1. Lecture 17 Proofs Compound Interest Ana Nora Evans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.edu/~ans5k/ Math 1140 Financial Mathematics

2. Math 1140 - Financial Mathematics What do you think about Wednesday’s class? A)Never do chalk again. B)I can tolerate chalk. C)I liked it. D)I want only chalk classes from now on. A)I wasn’t here. 2

3. Math 1140 - Financial Mathematics Questions About last class About homework 3

4. Math 1140 - Financial Mathematics Last time Brief review of powers, roots, exponential and logarithms. Introduction to inequalities. 4

5. Math 1140 - Financial Mathematics Exponential: 2 x The logarithm base 2 of x, denoted by log 2 x, is the number y with the property 2 y =x. 5

6. Math 1140 - Financial Mathematics Exponential: e x The logarithm base e of x, denoted by lnx, is the number y with the property e y =x. The logarithm base e of x is called the natural logarithm. 6

7. Math 1140 - Financial Mathematics Principal: P = $1 Nominal interest rate: 100%(n) The term: 1 year The amount when using n conversion periods a year S=(1+1/n) n When n gets bigger and bigger, the amount get closer and closer to e. Euler’s Constant (e) 7

8. Math 1140 - Financial Mathematics Logarithm base 5 of 5 3 is: A)0 B)1 C)3 D)5 Pledged quiz 8

9. Math 1140 - Financial Mathematics Today Homework 5. Wrap-up compound interest. 9

10. Math 1140 - Financial Mathematics T-Bills Observations The person buying the T-bill is the ‘lender’. The US Treasury is the ‘borrower’. A treasury bill has a fixed face value and term. The price is determined by an auction. 10

11. Math 1140 - Financial Mathematics Alice buys a $1,000 28-day T-Bill with a bid of 99.8. Today: Alice gives $998 to the US Treasury. 28 days from today: Alice receives $1,000 from the US Treasury. Bob borrows $998 from Shady Bank and pays back $1,000 in 28 days. Alice = Shady Bank US Treasury = Bob Example 11

12. Math 1140 - Financial Mathematics Is a high price of a T-bill good or bad for the US Government? Justify your answer with an explanation why. Fixed: the face value ( FV ), the number of days to maturity ( m ). Varies: the purchase price ( PP ) The interest charges paid by the US Treasury are: FV-PP. Homework 5 – Problem 2 12

13. Math 1140 - Financial Mathematics Given a simple interest loan, prove that when at least one partial payment is made, the balance calculated using the US Rule is higher than the balance calculated using the Merchant's Rule. Hint: Let n be the number of partial payments. First, prove it for n=1. Step 1: Understand the problem Given: P, t, i, Q, s Calculate: balance two ways. Homework 5 – Bonus Problem 1 13

14. Math 1140 - Financial Mathematics Given a simple interest loan, prove that when at least one partial payment is made the balance calculated using the US Rule is higher than the balance calculated using the Merchant's Rule. Hint: Let n be the number of partial payments. First, prove it for n=1. Step 2: Give names to the quantities in the problem. Let P be the principal, t be the term, and i be the interest rate of the simple interest loan. Let Q be the partial payment. Let s be the term from the partial payment to the due date. Homework 5 – Bonus Problem 1 14

15. Math 1140 - Financial Mathematics Given a simple interest loan, prove that when at least one partial payment is made the balance calculated using the US Rule is higher than the balance calculated using the Merchant's Rule. Hint: Let n be the number of partial payments. First, prove it for n=1. Step 3: Calculate the balance using the US Rule The balance after the first payment is -P(1+i(t-s))+Q The final balance is B US =(-P(1+i(t-s))+Q)(1+is) Homework 5 – Bonus Problem 1 15

16. Math 1140 - Financial Mathematics Given a simple interest loan, prove that when at least one partial payment is made the balance calculated using the US Rule is higher than the balance calculated using the Merchant's Rule. Hint: Let n be the number of partial payments. First, prove it for n=1. Step 3: Calculate the balance using the Merchant’s Rule The final balance is B M =-P(1+it) + Q(1+is) Homework 5 – Bonus Problem 1 16

17. Math 1140 - Financial Mathematics Given a simple interest loan, prove that when at least one partial payment is made the balance calculated using the US Rule is higher than the balance calculated using the Merchant's Rule. Hint: Let n be the number of partial payments. First, prove it for n=1. Step 3: Compare the two balances The final balance is B US =(-P(1+i(t-s))+Q)(1+is) B M =-P(1+it) + Q(1+is) Homework 5 – Bonus Problem 1 17

18. Math 1140 - Financial Mathematics B US =(-P(1+i(t-s))+Q)(1+is) B M =-P(1+it) + Q(1+is) 18

19. Math 1140 - Financial Mathematics Prove that the discount rate is smaller than the coupon equivalent. Step 1: What is coupon equivalent? Step 2: What is the problem asking? Show d < i. Homework 5 – Bonus Problem 2 19

20. Math 1140 - Financial Mathematics Prove that the discount rate is smaller than the coupon equivalent. Show d < i. More precisely: Homework 5 – Bonus Problem 2 20

21. Math 1140 - Financial Mathematics S = P(1+i) n P = S(1+i) -n Compound Interest 21

22. Math 1140 - Financial Mathematics Annual Effective Rate Given a nominal interest rate i(m), the annual effective rate is the interest rate i such that if the same principal P is deposited in two accounts: one with nominal interest rate i(m) and one with yearly interest rate i, compounded yearly; at the end of one year the two accounts have the same balance. 22

23. Math 1140 - Financial Mathematics The compounded amount formula is S = P(1+i) n The balance in an account with nominal interest rate i(m) after one year is: S = P(1 + i(m)/m) m The balance in an account with interest rate i per year, compounded yearly, after one year is S = P(1 + i) 1 23 To calculate i : P(1 + i) = P(1 + i(m)/m) m i = (1 + i(m)/m) m - 1

24. Math 1140 - Financial Mathematics Hal opened a savings account with a single deposit on June 1, 1960. On June 1, 2000, the account balance is $95,000. If the account pays an effective rate of 8% and no transactions other than the initial deposit have occurred, what was his original deposit? A)$95,000(1+0.08) 40 B)$95,000(1+0.08) -40 C)$95,000(1+0.08/12) -40 D)$95,000(1+0.08/3) -40 E)None of the above 24

25. Math 1140 - Financial Mathematics Hal opened a savings account with a single deposit on June 1, 1960. On June 1, 2000, the account balance is $95,000. If the account pays an effective rate of 8% and no transactions other than the initial deposit have occurred, what was his original deposit? The term is 40 years. The amount is $95,000. The interest rate is 8% per year compounded yearly. P = S(1+i) -n P = $95,000(1+0.08) -40 P = $4,372.94 Example 25

26. Math 1140 - Financial Mathematics Wednesday Homework 6 due Project Teams due next Friday. Charge 26