual fund five years ago. She has since made the following additional transactions: End of year 2, invested additional $2,500 in the fund End of year 3, invested additional $9,000 in the fund End of year 5, redeemed all of her shares in the fund, receiving $16,000 Kristy just told you she achieved an annual return of 9% in this investment. If Kristy is correct, what was her initial investment in the fund? 19. Loan Amortization Problems a. Willy and Ursha have just purchased a new car for $38,000. They financed $35,000 for five years at 6.75% (APR). What is the monthly payment and total interest expense? b. Assume an initial loan amount of $150,000, an APR of 6.5%, fixed monthly payments, and an amortization period of 20 years. What is the outstanding balance on the loan after 60 payments have been paid? 20. Kristin and Dan Peterson wonder how much they will need to save to fully fund four years of college for their daughter, Samantha, who is 12 years old today. She plans to attend college at age 18. They know that their first choice of college currently costs $13,000 per year. College costs are increasing 5% annually. They feel that it is possible to earn an effective annual rate of 8.3% (8% compounded monthly), both before and during college. a. How much will Samantha's first year of college cost? b. How much do the Peterson's need to fully fund college costs when Samantha begins college? c. How much do Kristin and Dan need to deposit in an account today to fully fund four years of expenses? d. How much will they need to save on a monthly basis starting at the end of this month to fully fund four years of expenses? 21. Tony has been eyeing a new car. Last weekend, he went to the dealership and noted that his dream car would cost $23,000 if purchased today. Tony currently has $9,000 saved. He does not want to go into debt to buy the car, so he has decided to save toward the purchase for three years. Tony estimates that inflation will average 4.5% per year. He earns 7% (EAR) on his savings. a. How much will the car cost in exactly three years? b. How much must he save per year (at end of each year) to purchase the car in three years? Kansas State University Module 2 - Week 2 - Comprehensive Math Review PFP 836 Case Studies in Personal Financial Planning Spring Semester 2016 The Case Approach to Financial Planning Chapter 2 Quantitative/Analytical Problems 1. Holding-Period Returns Last year Marla purchased 100 shares of stock for $8 per share. She paid a flat $75 to purchase the shares. Since making her purchase, she has received $200 in dividends. Marla is concerned that the stock price will fall below its current FMV of $7. Calculate her holding-period return if she sells today and pays a $75 commission. HINT: TWO transaction fees of $75 were paid. 2. Dollar-Weighted Returns Brad purchased 10 shares of stocks for $10 per share. He paid a $5 commission. One year later, he purchase another 10 shares at $9 per share. Again, he paid a $5 commission. In the second year, he got discouraged and sold 10 shares at $7 per share, paying another $5 commission. At the end of 3 years Brad liquidated his holdings, paying $5 in commissions, at $8 per share. What was his dollar-weighted rate of return? 3. Time-Weighted Returns Marybeth purchased one share of stock for $92. At the end of year 1, the stock was worth $95. At the end of year 2, the stock was worth $100; and at the end of year 3, the stock was worth $92. What was Marybeth's time-weighted return? How does this compare to her holding period rate of return? 4. Mean Returns Maurice purchased a coin collection several years ago. Each year he has had the collection appraised by a reputable coin dealer. Maurice has calculated the yearly percentage gain or loss based on the appraisal, as shown below: Year 1: 10% Year 2: 5% Year 3: 9% Year 4: -5% Year 5: 2% Year 6: -3% Year 7: 12% What is the average (mean) return of the coin collection over the seven-year period? Use the rate of return data provided by Maurice to calculate the geometric mean return. How does this compare to the mean return? 5. Weighted Average Returns a. Sherman is a novice investor. The following table shows his portfolio holdings and market values for each holding. Please calculate Sherman's weighted average return for his portfolio. Junk Bond Fund Gold Fund Small Cap Fund Index Fund CD TOTAL b. Market Value $ 13,000.00 $ 19,500.00 $ 36,000.00 $ 9,800.00 $ 41,000.00 Rate of Return 7.50% 12.00% 3.40% 6.20% 1.00% $ 119,300.00 Sonya and Josh own several pieces of expensive jewelry. The following table summarizes their most important gem assets: Asset Diamond Ring Sapphire Necklace Emerald Pendant Ruby Brooch TOTAL Market Value Purchase Price $ 12,000 $ 7,250 $ 23,000 $ 18,875 $ 19,500 $ 19,000 $ 21,200 $ 23,000 $ 75,700 $ 68,125 Use this information to calculate the holding-period return for all of their jewelry. 6. Real and Nominal Rates Consumer price increases have been averaging 3.50% per annum in South Korea. If SoHyun can earn a nominal rate of return on her investment portfolio of 10.25%, what is her real rate of return, assuming continuous compounding? 7. Tax-Exempt and Taxable Rates a. b. 8. Thomas Jones is in the 25% marginal federal tax bracket, the 4% marginal state tax bracket, and the 2% marginal city tax bracket. If he earns 9.25% on his portfolio, what is his after-tax rate of return? Stephanie is considering purchasing a fixed-income investment. She has narrowed her list of bond choice to two AAA-rated investments. The first is a corporate bond that matures in seven years. The bond yields 6.35%. The second bond also matures in seven years; however, this bond is a municipal bond issued by the state in which Stephanie resides. The bond has a current coupon rate at 4.79%. Stephanie is in the 25% marginal federal tax bracket and the 3.50% marginal state tax bracket. Which bond should she invest in to maximize her after-tax rate of return? Effective Rate Conversion Problems a. b. 9. A client was earning an EAR of 7%. The client was contributing to the account on a monthly basis. What would be the appropriate EPR? A client wants to compute the estimated future value of a bank account. The account pays a semiannual rate of 4% (EPR). The client is investing in the account on a biweekly basis. What is the appropriate EPR to be able to solve for an estimated future value for the client? Future Value Problems Lanisha currently earns $45,000 per year. She expects her salary to increase by 3.5% per year. She plans to work for another 20 years. How much will she earn in her final year of work? 10. Present Value Problems David and Iantha expect their 10-year-old daughter to get married someday. They estimate that her wedding would cost $30,000 today. Wedding costs will increase by 3% per year, and she will marry in 15 years. David and Iantha earn a pre-tax annual rate of 5%. They are in the 20% marginal tax bracket. What amount do they need to set aside today for their daughter's wedding? 11. Future Value of Annuity Problems Sue wants to save $1,000,000 in 20 years. She estimates she can earn 8.5% on savings. She intends to make deposits at the beginning of every year in a series of equal payments starting today. How much does she need to save each year to reach her goal? 12. Present Value of Annuity Problems Roshanna has just won the lottery. Her required rate of return is 6%. Should she take the annual annuity payment of $125,000 for the next 20 years or an immediate lumpsum payout of $1.5 million? 13. Future Value of Growing Annuity Problems Martha wants to start saving for college. She estimates that she will need $50,000 when she starts college four years from now. She plans to save $9,500 this year and increase deposits by 5% annually (payments at the end of each year). She can earn 7% on her savings. Will she meet her savings goal of $50,000 for college four years from now? 14. Present Value of Growing Annuity Problems Mike wants to receive $40,000 per year pre-tax from his investments for the next 30 years. He is concerned about inflation, which he expects to average 3%. He can earn a pre-tax rate of 6% on his money. How much does he need in an account to generate $40,000 in annual inflation-adjusted income (payments at the end of each year)? 15. Present Value of Delayed Annuity Problems Jonas desires fixed annual income of $85,000 beginning 20 years from now and lasting for 20 years. He plans to deplete the account. His annual required return is 9.5%. How much does he need to invest today to achieve his goal? 16. Present Value of Perpetuity Problems Jose Marie and his wife, Ayna, want to ensure that they do not outlive their money. They want to make end-of-year withdrawals that start at $80,000 after tax and increase by 4% per year. They can earn an annual effective rate of 7.2% after tax. How much do they need to have invested to make withdrawals that last forever? 17. Internal Rate of Return a. Ruth invested $16,000 in an exchange-traded fund six years ago. Dividends and earnings were automatically reinvested in new shares. Although Ruth considers herself to be a buy-and-hold investor, she nonetheless made a few trades during the time period, as follows: $3,025 redemption at the end of the second year $1,825 redemption at the end of the third year $4,200 additional investment into the fund at the end of the fifth year $19,885 received at the end of the sixth year, when Ruth redeemed the shares. What was Ruth's internal rate of return on the investment? 18. Net Present Value Kristy purchased shares of a mutual fund five years ago. She has since made the following additional transactions: End of year 2, invested additional $2,500 in the fund End of year 3, invested additional $9,000 in the fund End of year 5, redeemed all of her shares in the fund, receiving $16,000 Kristy just told you she achieved an annual return of 9% in this investment. If Kristy is correct, what was her initial investment in the fund? 19. Loan Amortization Problems a. Willy and Ursha have just purchased a new car for $38,000. They financed $35,000 for five years at 6.75% (APR). What is the monthly payment and total interest expense? b. Assume an initial loan amount of $150,000, an APR of 6.5%, fixed monthly payments, and an amortization period of 20 years. What is the outstanding balance on the loan after 60 payments have been paid? 20. Kristin and Dan Peterson wonder how much they will need to save to fully fund four years of college for their daughter, Samantha, who is 12 years old today. She plans to attend college at age 18. They know that their first choice of college currently costs $13,000 per year. College costs are increasing 5% annually. They feel that it is possible to earn an effective annual rate of 8.3% (8% compounded monthly), both before and during college. a. How much will Samantha's first year of college cost? b. How much do the Peterson's need to fully fund college costs when Samantha begins college? c. How much do Kristin and Dan need to deposit in an account today to fully fund four years of expenses? d. How much will they need to save on a monthly basis starting at the end of this month to fully fund four years of expenses? 21. Tony has been eyeing a new car. Last weekend, he went to the dealership and noted that his dream car would cost $23,000 if purchased today. Tony currently has $9,000 saved. He does not want to go into debt to buy the car, so he has decided to save toward the purchase for three years. Tony estimates that inflation will average 4.5% per year. He earns 7% (EAR) on his savings. a. How much will the car cost in exactly three years? b. How much must he save per year (at end of each year) to purchase the car in three years