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MBA 643 Managerial Finance Lecture 4: Time Value of Money

MBA 643 Managerial Finance Lecture 4: Time Value of Money. Spring 2006 Jim Hsieh. Goals. Calculate the amount of investment needed today to generate some (positive) value in the future Example: savings account Calculate the current value of cash flows expected (and known) in the future

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MBA 643 Managerial Finance Lecture 4: Time Value of Money

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  1. MBA 643Managerial FinanceLecture 4: Time Value of Money Spring 2006 Jim Hsieh

  2. Goals • Calculate the amount of investment needed today to generate some (positive) value in the future • Example: savings account • Calculate the current value of cash flows expected (and known) in the future • Example: bond prices Convention “Today” will always be donated time 0. Successive time points will be denoted time 1, time 2, …, etc.

  3. Future Values • Definition: The future value of a known current cash flow is obtained by compounding at a riskless rate or spot interest rate appropriate for the future period • FVt = C0(1+r)t • Example 1: How much wold $70,000 be worth in 14 years @7.5%? 0 1 2 C0 FV1=C0(1+r) FV2=FV1(1+r)=C0(1+r)2

  4. Present Values • Definition: The present value of a known future cash flow is obtained by discounting at a riskless rate or spot interest rate appropriate for the future period • PV = Ct/(1+r)t • Example 2: What is the maximum price you would pay today for a 2-year pure discount bond (also called a zero-coupon bond or Treasury strip) with interest rate r=0.08 and face value = $100? 0 1 2 PV1=C2/(1+r) C2 PV=PV1/(1+r)=C2/(1+r)2

  5. Additional Examples • Example 3: Suppose you need $10,000 in three years. If you earn 5% each year, how much money do you have to invest today to make sure that you have the $10,000 when you need it? • Example 4: What is the maximum price you’d be willing to pay for a promise to receive a $25,000 payment in 30 years? You can invest your money somewhere else with similar risk and make a 24% annual return.

  6. The Power of Compounding-- Longer compounding period with higher rates

  7. The Power of Compounding-- Compounding more often (Suppose you have $1,000 now, how much will you have after 1 year? r=0.1) • Annual Compounding: The interest is added to your investment once a year. • Semiannual Compounding: The interest is added to your investment twice a year. 0 1 1 0 2

  8. The Power of Compounding (cont’d)-- Compounding more often • Monthly Compounding: The interest is added to your investment 12 times a year. • Compounding n times: The interest is added to your investment n times a year. • A Generalized Formula: If you invest $PV in one account and the interest rate (r) is compounded n times per year, how much will you have after t years?

  9. The Power of Compounding (cont’d)-- Example 5 • Your bank representative gives you a quote of the interest rate in the savings account as 4% compounded semiannually. If you deposit $100 at the beginning of the year, how much will you have in your account after 2 years? After 10 years? What if the interest rate is compounded quarterly? Monthly?

  10. Quoted Interest Rates vs. Effective Annual Rates (EAR) • Example 6: Suppose you are trying to open a savings account and 3 banks quote you the following rates: Bank Annie: 15% compounded daily (365 days a year) Bank Booboo: 15.5% compounded quarterly Bank Charming: 16% compounded annually Which of these is the best? • Two different rates: • The quoted (stated) interest rate • The EAR: The interest rate expressed as if it were compounded once per year. If n is the number of times the interest is compounded per year, then : EAR = [1 + (Quoted rate/n)]n - 1

  11. Quoted Interest Rates vs. Effective Annual Rates (EAR) • Example 6: (cont’d) • Comments: • The highest quoted rate is not necessarily the best. • Compounding during the year can lead to a significant difference between the quoted rate and the effective rate. • The quoted rate is quoted by financial institutions, but the effective rate is what you really get or what you really pay.

  12. Annual Percentage Rate (APR) • Lenders are required by the Truth-in-lending laws in the U.S. to disclose an APR on almost all consumer loans. • Example 7: If your bank charges you 1.6% per month on your credit card, then the APR must be reported as 1.6%*12 = 19.2%. Thus, an APR is actually a quoted rate. To compare loans with different APRs, you still need to convert APRs to EARs. Remember: the EAR is the rate you actually pay. • Question: What is the EAR on your credit card?

  13. Multiple Cash Flows • We have a cash flow stream, C1, C2, …, CT, for T years • Rule: Discount (or compound) all cash flows to the present (or the future) and then add them up. PV0 = C1/(1+r) + C2/(1+r)2 + … + CT/(1+r)T = FVT = C1(1+r)T-1 + C2(1+r)T-2 + … + CT 0 1 2 … T … C1 C2 CT

  14. Investing for More than One Period:Present Values and Multiple Cash Flows • Example 8: Suppose your firm is trying to evaluate whether to buy an asset. The asset pays off $2,000 at the end of years 1 and 2, $4,000 at the end of year 3 and $5,000 at the end of year 4. Similar assets earn 6% per year. How much should your firm pay for this investment?

  15. Multiple Cash Flows and Future Value • Example 9: Suppose your rich uncle offers to help pay for your business school education by giving you $5,000 each year for the next three years beginning today (year = 0). You plan to deposit this money into an interest-bearing account so that you can attend business school six years from today. Assume you earn 4.25% per year on your account. How much will you have saved in six years (year=6)?

  16. Present Values and Multiple Cash Flows • The price of an asset is the present value of the CFs produced by the asset. • Ex. Stock  dividends; Bond  interests and principal • Two cash flows with the same PV are economically equivalent. • We always prefer the CF stream with the highest PV. • Example 10: Suppose r=0.1. Which investment would you take?

  17. Perpetuity: Multiple and Infinite Identical Cash Flows • PV(Perpetuity) = C/(1+r) + C/(1+r)2 + … = C/r • Example 11: Suppose Martin Co. wants to sell preferred stock at $60 per share and offers a dividend of $3 every quarter. What rate of return will be for Martin’s preferred stock? … … 0 1 2 T  … … C C C

  18. Annuity: Finite Stream of Identical Cash Flows • PV(Annuity) = C/(1+r) + C/(1+r)2 + … + C/(1+r)T • FV(Annuity) = PV(Annuity)*(1+r)T … 0 1 2 T … C C C

  19. Annuity – Example 12 (Car Loan Payments) • You want to buy a new sports coupe for $48,250, and the finance office at the dealership has quoted you a 9.8% APR loan for 60 months. What will your monthly payments be? What is the effective annual rate on this loan?

  20. Growing Perpetuity: Multiple and Infinite Cash Flows That Grows at a Fixed Rate (“g”) • PV(Grow. Perp.) = C/(1+r) + C(1+g)/(1+r)2 + … = C/(r-g) … … 0 1 2 T  … … C(1+g)T-1 C C(1+g)

  21. Growing Annuity: Multiple and Finite Cash Flows That Grows at a Fixed Rate (“g”) • PV(Grow. Annuity) = C/(1+r) + C(1+g)/(1+r)2 + … + C(1+g)T-1/(1+r)T … 0 1 2 T … C(1+g)T-1 C C(1+g)

  22. Multiple Growing Cash Flows – Example 13 • A wealthy GMU-grad entrepreneur wishes to endow a chair in finance at the SOM. The first payment is $50,000, occurring at the end of the first year. The amount grows at 3% afterward. The rate of interest is 10%. If she wants to provide an annual payment perpetually, what is the amount that must be set aside today?

  23. Example 13 (cont’d) • If she wants to provide an annual payment for the next 20 years only, what is the amount that must be set aside today?

  24. Additional Examples – Example 14 (Financing or Rebate?) Option 1: Rebate Option 2: 5% Financing SALE! SALE! 5%* FINANCING OR $500 REBATEFULLY LOADED MUSTANG only $10,999 *5% APR on 36 month loan. If United Bank is offering 10% car loans, should you choose the 5% financing or $500 rebate?

  25. Additional Examples – Example 15 (Mortgage) • You have just signed closing documents associated with the purchase of a house for $350,000, and have arranged a 30-year, fixed rate mortgage bank loan at a 7% stated (quoted) annual rate. Because you have made a 30% down payment, the loan amount is 70% of the purchase price. Mortgage payments will be made at the end of each month, and your first payment will be due exactly one month from today. What is your monthly mortgage payment?

  26. Additional Examples – Example 16 (Savings for retirement) • Your current salary is $60,000 per year, and is expected to grow by 5% per year until retirement. In 30 years (t=30) you plan to retire and hope that at t=30 to have amassed a $3 million retirement balance. If you invest part of your income each year in an account earning 9% per year, compounded annually, how much of your income must be invested to attain your retirement goal? Note that your first deposit will occur one year from today (t=1) and your last deposit occurs at t=30.

  27. Additional Examples – Example 17 (College planning) • Your child will start school 18 years from today (t=18). You have decided to start a college savings plan. You want to follow an aggressive investment strategy and plan to invest a lump sum at the end of each year for the next seventeen years (t=1~17) into the Trust Me Mutual Fund with a discount rate of 10%. At the end of the seventeenth year, you will deposit the money into a savings account that earns 4% annually. You will make tuition payments from this account. You estimate that tuition and room and board will cost $40,000 per year for four years. Assume the following: 1. The expected return on the mutual fund will be the same each year for the next seventeen years. 2. The first deposit to the mutual fund will be made one year from today (t=1). • The first tuition payment is made 18 years from today (t=18). Given your investment strategy, how much will you need to deposit in the Trust Me Mutual Fund each year?

  28. Example 17 (Cont’d) 0 1 2 … 17 18 19 20 21 … C C C P P P P r=0.1 r=0.04

  29. Exercise (“It pays to start early”) • Mei Xiang and Tian Tian have different investment strategies for their retirement. Mei deposits $4,000 in her IRA each year from t=21 to 41 (20 deposits) and keeps her savings in the account until t=60. Tian starts his investment ($4,000) at t=31 until t=60 (30 deposits). Assume both accounts earn 10% rate of return. How much will they have when they reach the age of 60?

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