CE 203 More Interest Formulas (EEA Chap 4)

1. CE 203 More Interest Formulas(EEA Chap 4)

2. Formulas for non-uniform payments • Arithmetic Gradient: payments increase by a uniform AMOUNT each payment period • Geometric Gradient: payments increase by a uniform RATE each payment period

3. Arithmetic Gradient Series A+(n-1)G Amount increases by “G” each period A+3G A+2G A+G A = (can be divided into) P

4. Arithmetic Gradient Series Components (n-1)G A A A A A 3G 2G G 0 + P’ P’’

5. Arithmetic Gradient Future Worth The Future Worth of an arithmetic gradient cash flow is given by F’’ = [ - n] (1 + i)n - 1 G (n-1)G i i 3G 2G = G [ ] (1 + i)n – in - 1 G i2 0 (see text p. 99-100 for derivation of formula) F’’

6. Arithmetic Gradient Uniform Series Factor If the arithmetic gradient factor is multiplied by the sinking fund factor, the result is called the Arithmetic Gradient Uniform Series Factor: A = G [ ] = G (A/G, i, n) (1 + i)n – in - 1 i {(1 + i)n - 1} (gives the uniform series cash flow payment equivalent to that of an arithmetic gradient cash flow) (see text p. 100 for derivation of formula)

7. In-class Example 1 (arithmetic series) Your company just purchased a piece of equipment. Maintenance costs are estimated at $1200 for the first year and are expected to rise by $300 in each of the subsequent four years. How much should be set aside in a “maintenance account” now to cover these costs for the next five years? Assume payments are made at the end of each year and an interest rate of 6%.

8. Geometric Gradient Series A1(1+g)4 A1(1+g)3 A1(1+g)2 A1(1+g) A1 - Amount changes at the uniform RATE, g- Useful for some types of problems such as those involving inflation P

9. Present Worth for Geometric Gradient Series A1(1+g)4 P = A1[ ] 1 - (1 + g)n (1 + i)-n i - g A1(1+g)3 = A1 (P/A, g, i, n) A1(1+g)2 A1(1+g) for i ≠ g A1 Can be very complex unless programmed on a computer P

10. In-class Example 2 (geometric series) You have just begun you first job as a civil engineer and decide to participate in the company’s retirement plan. You decide to invest the maximum allowed by the plan which is 6% of your salary. Your company has told you that you can expect a minimum 4% increase in salary each year assuming good performance and typical advancement within the company. 1) Choose a realistic starting salary 2) Assuming you stay with the company, the company matches your 6% investment in the retirement plan, expected minimum salary increases, and an interest rate of 10%, how much will you have in your retirement account after 40 years?

11. Nominal vs. Effective interest rate Nominal interest rate, r:annual interest rate without considering the effect of any compounding at shorter intervals so that i (for use in equations) = r/m where m is number of compounding periods per year (this is what we have been doing)

12. Nominal vs. Effective interest rate Effective interest rate, ia:annual interest rate taking into account the effect of any compounding at shorter intervals; also called “yield” An amount of $1, invested at r%, compounded m times per year, would be worth $1(1 + r/m)m and the effective interest 1(1 + r/m)m - 1 ia = (1 + i)m – 1 (see text p. 110, 9th Ed for derivation)

13. Example A bank pays 6% nominal interest rate. Calculate the effective interest with a) monthly, b) daily, c) hourly d) secondly compounding ia = (1 + i)m – 1 ia monthly = (1 + .06/12)12 -1 = 6.1678 % ia daily = (1 + .06/365)365 -1 = 6.183 % ia hourly = (1 + .06/8760)8760 -1 = 6.1836 % ia secondly = (1 + .06/31.5M)31.5M -1 = 6.18365 %

14. Continuous Compounding The effective interest rate for continuous compounding(i.e., as the length of the compounding period → 0 and the number of periods →∞) ia = er – 1 (see text pp. 116-117 for derivation and equations involving continuous compounding) In our previous example, ia = e0.06 – 1 = 6.183655%

15. Nominal vs. Effective interest rate Comparison of nominal and effective interest rates for various APR values and compounding periods

16. Adjustments may be needed because… • Cash flow period is not equal to compounding period • There is cash flow in only some of periods • Pattern does not exactly fit any of basic formulas • Etc.

17. Partial review • Single payment compound amount F = P(1+i)n = P(F/P,i,n) • Uniform series (sinking fund) • Arithmetic gradient series A = F [ ] = F (A/F, i, n) i (1 + i)n - 1 A = G [ ] = G (A/G, i, n) (1 + i)n – in - 1 i {(1 + i)n - 1}