Topics Covered. Compounding and Discounting Single Sums. Ch. 4 - The Time Value of Money. The Time Value of Money

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1 Ch. 4 - The Time Value of Money Topics Covered Future Values Present Values Multiple Cash Flows Perpetuities and Annuities Effective Annual Interest Rate For now, we will omit the section 4.5 on inflation and time value of money. We will visit this topic later in chapter 10. Omit problems and The Time Value of Money Compounding and Discounting Single Sums

2 The greatest mathematical discovery of all time is compound interest. Albert Einstein Future Values Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment. Example: Simple vs. Compound Interest Compare $100 invested at 10% simple annual interest for 3 years vs. 10% interest compound annually.

3 Compound Interest Future Value of Single Cash Flow FV = PV ( 1+ r) t Future Values FV = $100 ( 1+ ) r t Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for three years? FV = $100 (1+.06) 3 = $119.10

4 Future Values of $100 with Compounding FV of $ Interest Rates 0% 5% 10% 15% Number of Years Example: Taxes and Retirement Savings Both Ned Flanders and Homer Simpson have $4,000 they want to save today for 30 years toward their future retirement in the same investment fund expected to earn 8% compounded annually. Ned sets up an IRA account for his savings where his investment earnings build up tax free. Homer doesn t invest his money through an IRA and will see his annual investment earnings taxed at 15%. What is the difference in Ned and Homer s expected future retirement savings?

5 Ned s expected future retirement savings Homer s expected future retirement savings Present Values Present Value Value today of a future cash flow. Discount Factor Present value of a $1 future payment. Discount Rate Interest rate used to compute present values of future cash flows.

6 Present Values Present Value = PV PV = Future Value after t periods (1+r) t Example: Paying for Baby s MBA Just had a baby. You think the baby will take after you and earn academic scholarships to attend college to earn a Bachelor s degree. However, you want send your baby to a topnotch 2-year MBA program when baby is 25. You have estimated the future cost of the MBA at $95,000 for year 1 and $100,000 for year 2. Example: Paying for Baby s MBA Today, you want to finance both years of baby s MBA program with one payment (deposit) into an account paying 7% interest compounded annually. How large must this deposit be?

7 Time Value of Money (applications) Value of Free Credit Implied Interest Rates Internal Rate of Return Time necessary to accumulate funds

8 Example : Finding Rate of Return or Interest Rate A broker offers you an investment (a zero coupon bond) that pays you $1,000 five years from now for the cost of $740 today. What is your annual rate of return? Important Time Value Relationships Increasing interest rate and time increases future value. POSITIVE RELATIONSHIP. Increasing interest rate and time decreases present value. INVERSE RELATIONSHIP.

9 The Time Value of Money Compounding and Discounting Cash Flow Streams Annuities Annuity: a sequence of equal cash flows, occurring at the end of each period. This is known as an ordinary annuity PV FV Examples of Ordinary Annuities: If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. If you borrow money to buy a house or a car, you will pay a stream of equal payments.

10 Annuity-due A sequence of periodic cash flows occurring at the beginning of each period PV FV Examples of Annuities-due Monthly Rent payments: due at the beginning of each month. Car lease payments. Cable TV and most internet service bills. Annuities PV of Ordinary Annuity Formula [ ] 1 1 ( 1+ ) PV = C r r r t C = cash payment r = interest rate t = Number of years cash payment is received

11 Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. 1 1 [ ( 1+ ) ] PVAF = r r r t Annuities Applications Value of payments Implied interest rate for an annuity Calculation of periodic payments Mortgage payment Annual income from an investment payout Future Value of annual payments [ ] FV = C PVAF ( 1+ r) t Annuities PV (and FV) of Annuity-dues = PV (or FV) of ordinary annuity x (1 + r) or BGN mode on financial calculator. [ ] 1 (1 ) PV = C 1 r r t + r(1+ r) C = cash payment r = interest rate t = Number of years cash payment is received

12 Example: Invest Early in an IRA How much would you have at age 65 if you deposit $2,500 at the end of each year in an account paying 9% annually starting at: (A) age 42? (B) age 22?

13 A Friendly Car Dealer? A car dealership offers a 5-year car loan of $15,000 on new car at an annual interest rate of 7% with no payments for 3 years. In 3 years, the first of 3 equal annual payments will be made to repay the loan. What is this annual loan payment? Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. PV of Perpetuity Formula C PV = r C = cash payment r = interest rate

14 Perpetuities & Annuities Example - Perpetuity You want to create an endowment to fund a football scholarship, which pays $15,000 per year, forever, how much money must be set aside today if the rate of interest is 5%? PV = 15,000 PV = $300, = $300, , A-Rod s retirement plans At age 45 when the deferred payments from his current contract ends, all-star shortstop Alex Rodriguez plans to have $175 million in savings from his baseball playing days. He wants two things from his savings: a perpetuity of $6 million per year and a sum of money at age 55 in order to purchase majority ownership in his native Miami s Florida Marlins. How large of an offer can he make for the Marlins at age 55 based on this information and assuming his savings can earn 8% annually after age 45?

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