Lecture 3

1. Lecture 3

2. Sinking Fund Method of Valuation Pay an amount P for an annuity of K level payments over n periods Receive periodic return of i per period, value P*i Excess of K-Pi reinvested at a rate j into a “sinking fund” Recursive equation of

3. Loan Repayment Imagine one takes out a loan of value L, and have periodic payments Ki over a period of n total payments This has present value L :

4. Amortization In this method, we track the outstanding balance on a loan, denoted by OB(i) or OBi

5. Splitting into Interest and Principal Paid Interest paid at time t+1 is Principal repaid at time t+1 is Total Interest and Cash paid are , respectively

6. Retrospective Form Outstanding Balance:

7. Prospective Form Outstanding Balance at time t is present value of all future payments until loan is paid off

8. Loan With Level Payments of Principal Ex 3.2: A loan of 3000 at an effective quarterly rate of j =0.02 is amortized by means of 12 quarterly payments, beginning one quarter after the loan is made. Each payment consists of a principal repayment of 250 plus interest due on the previous quarter’s outstanding balance. Construct the amortization schedule.

9. Ex 3.2

10. Amortization of a Loan with Level Repayments In this case, which is common, the payments are held constant, say at K, but the proportion of interest paid versus principal paid varies with time. Examples such as mortgages, car payments, etc.. Can use retrospective method to value outstanding balance at time t.

11. Amortization with Level Payments

12. Graph of Outstanding Balance

13. The Sinking Fund Method of Loan Repayment Make level, periodic payments of interest only on an outstanding loan at rate i : L*i At the end of loan, pay back L in one lump sum Can set aside a bit extra every period to pay off this lump sum: add to an account earning interest at rate j Usually have i > j

14. Sinking Fund Method continued Even though the outstanding balance is paid off only at the end of the term of the loan, we can still interpolate and say that the outstanding balance decreases as wherein the balance decreases by the amount paid into the account earning at a rate j

15. Sinking Fund Method continued Note that we can also calculate the net interest payment each month via

16. Makeham’s Formula Consider the following scenario: An investor makes a loan at interest rate i, with only interest payments for the term of the loan, and then a lump sum L at the end of term. The investor then sells the loan to a speculator, who values the expected cash flows at a rate j. Then the value the speculator puts on the investment is

17. Makeham’s Formula ctd This idea generalizes to the case where the principal L is paid over a series of lump sum payments: Each of these lump sum payments can be viewed as corresponding to an individual loan:

18. Makeham’s Formula ctd We can find the total value now by summing up the individual loan values: