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Chapter 5 Discounted Cash Flow Valuation: Valuing Multiple CFs 1 Chapter Outline 1. FV and PV of Multiple Cash Flows – Using the calculator 2. Valuing Annuities and Perpetuities 3. Comparing Rates of Different Compounding Periods – Comparing “apples to apples” given the different ways rates are quoted (annual, semi-annual, monthly…) 4. Loan Types and Loan Amortization – Definitions of different finance contracts 2 5.1 Multiple CF’s • To find the FV of multiple CF’s: • Calculate the FV of each individual CF • Then ADD the individual CFs FVs together: Example: • Receive $100 at t = 0 and t = 1. • Calc value the Future Value at t = 2. • The first $100 increases twice. • The second $100 increases once. • $100(1.08)2 + $100(1.08) = $224.64 3 5.1 Multiple CF’s Figure 5.1: 4 Example 5.1 Page 117 • You currently have $7,000 in an account (at t = 0) • You will deposit $4,000 at the end of each of the next 3 years (at t = 1, t = 2 and t = 3) • How much will you have at time 3 at 8%? $7,000 at t = 0 with 3 years of interest $7,000(1.08)3 = $8,818 $4,000 at t = 1 with 2 years of interest $4,000(1.08)2 = $4,666 $4,000 at t = 2 with 1 year of interest $4,000(1.08)1 = $4,320 $4,000 at t = 3 $4,000 = $4,000 5 Example continued • Same Example, but now… • How much will you have at time 4 at 8%? $7,000 at t = 0 with 4 years of interest $7,000(1.08)4 = $9,523 $4,000 at t = 1 with 3 years of interest $4,000(1.08)3 = $5,039 $4,000 at t = 2 with 2 years of interest $4,000(1.08)2 = $4,666 $4,000 at t = 3 with 1 year of interest $4,000(1.08)1 = $4,320 6 Calculations: FV at t = 3: FV at t = 4: $7,000(1.08)3 = $8,818 $7,000(1.08)4 = $9,523 $4,000(1.08)2 = $4,666 $4,000(1.08)3 = $5,039 $4,000(1.08)1 = $4,320 $4,000(1.08)2 = $4,666 $4,000(1.08)0 = $4,000 $4,000(1.08)1 = $4,320 $21,804 $23,548 7 Clicker Question: • You currently have $600 in an account • You will deposit $1,000 at time 1and at time 2. • How much will you have at time 2 if your account earns 10%? A. $1,600 B. $2,600 C. $2,826 D. $3,600 E. $3,826 8 Clicker Answer: • The $600 in the account now (at time 0) will earn 10% for 2 years $600(1 + 0.10)2 = $726 • The first $1,000 (deposited at time 1) will earn 10% for 1 year $1,000(1 + 0.10) = $1,100 • The second $1,000 (deposited at time 2) will earn no interest $1,000 Sum = $726 + $1,100 + $1,000 = $2,826 The Answer is c 9 Another FV Example: $2,000 at the end of each year for 5 years Calculate FV at time 5 at 10% 10 Now Calculate the PV of Multiple CFs: You need $1,000 at t = 1 and $2,000 at t = 2 How much do you need to invest today if you earn 9%? Or what is the PV of these cash flows at 9%? 0 1 2 $917.43 $1,000 $1,683.36 $2,000 $2,600.79 $1,000/(1.09) + $2,000/(1.09)2 = $2,600.79 11 Think about the PV this way: • Invest $2,601 at 9%. • Show that you can withdraw $1,000 at t = 1 and withdraw $2,000 at t = 2: $2,600.79(1.09) = $2,834.86 (at t = 1) $2,834.86 - $1,000 = $1,834.86 (withdraw $1,000 at t = 1) $1,834.86(1.09)2 = $2,000 (available at t = 2) • So if you invest $2,601 at 9%, you can withdraw $1,000 at time 1 and $2,000 at time 2 • The PV of $1,000 at time 1 and $2,000 at time 2 is $2,601 12 Clicker Question: • If your investment earns 10%, how much do you need to invest now to be able to withdraw $500 in one year and $800 in two years? A. $1,000 B. $1,116 C. $1,200 D. $1,226 E. $1,300 13 Clicker Answer: • In order to withdraw $500 in one year and then $800 in two years, you must invest the sum of the PVs of these withdrawals. PV of $500 in one year = $500/(1.1) = $455 PV of $800 in two years = $800/(1.1)2 = $661 Sum = $455 + $661 = $1,116 The answer is B. 14 5.2 Annuities The word Annuity has two definitions Economic Definition: 1. All CFs are the same 2. CFs occur at regular intervals (Annually, Semi-annually, Quarterly, Monthly…) 3. All CFs are discounted at the same rate The Financial Product: 1. Pay an insurance company or a bank a lump sum today 2. Receive CFs at regular intervals for a fixed period or until you die 3. Sometimes you pay now (or make regular payments starting now) and then receive payments when you retire at 65 Same pattern of Cash Flow rules for: • Loans (you pay) • Purchased Annuities (you are paid) 15 Formula for PV of an Annuity (PVA) 1 PVA C 1 r 1 r t PVA = C{[1 - 1/(1 + r)t]/r} (Same thing but typed) Text Book’s Notation: 1/(1 + r)t = Present Value Factor (PVF) PVA = C{[1 - PVF]/r} Other Notation: {[1 - 1/(1 + r)t]/r} = Present Value Annuity Factor (PVAF) PVA = C{PVAF} 16 PV of $1,000 per for 5 years @ 6%: We’ll use your calculator’s TVM function: N = 5 I/YR = 6 PMT = 1000 PV = -4,212 (You don’t have to enter FV since it is zero) 17 Clicker Question: • An investment pays $15,000 at the end of each of the next four years. • Assume a discount rate of 8% • Calculate the present value of the investment. (Or you can say “calculate the PV of this annuity?”) A. $15,000 B. $20,000 C. $29,682 D. $49,682 E. $60,000 18 Clicker Answer: • Use the TVM Function to calculate the PV the annual payments of $15k per year for 4 years: N = 4 PMT = 15,000 I/YR = 8 FV = 0 PV = -49,682 The answer is D 19 Consumer Loans • Consumer loans by custom have monthly payments • Credit card loans • Mortgages • Similar structure to rent payments and traditional worker pay periods 20 Clicker Question (that does not count): • What is the monthly payment for a thirty year, 6.00% fixed rate $250,000 mortgage loan? • We have not covered how to calculate this yet, so just give me your best guess. A. $299 B. $499 C. $1,099 D. $1,499 E. $1,899 21 Clicker Answer (that does not count): • What is the monthly payment for a thirty year, 6.00% fixed rate $250,000 mortgage loan? • Use the TVM function on your calculator. (We’ll cover this procedure in more detail later) N = 30 x 12 = 360 I/YR = 6/12 = 0.5 PV = 250,000 PMT = -1,499 So the monthly payment is $1,499. 22 Consumer Loans • Consumer loans have monthly payments • Mortgages • Credit card loans • Car loans • Rates are quoted at the monthly rate times 12 • Called the Annual Percentage Rate (APR) • Or APR-Monthly • 12% APR-Monthly really means 1% per month • 18% APR-Monthly really means 1.5% per month 23 Example: The bank quotes a rate of 12% APR on a 5-year car loan You can afford to pay $500 per month. How much can you borrow? • t = 5 x 12 = 60 • r = 12%/12 = 1% • C = $500 PVA = C{[1 - 1/(1 + r)t]/r} = $500{[1 - 1/(1 + 0.01)60]/0.01} = $500{[1 - 1/(1 + 0.01)60]/0.01} = $500{44.9550} = $22,477.52 24 Using the TVM Calculator Function: Same Example: Calculate the loan size for a 5-year, 12% APR loan with $500 monthly payments. Two ways to do this: Set to 1 payments per year and enter 1%: N = 60 PMT = 500 I/Y = 1 FV = 0 PV = -22,477.52 Set to 12 payments per year and enter 12%: N = 60 PMT = 500 I/Y = 12 FV = 0 PV = -22,477.52 • I recommend you keep your calculator set to 1 payment per year and adjust your inputs 25 Finding Loan Payments: You want to borrow $100,000 to buy a house Calculate the monthly payments on a 30 year 9% APR loan r = 9%/12 = 0.75% t = 30 x 12 = 360 PV = C{[1 - 1/(1 + r)t]/r} = PV = C/PVAF C = PV/{[1 - 1/(1 + r)t]/r} = C = PV/PVAF C = $100,000{[1 - 1/(1 + 0.0075)360]/0.0075} C = $100,000/124.2819 = $804.62 Can also use this notation: PV = C{PVAF} C = PV/PVAF Using TVM Function (Set to 1 payment per year): N = 360 I/YR = 0.75 PV = 100,000 PMT = -804.62 26 Clicker Question: • You put $8,000 on your credit card. • The card has a stated rate or 18% APR-Monthly • Calculate the monthly payments assuming the credit card uses a 30 year payback period. A. $30 B. $121 C. $333 D. $3,284 E. $8,120 27 Clicker Answer: N = 30 x 12 = 360 PV = 8,000 I/YR = 18/12 = 1.5 PMT = 121 The answer is B. Bonus Question: • Now assume that you actually read the fine print and find that a single late payment will cause the credit-card rate to increases to 30% APR-Monthly. • Calculate the new monthly payment: N = 360 PV = 8,000 New I/YR = 30/12 = 2.5 New PMT = 200 28 Finding Annuity Payments: (Before we did PV of an annuity) • You want to buy an annuity from an insurance company • You will pay $100,000 today for equal monthly payments for the next 30 years • The Annuity is offered at 9% • Calculate the payments Same Calculation: N = 30 x 12 = 360 I/YR = 9/12 = 0.75 PV = 100,000 PMT = -804.62 29 Finding the Number of Periods • Put $2,000 on your credit card • Choose to pay $40 per month. • How many months to payoff at 18% APR-Monthly? PV = C{[1 - 1/(1 + r)t]/r} PV/C = [1 - 1/(1 + r)t]/r r(PV/C) = 1 - 1/(1 + r)t (1 + r)t = 1/[1 - r(PV/C)] t [ln(1 + r)]= ln{1/[1 - r(PV/C)]} t = ln{1/[1 - r(PV/C)]}/[ln(1 + r)] = ln{1/[1 – 0.015($1,000/$40)]}/[ln(1.015)] = 93.11 But we’ll use the TVM function? 30 Finding # of Periods using TVM Function 18% annual = 18/12 = 1.5 per month: I/YR = 1.5 PV = 2,000 PMT = -40 N = 93.11 I/Y R= 1.5 PV = -2,000 PMT = 40 N = 93.11 (One must be negative) Potential Errors when using the TVM function: • Set at 1 P/YR: (Neither PV or PMT are negative) I/YR = 1.5 PV = 2,000 PMT = 40 N = -37.59 (wrong) • Set at 12 P/YR: I/YR = 1.5 PV = 2,000 PMT = -40 N = 51.66 (wrong) I/YR = 18 PV = 2,000 PMT = -40 N = 93.11 31 Clicker Question: • You charge $5,000 on your credit card. • The card has a stated rate of 24% APR-Monthly • Calculate the number of months needed to payoff the debt assuming monthly payments of $150 A. 24 B. 36 C. 56 D. 60 E. 94 32 Clicker Answer: PV = 5,000 I/YR = 24/12 = 2 PMT = -150 N = 55.48 So you will payoff the loan in 56 months The Answer is C 33 Future Value of Annuities FVA = C{[(1 + r)t -1]/r} FVA = C{FVAF} Example: Make $5,000 annual contributions to retirement fund for 40 years. Earn 10% per year. t = 40 r = 10% C = $5,000 FVA = $5,000{[(1.10)40 -1]/0.10} = $5,000{442.59} = $2,212,963 • 442.59 is called the Future Value Annuity Factor (FVAF) Calculator TVM Function: N = 40 I/YR = 10 PMT = $5,000 FV = $2,212,963 34 Clicker Question: • To save for your child’s education, you invest $1,000 a year for 18 years in a college savings account. • The account earns 10% per year. • What will be the value of the account in 18 years? • What if you don’t start saving until your kid is 5 (save for 13 years)? A. $18,000 over 18 years and $13,000 over 13 years B. $36,000 over 18 years and $26,000 over 13 years C. $36,900 over 18 years and $26,300 over 13 years D. $45,599 over 18 years and $24,523 over 13 years E. $45,599 over 18 years and $30,250 over 13 years 35 Clicker Answer: • $1,000 per year over 18 years at 10%: N = 18 I/YR = 10 PMT = 1000 FV = 45599 • $1,000 per year over 13 years at 10%: N = 13 I/YR = 10 PMT = 1000 FV = 24523 The answer is $45,599 over 18 years and $24,523 over 13 years (D) • So roughly half as much ($24,523/$45,599 = 54%) if you only save for 13 years. • How much saved each year for 18 years to have $100k? N = 18 I/YR = 10 FV = 100,000 PMT = 2,193 Note that $100,000/$45,599 = 2.193 36 Annuities Due An Annuity Due means the payments are made at the beginning of each period, not at the end of each period: So the First payment is made immediately, not at the end of the first period. The figure below shows the payment timing of a four year $100 Annuity and an Annuity Due: 0 1 2 3 4 Annuity $100 $100 $100 $100 Annuity Due $100 $100 $100 $100 37 Compare an Annuity Due to an Annuity • Is the PV of a 4 yr Annuity Due greater than a regular 4 yr annuity? – Would you rather get the Annuity or the Annuity Due? • The 4 yr Annuity Due is the same as a 3 yr (regular) Annuity plus an extra $100 now (at time zero): 0 1 2 3 3 yr Annuity $100 $100 $100 4 yr Annuity Due $100 $100 $100 $100 • PV 3 yr Annuity: N=3 I/Y=10 PMT=100 PV=248.69 • PV 4 yr Annuity Due = 248.69 + 100 = 348.69 • PV 4 yr (regular) Annuity: N=4 I/Y=10 PMT=100 PV=316.99 38 Clicker Question: • You will pay $1,000 per month to rent apartment for a year • The lease requires monthly payments at the beginning of each month. • Assume a 12% APR-Monthly discount rate. • Calculate the NET BENEFIT (in present value terms) to the landlord of receiving the rent payments at the beginning of each month as opposed to the end of each month. A. $1,343 B. $1,000 C. $743 D. $500 E. $113 39 Clicker Answer: • PV of a 12 month, $1,000 Annuity discounted at 12% APR: N = 12 I/Y = 12/12 = 1 PMT = 1,000 PV = 11,255 • PV of a 12 month, $1,000 Annuity Due discounted at 12% APR: N = 11 I/Y = 12/12 = 1 PMT = 1,000 PV = 10,368 PV of Annuity Due = $10,368 + $1,000 = $11,368 • Net Benefit = $11,368 – 11,255 = $113 • The Answer is E 40 Perpetuities: Level stream of cash flows forever: PV Perpetuity = C/(1+r) + C/(1 + r)2 + C/(1 + r)3 + … = C[1/(1+r) + 1/(1 + r)2 + 1/(1 + r)3 + …] • The term in brackets is a “convergent sequence” for any value of r between 0 and 1 • Since each subsequent term is divided by (1+ r) taken higher and higher power, the ratio becomes a smaller number, approaching zero • Eventually, you are just adding zeros • Economically, payments made 100 years out (divided by 1+r taken to the 100th power) aren’t really adding anything For 0 < r < 1: [(1+ r)-1 + (1 + r)-2 + (1 + r)-3 + …] = 1/r PV Perpetuity = C[1/r] = C/r The PV of $500 per year forever discounted at 8% = $500/.08 = $6,250 41 Review of Formulas and Symbols Page 137 42 5.3 Comparing Rates: APR vs. EAR Banks (and other financial institutions) must quote loan and deposit rates as Annual Percentage Rate (APR) An APR is the periodic rate times the number of periods: • 5% every six months is 10% APR Semi-Annual • 1% every month is 12% APR Monthly • 2% every quarter is 8% APR Quarterly An EAR (Effective Annual Rate) is the annual rate that is equivalent to a given APR • What annual rate is the same as 5% compounded twice a year? (10% APR Semi-Annual) • (1.05)(1.05) = 1.1025 • So the equivalent is 1.1025 – 1 = 0.1025 = 10.25% • So either pay me 10% APR S-A or 10.25% per year • They are the same! 43 APR and EAR Formulas APR: Annual Percentage Rate • APR = (Periodic Rate) x (# of periods) • 5% every six months = 10% APR SA • 12% APR Monthly = 1% every month EAR: Effective Annual Rate • Annual rate that is equivalent to the compounded APR • 10% APR-Semi Annual = 5% compounded twice • (1.05)(1.05) – 1 = 10.25% EAR • 12% APR-Monthly = (1.01)12 – 1 = 12.68% EAR Let m = # of periods Calculate the EAR if the quoted rate is 18% APR monthly: APR = 18% m = 12 EAR = (1 + APR/m)m – 1 = (1 + 0.18/12)12 - 1 = 1.01512 – 1 = 19.56% You would be indifferent between paying/earning 19.56% per year and 18% APR monthly (1.5% per month) 44 APR and EAR Formulas Convert EAR to APR: APR = m[(1 + EAR)1/m -1] What semi-annual APR is equivalent to 10% EAR? EAR = 10% m = 2 APR = 2[(1.1)1/2 -1] = 9.76% Show this works: 9.76% APR means 4.88% each 6 month period: EAR = (1 + APR/m)m – 1 = (1.0488)2 – 1 = 10% 45 Terminology and Calculators EAR is also called: • Effective Rate (EFF) • Effective Annual Yield (EAY) • Annual Percentage Yield (APY) APR is also called: • Nominal (NOM) Using the function in your HP Calculator: Calculate EAR for 18% APR monthly: Function Buttons Display Set payments per year to 12 12 {Yellow Shift} {P/Y} 12. Input 18% APR 18 {Yellow Shift} {NOM%} 18. Calculate EAR {Yellow Shift} {EFF%} 19.5618 46 Clicker Question: • You want to invest for one year • A bank is offering 3.10% APR-Quarterly on a three- month CD and 3.25% on a one-year CD • Assume you can roll-over the three-month CD at the same 3.10% rate three more times – So assume rates will not change until you have rolled over the three-month CD three more times • Which investment alternative is better? A. The three-month CD 4 times B. The one-year CD once C. There is not enough information 47 Clicker Answer: • Compare 3.10% APR-Quarterly to 3.25% Annual • So calculate the EAR of 3.10% APR-Quarterly: m = 4, APR = 3.10% EAR = (1 + APR/m)m – 1 = (1 + 0.0310/4)4 – 1 = (1.00775)4 – 1 = 0.0314 = 3.14% Since 3.10 APR-Quarterly is the same as 3.14% Annual, 3.25% Annual is better than 3.10% APR-Quarterly. The Answer is B. 48 5.4 Loan Types and Amortization There are a number of different ways a loan can be repaid: 1. Ways in which interest can be paid 2. Ways in which the loan amount (called the principal) can be repaid There are common loan structures for different types of loans • Loans made to people – called consumer or retail loans) • Loans make to business by banks – Called institutional or business loans) • Loans make to business by “the market” – The company sells Commercial Paper to the market – The company sells Bonds to the market (long term loans) • The way the loan is repaid is called the Amortization Schedule 49 5.4 Loan Types and Amortization Amortization refers to how the loan is repaid 1. Pure Discount Loans • Only one payment • Payment Includes both the principal (loan amount) and interest • Amount loaned is the Present Value of the one payment • The payment is the Future Value of the loan amount • The payment is also called the Face Value of the loan (also FV) 50 1. Pure Discount Loans (Continued) Common loan structure for a Bank CD: Calculate the payment for a $100 2 year loan at 5% annual rate: • FV = PV(1 + r)t = $100(1.05)2 = $110.25 • $100 is the loan amount and $10.25 is the interest Common loan structure for a T-Bill: Calculate the amount loaned for a $10,000 FV 1 year loan at 6% • PV = FV/(1 + r)t = $10,000/(1.06) = $9,433.96 • $10,000 - $9,433.96 = $566.04 is the interest Common loan structure Zero-Coupon corporate bond (a “zero”): Calculate the amount loaned for a $1,000 FV five-year loan at 10% • PV = FV/(1 + r)t = $1,000/(1.1)5 = $620.92 • $1,000 - $620.92 = $379.08 is the interest 51 2. Bullet Maturity Loans • Also called “Interest Only Loans” • Periodic interest payments are made • Quoted as a percentage of the Face Value • The loan amount is repaid at the end This is the common structure for corporate bonds • A $1,000 four-year loan is structured to make 8% interest-only ANNUAL payments • Payments = $1,000(0.08) = $80 made at the end of each year 0 1 2 3 4 $80 $80 $80 $80 $1,000 $1,080 52 3. Amortized Loans or Self-Amortizing Loans • Periodic payments include both interest and principal One possible type of amortizing loan: • A small business borrows $5,000 for 5 years at 9% • The contract requires borrower to repay $1,000 each year Amortization Table: Beginning Interest Principal Total Ending Year Balance at 9% Payment Payment Balance 1 $5,000 $450 $1,000 $1,450 $4,000 2 $4,000 $360 $1,000 $1,360 $3,000 3 $3,000 $270 $1,000 $1,270 $2,000 4 $2,000 $180 $1,000 $1,180 $1,000 5 $1,000 $90 $1,000 $1,090 $0 Totals $1,350 $5,000 $6,350 Notice the total payment decreases as the balance decreases 53 Another type of amortizing loan: • The contract calls for equal payments each period • Solve for the equal annuity payments: • PVA = C{[1 - 1/(1 + r)t]/r} PVA = C{PVAF} C = PVA/{PVAF} Same loan 5 yr 9% $5,000 loan but with equal payments: C= $5,000/{[1 - 1/(1.09)5]/0.09} = $1,285.46 N = 5 I/Y = 9 PV = $5,000 PMT = -1,285.46 Amortization Table: Beginning Interest Principal Ending Year Balance Payment Owed Paid Balance 1 $5,000 $1,285 $450 $835 $4,165 2 $4,165 $1,285 $375 $911 $3,254 3 $3,254 $1,285 $293 $993 $2,261 4 $2,261 $1,285 $204 $1,082 $1,179 5 $1,179 $1,285 $106 $1,179 $0 Totals $6,427 $1,427 $5,000 • Note: Since the loan is repaid slower, the total interest paid is higher 54 than the loan described on the previous slide Recap Loan Types: 1. Pure Discount • Also called Zero Coupon • Single payment consisting of both interest and principal at maturity • Very common for short-term debt and some corporate bonds 2. Interest Only • Also called Bullet Maturity or Coupon Bond • Periodic interest payments with entire principal repaid at maturity • Very common for long-term debt • Often at maturity, new bonds are sold and the proceeds are used to repay the maturing bonds 3. Amortizing or Self-Amortizing • Common for consumer loans • Common for small business loans • Each payment consists of interest and principal • Balance of loan decreases over time 55 Personal Finance – Saving for Retirement Assume you will: • Save $1,000 per year • Earn 10% per year • Retire at age 65 How much more will you have if you start saving: • At age 30 (save for 35 years)? • At age 20 (save for 45 years)? • Will it be 10% more? 20% more? 50% more? N = 35 I/Y = 10 PMT = 1,000 FV = 271,024 N = 45 I/Y = 10 PMT = 1,000 FV = 718,905 Dollar difference: 718,905 – 271,024 = 447,880 Percent difference: 718,905/271,024 - 1 = 165% So starting 10 years earlier increases retirement savings by 165% Or gives you over two and a half times more money 56 Multiplication • These future values are “per thousand saved” • Doubling your savings doubles your return: N = 45 I/Y = 10 PMT = 1,000 FV = 718,905 N = 45 I/Y = 10 PMT = 2,000 FV = 1,437,810 Percent difference: 1,437,810/718,905 - 1 = 100% • But how much higher does return have to be to double savings? N = 45 PMT = -1,000 FV = 1,437,810 I/Y = 12.18 An increase in return from 10% to 12.18% doubles the FV This leads to Fees 57 The Effect of Fees on Return • These future values are “per thousand saved” at 10% • But what if you were charged a 2% fee so that you earned only 8% per year? N = 45 I/Y = 10 PMT = 1,000 FV = 718,905 N = 45 I/Y = 8 PMT = 1,000 FV = 386,506 Dollar difference: 718,905 – 386,506 = 332,399 Percent difference: 718,905/386,506 - 1 = 86% So not paying a fee increases retirement savings by 86% Or gives you over 1.8 times more money 58 What to do with the lump sum? Buy an annuity! • How much can you expect to get per year? • Depends on the number of years and rate of return • Assume an 8%, 20 fixed year annuity N = 20 I/Y = 8 PV = 100,000 PMT = 10,185.22 • This is per $100,000. So a starting value of 718,905 N = 20 I/Y = 8 PV = 718,905 PMT = 73,222 Note that 73,222/10,185 = 7.18905 59

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