1. Interest, Discount & Return Rates

2. Learning Objectives  Present and future value  Discount rates  Rate compounding  Nominal and effective rates  Interest rates  Inflation adjustments  Nominal and real rates  Return rates  Simple and natural log  Mean return rates  Arithmetic and Geometric 2

3. Present Value: No Intermediate Cash Flow 3 0 1 2 N PV FV FV: Future value PV: Present value k: effective periodic discount or future value rate N: number of periods : Discount factor : Future value factor

4. Present Value w/ No Intermediate Cash Flow  Example  k = annual effective discount rate = 5.116%  N = 5 years  PV =$100.00  FV = PV·(1+.05116) 5 = $128.33 i=0 1 2 3 4 5 PV FV 4

5. Present Value w/ periodic compounding and no intermediate cash flow  Annual effective rate includes effect of periodic compounding  Annual nominal rate does not include effect of periodic compounding  Example  5% annual compounded monthly k = 5%, annual nominal rate m = 12, compounding frequency  Annual effective rate is  N is number of years  Effective and nominal monthly rate 5 Using annual nominal rate Using annual effective rate

6. k i is effective annual rate k j is nominal annual rate Present Value w/ periodic compounding and intermediate cash flow 6 i 0 1 2 m·N PV CF i

7. Interest Rates  Rate of return on debt securities  Bonds Fixed ‘coupon’ rate  Certificates of deposit  Notes Floating rate  Mortgages  Commercial paper 7 Govt Rates BLS CPI BLS CPI Chart BLS FAQs CD Rates

8. Interest Rates BBA Libor rates for 1 June April 2011BBA Libor rates for 1 June April 2011 (Simple annual rates) s/n is spot/next and o/n is overnight

9. Inflation Adjustments  n = nominal rate  r = real rate  i = inflation rate  Example  n=3%  i=2%  r =0.98%  1%  Cash flows and discount rates must be congruent  Nominal is typical 9

10. Continuous Compounding 10 k is annual nominal rate, m is number of compounding periods per year 5% annual nominal rate is e.05 – 1 continuously compounded annual effective rate: 5.1271%

11. Continuous Compounding 11 FV = PV · e k k = 5% e k is the future value factor e.05 = 1.051271 e -k is the discount factor e -.05 = 0.951229 e k -1 is the continuously compounded rate e.05 -1 = 0.051271 Now utilize with the stock price model Continuously compounded future value factor Natural log rate of return

12. Mean Rate: Simple Return Rates 12 What’s the average or mean quarterly simple rate of return? Example: Quarterly historical price record for 1 year Compute the sequence of simple rates of return from security price, S n = number of periods in a historical return record, associated with n+1 prices m = number of periods in a year

13. Mean Rate: Simple Return Rates 13 No, it over estimates the price What’s the mean rate of return that results in the actual price, S 4 ? Does this mean rate over 4 quarters reproduce the stock price at the end of 1 year ? That’s the geometric mean rate of return, g

14. Mean Rate: Simple Return Rates a is the periodic (e.g., quarterly) arithmetic mean rate of return g is the periodic (e.g., quarterly) geometric mean rate of return ‘Periodic’ herein means daily, weekly, monthly, quarterly, but not annual So how do we time scale these periodic mean return rates? For example: Scale the quarterly mean rates to an annual mean return Via multiplication ? Via compounding NO But compounding the geometric mean rate does produce the annual rate – by definition - but ignores the intermediate rate fluctuations

15. Mean Rate: Log Return Rates 15 The periodic mean natural log return rate is Now the natural log rate of return Multiply the quarterly natural log mean return rate by 4 to get the annual log mean return rate?

16. Mean Rate of Return 16 Now check whether the natural log mean return rate reproduces the year end stock price Annual and other accumulated rates of return can be determined by multiplying the log mean periodic rate of return

17. Another Example 17

18. SPX (^GSPX) Monthly Prices: 1950 - 2012 18 753 Monthly Prices January 1950 to September 2012

19. SPX Monthly Return Rates: 1950 - 2012 19 752 monthly return rates January 1950 to September 2012

20. SPX Monthly Return Rates: 1950 - 2012 20

21. Simple mean rate of return Future value factor Natural log mean rate of return SPX Monthly Ln Return Rates: 1950 - 2012 21

22. SPX Monthly Mean Rates: 1950 - 2011 22

23. SPX Monthly Prices: 1950 - 2011 23

24. SPX Monthly Variance Rates: 1950 - 2011 24

25. Next Topics Building These Concepts  Equities  Stochastic Processes Martingales  PDF (probability density function) for future value factors Central Limit Theorems  PDF for return rates  Convert statistics between return rate PDFs  Bonds  Yield to maturity  Mortgages  etc 25