1111 McGraw-Hill Ryerson© 11-1 Chapter 11 O rdinary A nnuities McGraw-Hill Ryerson©

Calculate the… Learning Objectives After completing this chapter, you will be able to: … number of payments in ordinary and deferred annuities … payment size in ordinary and deferred annuities … interest rate in ordinary annuities LO-1 LO-2 LO-3

1111 McGraw-Hill Ryerson© 11-3 Using your financial calculator we need to reorganize the formulae to solve algebraically … solve for payment number or size or interest rate using the same steps as before …

1111 McGraw-Hill Ryerson© 11-4 Finding the Payment Size…. PMT

1111 McGraw-Hill Ryerson© 11-5 Your life partner somehow convinced you that you can’t afford the car of your dreams, priced at $ You are advised to… “Save up for 4 years and then buy the car for cash.” How much would you have to save each month, if you could invest with a return of 10% compounded monthly? You need to decide if this situation involves … a PV or a FV and then use the appropriate formula... PMT As you have to save up the $28,800, i.e. in the future, FV = $28,800 Assume you have no savings … PV = 0 Finding Payment Size of an Ordinary Simple Annuity

1111 McGraw-Hill Ryerson© 11-6 Your life partner somehow convinced you that you can’t afford the car of your dreams, priced at $ (At least not right now). You are advised to… “Save up for 4 years and then buy the car for cash.” How much would you have to save each month, if you could invest with a return of 10% compounded monthly? Finding Payment Size of an Ordinary Simple Annuity PMT = Formula solution

1111 McGraw-Hill Ryerson© 11-7 [ FV= PMT (1+ i) n - 1 i ] i PV= PMT 1-(1+ i) -n [ ] Which Formula? Algebraic Method of Solving for PMT (a) If the payments form a Simple Annuity go directly to If the annuity’s PV is known, substitute values of PV, n, and i into PV formula. If the annuity’s FV is known, substitute values of FV, n, and i into FV formula. 3. & & 4. (b) If the payments form a General Annuity, find c and i 2 2.

1111 McGraw-Hill Ryerson© 11-8 Calculate the quantity within the square brackets. Rearrange the equation to solve for PMT. i [ FV= PMT (1+ i) n - 1 ] i PV= PMT 1-(1+ i) -n [ ] Which Formula? Algebraic Method of Solving for PMT Applying Method…

1111 McGraw-Hill Ryerson© 11-9 Finding Payment Size of an Ordinary Simple Annuity Which Formula? Your life partner somehow convinced you that you can’t afford the car of your dreams, priced at $ (At least not right now). You are advised to… “Save up for 4 years and then buy the car for cash.” How much would you have to save each month, if you could invest with a return of 10% compounded monthly? [ FV= PMT (1+ i) n - 1 i ] As the annuity’s FV is known, therefore, the FV formula is used 2. Extract necessary data... FV = n = 4*12 = 48 i =.10/12 c = 1 PMT = ? PV = 0

1111 McGraw-Hill Ryerson© Your life partner somehow convinced you that you can’t afford the car of your dreams, priced at $ (At least not right now). You are advised to… “Save up for 4 years and then buy the car for cash.” How much would you have to save each month, if you could invest with a return of 10% compounded monthly? [ FV= PMT (1+ i) n - 1 i ] Formula FV = n = 4*12 = 48 i =.10/12 c = 1 PMT = ? PV = 0 …another example

1111 McGraw-Hill Ryerson© The

1111 McGraw-Hill Ryerson© Your parents are discussing the terms of the $ mortgage that they have offered to hold in the purchase of your first home. They are considering an interest rate of 5% compounded monthly. If you were to take 20 years to repay the mortgage, find the size of the monthly payment. Your parents are discussing the terms of the $ mortgage that they have offered to hold in the purchase of your first home. They are considering an interest rate of 5% compounded monthly. If you were to take 20 years to repay the mortgage, find the size of the monthly payment PMT = n = 12 *20 = 240 PV = $100000FV = 0 Formula solution

1111 McGraw-Hill Ryerson© Your parents are discussing the terms of the mortgage that they have offered to hold in the purchase of your first home. They are considering an interest rate of 5% compounded monthly. If you were to take 20 years to repay the mortgage, find the size of the monthly payment. i =.05/12 Extract necessary data... n = 12* 20 = 240 PV = $ FV = 0 C =1

1111 McGraw-Hill Ryerson© Choose appropriate formula and Solve As the annuity’s PV is known, the PV formula is used 2. i =.05/12n = 12 *20 =240PV = $ i PV= PMT 1-(1+ i) -n [ ] Formula Size of monthly mortgage payment

1111 McGraw-Hill Ryerson© Amount $ How much interest will you pay your parents over the 20 year period? Monthly PaymentNumber of Payments , Amount Borrowed100, Total Interest Paid 58, x

1111 McGraw-Hill Ryerson© PMT = -700 As this amount of interest shocks you, you discuss the possibility of making payments of $700/month, to save some time and interest costs. Determine the time it will take you to repay your mortgage at this new rate. 700 N = Formula solution 218 payments = 18 yrs 2months

1111 McGraw-Hill Ryerson© Formula  i PMT PVi   1ln 1 [] n i =.05/12PMT = $700PV = $100,000 C = 1 n payments = 18 yrs 2months

1111 McGraw-Hill Ryerson© Base formula   i -n i)i) PMT PV (11 [] 2. To isolate n, divide both sides by PMT PMT …Continue… Developing the Formula PMT  PV   i -n i)i) (11 []   i -n i)i) PMT PV (11 ][ Formula  i PMT  PV i * 1ln 1 [] n

1111 McGraw-Hill Ryerson© (a) Multiply both sides by i 3. Continue to isolate n. PMT  PV   i -n i)i) (11 [] PMT  PV   -n i)i) (11 i [] * i *i*i (b) Reorganize equation (c) Now Take the natural logarithm (ln or lnx) of both sides -n* ln PMT * i  PV  -n 1  i)i) (1 -n  i)i) (1  i)i) (1  ln (d) Solving for n… divide both sides by ln(1+i) ln(1+i) …from 2.  PMT *i*i PV 1  [] PMT *i*i PV 1  -n* ln  i)i) (1  ln [] PMT *i*i PV 1  PMT PV * i  i  1ln 1 n []

1111 McGraw-Hill Ryerson© , Total Interest Saved 6, Approximately how much money do you save in interest charges by paying $700/month, rather than $659.91/month? Amount $ Monthly PaymentNumber of Payments x 158,

1111 McGraw-Hill Ryerson© If you could see your way to a further increase of $25/month, (a) how much faster would you pay off the mortgage, and (b) approximately how much less interest would be involved? 725 PMT = -725N = Paying $ payments = 17 yrs 2months Formula solution

1111 McGraw-Hill Ryerson© i =.05/12PMT = $725PV = $100,000 C = 1 n payments = 17 yrs 2months Formula  i PMT  PV i * 1ln 1 [] n

1111 McGraw-Hill Ryerson© (b)Total Interest Saved 3, or , Amount $ Monthly PaymentNumber of Payments x or , (a) Payments Saved 12

1111 McGraw-Hill Ryerson© 11-24

1111 McGraw-Hill Ryerson© York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? Since you want the furniture now, this involves a PV PMT PV = $2250 Once you repay the loan, FV = 0 Payments are deferred for 11 months. DEFERRAL Finding Payment Size in a Deferred Annuity

1111 McGraw-Hill Ryerson© York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? In effect, York furniture has given a loan to a buyer of $2,250 on the day of the sale! When the payments begin, the buyer owes $2,250 plus accrued interest!

1111 McGraw-Hill Ryerson© d = 11 i = 0.10/12 n = 24 $2250 PMT Payments $2250 PV Annuity FV PV of the payments at the end of month 11 FV of the $2,250 loan at the end of month 11 = Months

1111 McGraw-Hill Ryerson© FV = 2, Find the amount owed after 11 months: 2250 $2, is the PV of the annuity York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? Finding Payment Size in a Deferred Annuity

1111 McGraw-Hill Ryerson© FV = 2, Now find the PMT of the annuity … 0 24 monthly payments of $ will repay the loan. PV = - 2, PMT = York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? Formula solution Finding Payment Size in a Deferred Annuity

1111 McGraw-Hill Ryerson© FV = PV(1 + i) n Formula FV = 2250( /12) 11 = $2,   i -n i)i) PMT PV (11 [] = PMT [1-(1+.10/12) -24 ].10/12 = $ PMT York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period and the repayment period? 24 monthly payments of $ will repay the loan. Find the amount owed after 11 months: Finding Payment Size in a Deferred Annuity

1111 McGraw-Hill Ryerson© i.e....Number Of Payments

1111 McGraw-Hill Ryerson© $20,000 is invested in a fund earning 8% compounded quarterly. The first quarterly withdrawal of $1,000 will be taken from the fund five years from now. How many withdrawals will it take to deplete the fund? N Payments are deferred for 19 quarters DEFERRAL Finding Number Of Payments in a Deferred Annuity The FV of $20,000 after the deferral, becomes the PV of the annuity...

1111 McGraw-Hill Ryerson© Years d = 19 $20,000 PV 1 n = ? This FV 1 then becomes the PV of the annuity of $1000/quarter The $20000 earns interest for 4 years 9 months Payments of $1000 / quarter FV 1 i = 0. 08/4 =.02PMT = $1000

1111 McGraw-Hill Ryerson© $20,000 is invested in a fund earning 8% compounded quarterly. The first quarterly withdrawal of $1000 will be taken from the fund five years from now. How many withdrawals will it take to deplete the fund? Find the FV of $20,000 in 4.75 years $29, is the PV of the annuity FV = 29, Finding Number Of Payments in a Deferred Annuity

1111 McGraw-Hill Ryerson© Now find the PMT of the annuity … quarterly payments will deplete the fund(44 full payments and 1 partial) FV = 29, PV = N = 44.1 Formula solution $20,000 is invested in a fund earning 8% compounded quarterly. The first quarterly withdrawal of $1000 will be taken from the fund five years from now. How many withdrawals will it take to deplete the fund? Finding Number Of Payments in a Deferred Annuity

1111 McGraw-Hill Ryerson© FV = PV(1 + i) n Formula FV = 20000( /4) 19 = $29, Find the FV of $20,000 in 4.75 years PMT PV * i  i  1ln 1 n [] ln(1.02) = 44.1 payments or 11 years $20,000 is invested in a fund earning 8% compounded quarterly. The first quarterly withdrawal of $1000 will be taken from the fund five years from now. How many withdrawals will it take to deplete the fund? Finding Number Of Payments in a Deferred Annuity [ ln 1 - ] *

1111 McGraw-Hill Ryerson© When… number of compoundings per year number of payments per year 

1111 McGraw-Hill Ryerson© Since you get paid every second Thursday you decide to pay $350 every two weeks to make your budgeting easier. Find the new term of your mortgage if the interest charges remain at 5% compounded monthly P/Y = bi-weekly payments or 15 yrs 11.4 months C/Y= 12 PMT = -350 N = Formula solution

1111 McGraw-Hill Ryerson© Since you get paid every second Thursday you decide to pay $350 every two weeks to make your budgeting easier. Find the new term of your mortgage if the interest charges remain at 5% compounded monthly. Determine c Step 1 C= 12 / 26 =.4615 i 2 = (1+i) c - 1 i 2 = (1+.05/12) i 2 = Use c to determine i 2 Step 2 C = number of compoundings per year number of payments per year Step 3

1111 McGraw-Hill Ryerson© as the value for “ i ” in the appropriate annuity formula Step 3 Use this rate i 2 = Formula  i PMT  PV i * 1ln 1 [] n payments or 15 yrs 11.4 months

1111 McGraw-Hill Ryerson© # of Payments Payment Amount Payment Amount Total Cost Scenario $100,000 Twenty-year Mortgage – Interest 5% per annum $ $158, $152, $149, $ $ $145, $ Terms Per month Every two weeks $145, Every two weeks 415 Best Scenario

1111 McGraw-Hill Ryerson© You are now considering delaying the purchase of your first house to allow for a larger down payment. If you save $350 per pay, how long would it take to have an additional $15000, if you can earn 8% compounded monthly on your savings? N = = FV New required Formula

1111 McGraw-Hill Ryerson© Determine c Step 1 C= 12 / 26 =.4615 i 2 = (1+i) c - 1 i 2 = (1+.08/12) i 2 = Use c to determine i 2 Step 2 C = number of compoundings per year number of payments per year Step 3 You are now considering delaying the purchase of your first house to allow for a larger down payment. If you save $350 per pay, how long would it take to have an additional $15000, if you can earn 8% compounded monthly on your savings?

1111 McGraw-Hill Ryerson© You are now considering delaying the purchase of your first house to allow for a larger down payment. If you save $350 per pay, how long would it take to have an additional $15000, if you can earn 8% compounded monthly on your savings? Formula n  i PMT  FV i * 1ln 1 [] bi-weekly payments = approx 1yr 7months 1 350

1111 McGraw-Hill Ryerson© Base formula   i PMT FV  n i)i) (1 1 [] 2. To isolate n, divide both sides by PMT PMT …continued… Developing the Formula PMT  FV Formula  i PMT  FV i * 1ln 1 [] n +   i PMT FV  n i)i) (1 1 []  i  n i)i) (1 1 []

1111 McGraw-Hill Ryerson© …from 2. PMT  FV  i  n i)i) (1 1 [] (a) Multiply both sides by i 3. Continue to isolate n … PMT FV   i  n i)i) (1 1 [] * i []  PMT FV  n i)i) (11 * i (b) Reorganize equation  PMT FV  n i)i) (11 * i (c) Now Take the natural logarithm (ln or lnx) of both sides nln(1+ i)  ln [] PMT * i FV 1 + (d) Solving for n… divide both sides by ln(1+i) ln(1+i) nln(1+ i)  ln [] PMT * i FV 1 + PMT FV * i  i  1ln 1 +ln n []

1111 McGraw-Hill Ryerson© You already have $10000 saved for your down payment. If you save $350 per pay, how long would it take to have an additional $15000? Assume you can earn 8% compounded monthly on all of your savings Already entered N = bi-weekly payments = approx 1 yr 5months

1111 McGraw-Hill Ryerson© You already have $10000 saved for your down payment. If you save $350 per pay, for the next 2 years, find the size of your available down payment. Assume you can earn 8% compounded monthly on all of your savings. Already entered FV = Formula solution 350

1111 McGraw-Hill Ryerson© Steps Formula Solution This is more complicated to solve when using algebraic equations! Find the FV of the $ in 2 years Find the FV of the $350 per pay Add totals together The $ continues to earn interest during the new savings period! You already have $10000 saved for your down payment. If you save $350 per pay, for the next 2 years, find the size of your available down payment. Assume you can earn 8% compounded monthly on all of your savings.

1111 McGraw-Hill Ryerson© Formula Solution FV = PV(1 + i) n Formula = 10000( /12) 24 = $11, i 2 = (1+i) c - 1 = (1+.08/12) = $ 11, = 350 [( ) 52 –1].0031 = $ , ,430.12Total  PMT FV  i  n i)i) (1 1 [] You already have $10000 saved for your down payment. If you save $350 per pay, for the next 2 years, find the size of your available down payment. Assume you can earn 8% compounded monthly on all of your savings.

1111 McGraw-Hill Ryerson© 11-51

1111 McGraw-Hill Ryerson© A life insurance company advertises that $50,000 will purchase a 20-year annuity paying $ at the end of each month. What nominal rate of return does the annuity investment earn? C/Y = 1 I/Y = 5.54 The annuity earns 5.54% pa

1111 McGraw-Hill Ryerson© …to solve for i without a financial calculator

1111 McGraw-Hill Ryerson© This completes Chapter 11