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## Ch. 4 - The Time Value of Money

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**Topics Covered**• Future Values • Present Values • Multiple Cash Flows • Perpetuities and Annuities • Effective Annual Interest Rate • Inflation & Time Value**The Time Value of Money**Compounding and Discounting Single Sums**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.**Future Values**Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Interest Earned Per Year = 100 x .06 = $ 6**Future Values**Example - Simple Interest Interest earned at a rate of 6% for three years on a principal balance of $100. Today Future Years 123 Interest Earned 6 6 6 Value 100 106112 118 Value at the end of Year 3 = $118**Future Values**Example - Compound Interest Interest earned at a rate of 6% for three years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x .06**Future Values**Example - Compound Interest Interest earned at a rate of 6% for three years on the previous year’s balance. Today Future Years 1 2 3 Interest Earned 6.00 6.36 6.74 Value 100 106.00 112.36 119.10 • Future Value of $100 compounded at 6% for three years = $119.10**Future Values**Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for three years?**Future Values with Compounding**Interest Rates**Example: Mutual Fund Fees and Retirement Savings**• Prof. Finance moves to a new university and has $100,000 in retirement savings to invest (rollover) into a new retirement account. • Prof. Finance wants to invest this money for 25 years into an indexed stock fund, which is expected to return 9% annually. • Prof. has two choices: Vanguard Total Equity Fund with a 0.4% annual expense fee and Onguard Total Fencing Fund with an 1.2% annual expense fee. • What is the difference in Prof. Finance’s expected future retirement savings between the two funds?**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.**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 top-notch 2-year MBA program when baby is 25. You have estimated the future cost of the MBA at $85,000 for year 1 and $89,000 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 8% interest compounded annually. • How large must this deposit be?**Time Value of Money(applications)**• Value of Free Credit • Implied Interest Rates • Internal Rate of Return • Time necessary to accumulate funds**Example : Finding Rate of Return or Interest Rate**• A broker offers you an investment (a zero coupon bond) that pays you $5,000 five years from now for the cost of $3,700 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 decreasespresent value. INVERSE RELATIONSHIP.**0**1 2 3 4 The Time Value of Money Compounding and Discounting Cash Flow Streams**Perpetuities**• Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.**Perpetuities**PV of Perpetuity Formula C = cash payment r = interest rate**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 in the rate of interest is 5%?**Perpetuities & Annuities**Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?**0**1 2 3 4 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.**Annuity-due**• A sequence of periodic cash flows occurring at the beginning of each period. 0 1 2 3 4 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.**Perpetuities & Annuities**PV of Ordinary Annuity Formula C = cash payment r = interest rate t = Number of years cash payment is received**Perpetuities & Annuities**PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.**Perpetuities & 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**Perpetuities & Annuities**PV (and FV) of Annuity-dues = PV (or FV) of ordinary annuity x (1 + r) or BGN mode on financial calculator. C = cash payment r = interest rate t = Number of years cash payment is received**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 41? • (B) age 22?**Why an IRA?**• Imagine in the last example, you didn’t take advantage of the tax-sheltered environment of an IRA. • Your annual investment return would be taxed! • With a 28% tax rate, our annual after-tax return would fall from 9% to 6.48% (=9%(1-.28)). • At age 65: I would have $135,519 vs. $191,975. • You would have $535,392 vs. $1,102,114: 52% less!! The IRS killed Kenny,…!**Example: Enjoying your Retirement**• You go ahead and make the contributions starting at age 22 in the last example, giving you $1,102,114 at age 65. • You expect to live to age 85. So, you want to make 20 annual withdrawals from your IRA paying 9% at the beginning of each year starting at age 65. • How large can this annual withdrawal be?**More annuity fun, enjoying your release from baseball**• Bob B. is released from the last year of his guaranteed contract from a New York baseball team. He is due $5.9 million from the last year of this contract. Bob and the team agree to defer the $5.9 million at 8% interest for 15 years. At this time (15 years from today), the team will begin the first of 15 equal annual payments at 8% interest. • The press is reporting the payments will total $30 million. Are they correct?**Non-Annual Interest Compounding**• When interest is compounded more frequently than once a year. • Important non-annual compounding terms and things to know: • Quoted Annual Rate, or Annual Percentage Rate (APR): Stated nominal annual rate before compounding. • Effective Annual Rate (EAR): the actual (effective) annual interest rate earned or paid. • Periodic Interest Rate: the interest rate paid or charged each interest compounding period = quoted rate/m, where m = number of compounding periods per year.**Effective Interest Rates**example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?**FV and PV with non-annual interest compounding**• n = number of years • m = number of times interest is paid per year • APR = nominal annual rate (APR) • APR/m = periodic rate Single CF FVnm = PV(1+ARR/m)nm PV = FVnm/(1+APR/m)nm**Non-annual annuities**Ordinary: • PV= C(PVAFAPR/m,nm) • FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm Annuity-Due: • PV= C(PVAFAPR/m,nm)(1+APR/m) • FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm+1**Example: Low Rate or Rebate?**• The Frontier family want to buy a sport ut (SUV). They decide on a 4-wheel drive Jeep Grand Cherokee. The purchase price with tax of the vehicle is $32,500. The Frontiers have $4,000 as a down payment. • Jeep offers the choice of two incentives on the 4-door Grand Cherokee. • 0% APR Financing for 60 months, or • $3,000 rebate which would be applied toward the purchase price. If the Frontiers elect to take the rebate, they can get 4.49% APR financing for 60 months. • Question: Which incentive would give the Frontiers the lowest monthly payment?**Example: The $200 national ISP signup credit: good deal for**whom? • A national ISP all provide $200 for new customers to use at a particular electronics store chain if they sign-up for a 2-year internet service contract at $21.95/month. • What interest rate (APR) are you paying on this “free money” if you wanted internet service and could get it for free? (200 PV, -21.95 PMT, 24 N 0 FV, CPT I/Y = 9.8%/month x 12= 117.8% APR!!) • What interest rate (APR) are you paying on this “free money” if you wanted internet service and could get it for $9.95/month?(-12 PMT CPT I/Y = 3.15%/mo x 12 = 37.8%! Thanks, but no thanks!**Inflation**Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases.**Inflation**approximation formula**Inflation**Example If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? Savings Bond**Example: Real retirement income**• Going back to your retirement in 43 years, you expect 3% inflation along with your 9% nominal investment rate annually and want to withdraw $32,000 in real terms at the beginning of each year for 20 years once you retire. • How will this change your retirement saving plans?