1. Lecture 34 Sinking funds vs Amortization Ana Nora Evans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.edu/~ans5k Math 1140 Financial Mathematics

2. Math 1140 - Financial Mathematics Last Time: Owner’s Equity The owner’s equity in a property is the current value minus the current debt. The equity increases when: - the current value increases (the property appreciates) - the debt decreases trough periodic payments 2

3. Math 1140 - Financial Mathematics Last Time: Owner’s Equity The equity decreases when: - the current value decreases (the property depreciates) - the debt increases - no payments - home equity loan 3

4. Math 1140 - Financial Mathematics Last Time: Tax Deductions Under certain conditions, the mortgage interest can be deducted from the US federal taxes. 4

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6. Last Time: Sinking Funds A loan is paid back by making paying the interest periodically and paying the principal at the due date of the loan. To be able to meet the financial obligation at a future date, periodic payments are made in a fund. This method of repaying a debt is called a sinking fund. 6

7. Math 1140 - Financial Mathematics The total periodic charge has two components: 1.The interest charges 2.The sinking fund payment The future value of the sinking fund payments is the principal (the amount borrowed). 7

8. Math 1140 - Financial Mathematics A business borrows P dollars at interest rate i(12) due in n months. To repay this debt the business makes monthly payments in a fund paying interest rate j(12). What is the total monthly payment? The total monthly payment has two components: -the interest on the loan -the payment made in the fund Step 1: Calculate the interest charge 8

9. Math 1140 - Financial Mathematics A business borrows P dollars at interest rate i(12) due in n months. To repay this debt the business makes monthly payments in a fund paying interest rate j(12). What is the total monthly payment? Step 2: Calculate the payment made in the fund The payments form and ordinary annuity. 9

10. Math 1140 - Financial Mathematics A business borrows P dollars at interest rate i(12) due in n months. To repay this debt the business makes monthly payments in a fund paying interest rate j(12). What is the total monthly payment? The total monthly payment is 10

11. Math 1140 - Financial Mathematics Total monthly payment when using a sinking fund Monthly payment when using amortization 11

12. Math 1140 - Financial Mathematics Comparison The sinking fund is more expensive than amortization when the interest rate earned by the fund is smaller than the interest rate paid for the loan. ( j<i ) The sinking fund is less expensive than amortization when the interest rate earned by the fund is bigger than the interest rate paid for the loan. ( i<j ) 12

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14. Math 1140 - Financial Mathematics A church uses a bond issue to borrow $400,000 for a building program. The bonds pay 7%(2) and mature in 15 years. In order to redeem the bonds in 15 years, the church sets up a sinking fund with a local bank that pays 6%(12) on deposits. Find the monthly cost of retiring the debt with a sinking fund. The monthly payment must cover -the semiannual interest payments, every six months -the $400,000 needed to redeem the bonds in 15 years 14

15. Math 1140 - Financial Mathematics A church uses a bond issue to borrow $400,000 for a building program. The bonds pay 7%(2) and mature in 15 years. In order to redeem the bonds in 15 years, the church sets up a sinking fund with a local bank that pays 6%(12) on deposits. Find the monthly cost of retiring the debt with a sinking fund. Step 1: Calculate the part of the monthly payment that covers the semiannual interest payments Use an ordinary annuity with 6 payments. The semiannual interest payment is the future value of this annuity 15

16. Math 1140 - Financial Mathematics A church uses a bond issue to borrow $400,000 for a building program. The bonds pay 7%(2) and mature in 15 years. In order to redeem the bonds in 15 years, the church sets up a sinking fund with a local bank that pays 6%(12) on deposits. Find the monthly cost of retiring the debt with a sinking fund. The semiannual interest payment is $400,000×0.07/2=$14,000 The monthly payment covering the interest is R=$2,304.34 16

17. Math 1140 - Financial Mathematics A church uses a bond issue to borrow $400,000 for a building program. The bonds pay 7%(2) and mature in 15 years. In order to redeem the bonds in 15 years, the church sets up a sinking fund with a local bank that pays 6%(12) on deposits. Find the monthly cost of retiring the debt with a sinking fund. Step 2: Calculate the monthly payment needed to cover the $400,000 in 15 years These payments form an annuity with monthly payments for 15 years, whose maturity value is $400,000. R=$1,375.43 17

18. Math 1140 - Financial Mathematics A church uses a bond issue to borrow $400,000 for a building program. The bonds pay 7%(2) and mature in 15 years. In order to redeem the bonds in 15 years, the church sets up a sinking fund with a local bank that pays 6%(12) on deposits. Find the monthly cost of retiring the debt with a sinking fund. The monthly payment must cover -the semiannual interest payments, every six months -the $400,000 needed to redeem the bonds in 15 years The monthly payment is $2,304.34 + $1,375.43 18

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20. Math 1140 - Financial Mathematics Find the total periodic cost for borrowing $56,400 for 2 years and 6 months at 7%(4). Use an amortized loan and a sinking fund that earns 6%(12). Step 1: Amortized loan The payments are made quarterly. P=$56,400 n=4×2+2=10 i=0.07/4 R=$6196.97 20

21. Math 1140 - Financial Mathematics Find the total periodic cost for borrowing $56,400 for 2 years and 6 months at 7%(4). Use an amortized loan and a sinking fund that earns 6%(4). Step 2: Sinking fund The total periodic payment is the sum of the interest charge and the fund payment. The interest per quarter is I=Pi I=$56,400×0.06/4 = $5,269 21

22. Math 1140 - Financial Mathematics Find the total periodic cost for borrowing $56,400 for 2 years and 6 months at 7%(4). Use an amortized loan and a sinking fund that earns 6%(4). Step 2: Sinking fund The monthly payment covering the $56,400 due in 2 years and 2 months. S=$56,400 n=4×2+2=10 i=0.06/4 R=$987 22

23. Math 1140 - Financial Mathematics Find the total periodic cost for borrowing $56,400 for 2 years and 6 months at 7%(4). Use an amortized loan and a sinking fund that earns 6%(4). Step 2: Sinking fund The total periodic payment is the sum of the interest charge and the fund payment. $5,269.69+$987=$6,256 Sinking fund costs $6,256- $6,196.97 = $59.72 more per quarter. 23

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25. Math 1140 - Financial Mathematics Find the total periodic charge to retire a $38,600 debt with quarterly payments for 5 years, using the following plans: a)Amortized at 8%(4) b)Loan at 8%(4) with a sinking fund at 5%(4) c)Loan at 8%(4) with a sinking fund at 8%(4) d)Loan at 8%(4) with a sinking fund at 9%(4) 25

26. Math 1140 - Financial Mathematics Find the total periodic charge to retire a $38,600 debt with quarterly payments for 5 years, using the following plans: a)Amortized at 8%(4) The periodic payment is given by the formula P = $38,600 i = 0.08/4 n = 5×4 R=$2,360.65 26

27. Math 1140 - Financial Mathematics Find the total periodic charge to retire a $38,600 debt with quarterly payments for 5 years, using the following plans: b) Loan at 8%(4) with a sinking fund at 5%(4) The total periodic charge is given by formula P = $38,600 i = 0.08/4 j = 0.05/4 n = 5×4 R=$2,482.77 27

28. Math 1140 - Financial Mathematics Find the total periodic charge to retire a $38,600 debt with quarterly payments for 5 years, using the following plans: c) Loan at 8%(4) with a sinking fund at 8%(4) The total periodic charge is given by formula P = $38,600 i = 0.08/4 j = 0.08/4 n = 5×4 R=$2,360.65 28

29. Math 1140 - Financial Mathematics Find the total periodic charge to retire a $38,600 debt with quarterly payments for 5 years, using the following plans: d) Loan at 8%(4) with a sinking fund at 9%(4) The total periodic charge is given by formula P = $38,600 i = 0.08/4 j = 0.09/4 n = 5×4 R=$2,321.49 29

30. Math 1140 - Financial Mathematics Find the total periodic charge to retire a $38,600 debt with quarterly payments for 5 years, using the following plans: a) Amortized at 8%(4) Answer: $2,360.65 b) Loan at 8%(4) with a sinking fund at 5%(4) Answer: $2,482.77 c) Loan at 8%(4) with a sinking fund at 8%(4) Answer: $2,360.65 d) Loan at 8%(4) with a sinking fund at 9%(4) Answer: $2,321.49 30

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32. Math 1140 - Financial Mathematics Due today Homework 11 Project Progress Report (one per team) Team member evaluation (one per student) by email or in my mailbox by 5pm. Monday Read sections 6.5, 6.6 and 6.7 Dec 8 FINAL!!!! Charge 32