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

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Presentation transcript:

Mathe III Lecture 3 Mathe III Lecture 3 Mathe III Lecture 3 Mathe III Lecture 3

2 Mortgage Repayments Outstanding Balance At time t Repayment per period

3 Mortgage Repayments

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

5 We found that: and:

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

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

8 Linear Equations with a Variable Coefficient

9

10 example The solution to: is: or:

11 The general solution becomes:

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

13

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

15 Second Order Equations etc.

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

17 Example

18 Example

19 Example

20 Linear Equations The associated homogeneous equation:

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

22 Linear Equations

23 Linear Equations

24 Linear Equations

25 Linear Equations

26 Linear Equations

27 We have shown that:

28 Proof: Lemma:

29 Proof (cntd.)

30 Proof (cntd.)