1. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value and Compounding –After 1 year, your CD is worth $2,500 (1+0.0675) = $2,668.75 –After 2 years, the CD is worth $2,668.75 (1+0.0675) = $2,500 (1+0.0675) 2 = $2,848.89 –After 3 years, the CD is worth $2,848.89 (1+0.0675) = $2,500 (1+0.0675) 2 = 3,041.19 Suppose you have $2,500 that you can put in a three- year bank CD yielding 6.75% annually. How much money will you have when this CD matures?

2. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value and Compounding More generally, FV = PV (1+i) n, where FV = the future value of a lump sum PV = the initial principal, or present value of the lump sum i = the annual interest rate n = the number of years interest compounds

3. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value and Compounding A good way of understanding this process is through the use of a time line: 012 3 i = 6.75% PV = –2500 FV = 2500 (1.0675) 3 = 3041.19

4. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value and Compounding How much would this CD be worth at maturity if interest compounds quarterly? –The trick is to convert the interest rate into a periodic rate and compound each period, rather than annually. FV = PV (1+i/m) n  m, where m is the number of periods per year. –FV = $2,500(1+0.0675/4) 3  4 = $3,055.98.

5. CIMABusiness MathematicsMr. Rajesh Gunesh Present Value and Discounting Suppose you will receive $5,000 three years from now. If you can earn 4.5% on your savings, how much is this worth to you today? 012 3 i = 4.5% FV = 5000PV = ? $5,000 = PV (1+0.045) 3  PV = $5,000 / (1+0.045) 3 = $4,381.48

6. CIMABusiness MathematicsMr. Rajesh Gunesh Present Value and Discounting More generally, PV = FV / (1+i) n, where FV = the future value of a lump sum PV = the initial principal, or present value of the lump sum i = the annual discount rate n = the number of years

7. CIMABusiness MathematicsMr. Rajesh Gunesh Present Value and Discounting If discounting occurs at a frequency other than annually: PV = FV / (1+i/m) n  m, where m = the number of discounting periods per year

8. CIMABusiness MathematicsMr. Rajesh Gunesh Annuities An annuity is a series of payments or receipts made at regular intervals for a determined period of time 012 3 i PMT PV = ? FV = ?

9. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value of an Annuity If you will receive $100 at the end of each of the next 3 years and can invest it at 9%, how much will it be worth at the end of the 3 years? 012 3 9% 100 327.81 + 100(1.09) 2 = 118.81 + 100(1.09) = 109.00 = 100.00

10. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value of an Annuity More generally, FV = PMT + PMT(1+i) + PMT(1+i) 2 + PMT(1+i) 3 + … + PMT(1+i) n–1 n–1 = PMT  (1+i) n–t–1 t=0 = PMT [(1+i) n – 1] / i

11. CIMABusiness MathematicsMr. Rajesh Gunesh Future Value of an Annuity If compounding occurs at a frequency other than annually, FV = PMT [(1+i/m) n  m – 1] / (i/m)

12. CIMABusiness MathematicsMr. Rajesh Gunesh Present Value of an Annuity How much is this $100, 3-year annuity worth today, assuming a 9% discount rate? 012 3 9% 100 253.13 91.74 = 100 / (1.09) 84.17 = 100 / (1.09) 2 77.22 = 100 / (1.09) 3

13. CIMABusiness MathematicsMr. Rajesh Gunesh Present Value of an Annuity More generally, PV = PMT / (1+i) + PMT / (1+i) 2 + PMT / (1+i) 3 + … + PMT / (1+i) n n = PMT  1 / (1+i) t t=1 = PMT [1 – 1 / (1+i) n ] / i

14. CIMABusiness MathematicsMr. Rajesh Gunesh Present Value of an Annuity If compounding occurs at a frequency other than monthly, PMT [1 – 1 / (1+i/m) n  m ] / (i/m)

15. CIMABusiness MathematicsMr. Rajesh Gunesh Effective Annual Rates Which provides the highest total return, a savings account that pay 5.00% compounded annually or one that pays 4.75% compounded monthly? –One way to answer this is to calculate the future value of $100 invested in each PV 1 = $100 (1.05) = $105.00 PV 2 = $100 (1+0.0475/12) 12 = $104.85

16. CIMABusiness MathematicsMr. Rajesh Gunesh Effective Annual Rates Alternatively, you can calculate the effective annual rate associated with each account EAR = (1 + i/m) m – 1 –EAR 1 = (1 + 0.05/1) 1 – 1 = 0.0500 = 5.00% –EAR 2 = (1 + 0.0475/12) 12 – 1 = 0.0486 = 4.86% Effective annual rates are directly comparable in terms of total yield