Ch06(3).pdf

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION (FORMULAS)  ANNUITIES AND PERPETUITIES DEFINED Annuity – finite series of equal payments that occur at regular intervals § If the first payment occurs at the end of the period, it is called an ordinary annuity. § If the...

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION (FORMULAS)  ANNUITIES AND PERPETUITIES DEFINED Annuity – finite series of equal payments that occur at regular intervals § If the first payment occurs at the end of the period, it is called an ordinary annuity. § If the first payment occurs at the beginning of the period, it is called an annuity due. Perpetuity – infinite series of equal payments 6F-2  ANNUITIES AND PERPETUITIES – BASIC FORMULAS Perpetuity: PV = C / r Annuities: é 1 ù ê1 - (1 + r ) t ú PV = C ê ú ê r ú ê ë ú û é (1 + r ) t - 1 ù FV = C ê ú ë r û 6F-3  ANNUITY – EXAMPLE 6.5 After carefully going over your budget, you have determined you can afford to pay $632 per month toward a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow? To determine how much you can borrow, we need to calculate the present value of $632 per month for 48 months at 1 percent per month. 6F-4  ANNUITY – EXAMPLE 6.5 (CTD.) You borrow money TODAY so you need to compute the present value. § 48 N; 1 I/Y; -632 PMT; CPT PV = 23,999.54 ($24,000) Formula: é 1 ù 1 - ê (1.01) 48 ú PV = 632 ê ú = 23,999.54 ê.01 ú êë úû 6F-5  FINDING THE PAYMENT Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 =.66667% per month). If you take a 4 year loan, what is your monthly payment? § 20,000 = C[1 – 1 / 1.006666748] /.0066667 § C = 488.26 6F-6  FINDING THE RATE Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? § Sign convention matters!!! § 60 N § 10,000 PV § -207.58 PMT § CPT I/Y =.75% 6F-7 FUTURE VALUES FOR ANNUITIES Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years? § FV = 2,000(1.07540 – 1)/.075 = 454,513.04 6F-8 ANNUITY DUE TIMELINE 0 1 2 3 10000 10000 10000 32,464 35,016.12 6F-9  ANNUITY DUE ! # 𝑃𝑉 = 𝐶 + " ×(1 − #$" !"#) 6F-10  ANNUITY DUE You are saving for a new house, and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of year 3? § FV = [10,000 + 10000/0.08*(1-1/ 1.083)]* 1.083 = 45061.12 6F-11  GROWING PERPETUITY A growing stream of cash flows that lasts forever C C ´ (1 + g ) C ´ (1 + g ) 2 PV = + + + (1 + r ) (1 + r ) 2 (1 + r ) 3 C PV = r-g 6F-12 GROWING PERPETUITY: EXAMPLE The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? $ 1. 30 PV = = $ 26. 00. 10 -. 05 6F-13  GROWING ANNUITY A growing stream of cash flows with a fixed maturity t -1 C C ´ (1 + g ) C ´ (1 + g ) PV = + + + (1 + r ) (1 + r ) 2 (1 + r ) t C é æ (1 + g ) ö ù t PV = ê1 - çç ÷÷ ú r - g ê è (1 + r ) ø ú ë û 6F-14 GROWING ANNUITY: EXAMPLE A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent? $ 20 ,000 é æ 1.03 ö ù 40 PV = ê1 - ç ÷ ú = $ 265,121.57.10 -.03 êë è 1.10 ø úû 6F-15  ANNUAL PERCENTAGE RATE This is the annual rate that is quoted by law By definition APR = period rate times the number of periods per year. Consequently, to get the period rate we rearrange the APR equation: § Period rate = APR / number of periods per year You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate. 6F-16  EFFECTIVE ANNUAL RATE (EAR) This is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison. 6F-17  THINGS TO REMEMBER You ALWAYS need to make sure that the interest rate and the time period match. § If you are looking at annual periods, you need an annual rate. § If you are looking at monthly periods, you need a monthly rate. If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly. 6F-18  COMPUTING EARS – EXAMPLE Suppose you can earn 1% per month on $1 invested today. § What is the APR? 1(12) = 12% § How much are you effectively earning? FV = 1(1.01)12 = 1.1268 Rate = (1.1268 – 1) / 1 =.1268 = 12.68% Suppose if you put it in another account, you earn 3% per quarter. § What is the APR? 3(4) = 12% § How much are you effectively earning? FV = 1(1.03)4 = 1.1255 Rate = (1.1255 – 1) / 1 =.1255 = 12.55% 6F-19  EAR – FORMULA m é APR ù EAR = ê1 + ú - 1 ë m û Remember that the APR is the quoted rate, and mRemember is the number that of thecompounding periods APR is the quoted perand rate, year m is the number of compounding periods per year 6F-20  ANNUAL PERCENTAGE RATES (CONT'D) Table 5.1 Effective Annual Rates for a 6% APR with Different Compounding Periods With the same APR, EAR increases with the frequency of compounding. 6F-21  COMPUTING APRS FROM EARS If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: é APR = m (1 + EAR) 1 m -1 ù êë úû 6F-22  APR – EXAMPLE Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? [ APR = 12 (1 +. 12 ) 1 /1 2 ] - 1 =. 113 8655152 or 11.39% 6F-23 AMORTIZE LOANS WITH FIXED TOTAL PAYMENTS Computing Loan Payments Consider a $30,000 car loan with 60 equal monthly payments, computed using a 6.75% APR with monthly compounding. 6.75% APR with monthly compounding corresponds to a one-month discount rate of 6.75% / 12 = 0.5625%. P 30, 000 C = = = $590.50 1 æ 1 ö 1 æ 1 ö 1 - 1 - r çè (1 + r ) N ÷ø 0.005625 çè (1 + 0.005625) 60 ÷ø 6F-24  COMPUTING THE OUTSTANDING LOAN BALANCE Problem: Let’s say that you are now 3 months into a $30,000 car loan (at 6.75% APR, originally for 60 months) and you decide to sell the car. When you sell the car, you will need to pay whatever the remaining balance is on your car loan. After 3 months of payments, how much do you still owe on your car loan? 6F-25  COMPUTING THE OUTSTANDING LOAN BALANCE Execute: Using a financial calculator or Excel: Given: 57 0.5625 -590.50 0 Solve for: 28727.62 Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.005625,57,-590.50,0) 6F-26