1. Mathe III Lecture 3 Mathe III Lecture 3 Mathe III Lecture 3 Mathe III Lecture 3

2. 2 Mortgage Repayments Outstanding Balance At time t Repayment per period

3. 3 Mortgage Repayments

4. 4 The loan equals the present value of T payments of z

5. 5 We found that: and:

6. 6 For t-1 this becomes: Interest on last period’s principal Payment towards the principal

7. 7 For t=1: For t=T: Interest: principal repayment: Interest: principal repayment: small large small large

8. 8 Linear Equations with a Variable Coefficient

9. 9

10. 10 example The solution to: is: or:

11. 11 The general solution becomes:

12. 12 then: ( the present value, at t = 0, of $1 at t )

13. 13

14. 14 or: the present value at period 0 the present value at period t

15. 15 Second Order Equations etc.

16. 16 Existence and Uniqueness : The equation has a unique solution for any given

17. 17 Example

18. 18 Example

19. 19 Example

20. 20 Linear Equations The associated homogeneous equation:

21. 21 Linear Equations { {{ { { { 0 0 The set of solutions is a linear space !!!

22. 22 Linear Equations

23. 23 Linear Equations

24. 24 Linear Equations

25. 25 Linear Equations

26. 26 Linear Equations

27. 27 We have shown that:

28. 28 Proof: Lemma:

29. 29 Proof (cntd.)

30. 30 Proof (cntd.)