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1. Interest Rates Empirical Properties

2. The Nominal Interest Rate • Suppose you take out a \$1000 loan today. You agree to repay the loan with a \$1050 payment in one year. • Interest = Payment (Face Value) – Principal (Price) • Interest = \$1,050 - \$1,000 = \$50 • Interest Rate = (Interest/Principal) • Interest Rate = (\$50)/(\$1,000) = .05 (5%) Per Year • This is the one year spot rate • INTEREST RATES ALWAYS HAVE A TIME PERIOD ASSOCIATED WITH THEM!!!

3. Annualizing • Suppose that you invest \$1 at a quarterly interest rate of 2%. What is your annual return? \$1 \$1.02 \$1.04 \$1.06 \$1.082 X (1.02) X (1.02) X (1.02) X (1.02) (1.02)(1.02)(1.02)(1.02) = 1.082 = 8.2% Note: It is generally a safe approximation to multiply by 4

4. Annualizing • Suppose you earn a cumulative interest rate of 5% over a 4 year period. What is your annualized return? \$1 \$?? \$?? \$?? \$1.05 X (1+i) X (1+i) X (1+i) X (1+i) (1+i)(1+i)(1+i)(1+i) = 1.05 (1+i) = (1.05)^(.25) = 1.012 = 1.2% Note: Its generally a safe approximation to just divide by 4

5. The Yield Curve • Spot Rates are interest rates charged for loans contracted today: S(1), S(2), S(3), etc… • The Yield curve is a listing of current spot rates for different maturities (on an annualized basis)

6. Forward Rates • Forward rates are interest rates for contracts to be written in the future. (F) • F(1,1) = Interest rate on 1 year loans contracted 1 year from now • F(1,2) = Interest rate on 2 yr loans contracted 1 year • from now • F(2,1) = interest rate on 1 year loans contracted 2 years from now • S(1) = F(0,1) • Forward rates are not explicitly stated, but are implied through observed spot rates

7. Calculating Forward Rates • The current annual yield on a 1 yr Treasury is 2.0% while a 2 yr Treasury pays an annual rate of 2.6% • \$1(1.02) = \$1.02 (\$1 invested for 1 year) • \$1(1.026)(1.026) = \$1.053 (invested for two years) • (\$1.02)(1+F(1,1)) = \$1.053 • Therefore, the implied return from the 1st year to the second is \$1.053/\$1.02 = 1.032 = F(1,1) = 3.2%

8. Calculating Forward Rates • The current annual yield on a 2 yr Treasury is 2.6% while a 3 yr Treasury pays an annual rate of 2.9% • \$1(1.026)(1.026) = \$1.053 (invested for two years) • \$1(1.029)(1.029)(1.029) = \$1.09 (invested for 3 years) • (\$1.053)(1+F(2,1)) = \$1.09 • Therefore, the implied return from the 2nd year to the third is \$1.09/\$1.053 = 1.035 = F(2,1) = 3.5%

9. Spot Rates & Bond Prices • Zero Coupon (Discount) Bonds are convenient because they only involve one payment. • Maturity date (Term) • Face Value (Assume \$100) • A 90 Day T-Bill is currently selling for \$99.70 • Yield (Yield to Maturity) = (\$100 - \$99.70)/\$99.70 = .003 (.3%) • Annualized YTM = (1.003)^(365/90) = 1.012 (1.2%)

10. Spot Rates & Bond Prices • STRIPS (Separately Traded Registered Interest and Principal) were created by the Treasury department in 1985. • Maturity date (Term) • Face Value (Assume \$100) • A 10 Yr. STRIP is selling for \$63.69 • YTM = (\$100 - \$63.69)/\$63.69 = .5701 (57.01%) • Annual YTM = (1.5701)^(.1) = 1.0461 (4.61%)

11. Forward Rates and Bond Prices • STRIP prices also imply forward rates… • An August 2015 STRIP is currently selling for \$63.55 while an August 2014 STRIP is selling for \$68.07. • F(9,1) = \$68.07/\$63.55 = 1.07 = 7%

12. Consider a 1 year, \$100 discount bond with a price of \$98.00 i = (\$100 – \$98.00) *100 =2% \$98.00 Now, consider the same 1 year, \$100 discount bond with a price of \$94.00 i = (\$100 – \$94.00) *100 = 6.4% \$94.00 Interest Rates & Bond Prices Higher bond prices are associated with Lower Returns!!

13. Interest Rates & Bond Prices • What’s the difference between a bond price and an interest rate? • They are both relative prices • Interest Rate = Price of a current \$ in terms of foregone future dollars. • Bond Price = Price of a Future \$ in terms of foregone current dollars

14. Interest Rates in the US (1984 – 2004)

15. 1 Year Treasury Rate

16. Interest Rates in the US

17. Interest Rates in the US

18. Interest Rates in the US

19. Correlations

20. Interest Rates • Mean reverting (stationary) • Long term rates are less volatile than short term rates • Long term rates show more persistence than short term rates • High degree of persistence • Highly correlated with one another (long rates less correlated with shorter rates)

21. Interest Rates & Inflation

22. Interest Rates & Inflation

23. Interest Rates & Inflation • Inflation rates are highly correlated with interest rates (less so for longer term rates)

24. Characteristics of Business Cycles • All recessions/expansions “look similar”, that is, there seems to be consistent statistical relationships between GDP and the behavior of other economic variables. • Correlation (procyclical, countercyclical) • Timing (leading, coincident, lagging) • Relative Volatility

25. Interest Rates vs. GDP • Nominal Interest Rates tend to be Procyclical and lagging

26. Interest Rates vs. Money • Interest rates tend to be negatively correlated with changes in money (in the short run)

27. Nominal vs. Real Interest Rates • A \$1000 investment at a 10% annual interest rate will pay out \$1100 in one year. • Nominal Return (i) = (\$1100 - \$1000)/\$1000 = .10 (10%) or (1+i) = \$1100/\$1000 = 1.10

28. Nominal vs. Real Interest Rates • A \$1000 investment at a 10% annual interest rate will pay out \$1100 in one year. To get a real (inflation adjusted) returns, we must divide by the price level (current and future) • Real Return (r) = ((\$1100/P’) – (\$1000/P))/(\$1000/P) or (1+r) = (\$1100/\$1000)/(P’/P) (1+r) = (1+i) / (1+ inflation rate)

29. Nominal vs. Real Interest Rates • A \$1000 investment at a 10% annual interest rate will pay out \$1100 in one year. To get a real (inflation adjusted), we must divide by the price level (current and future). • Suppose that the inflation rate is equal to 5% annually • Real Return (1+r ) = (1.10) / (1.05) = 1.048%

30. An Easy Approximation • We have the following: (1+i) = (1+r)(1+inflation) (1+i) = 1 + r + inflation + r*inflation i = r + inflation. + r*inflation ( usually r*inf is small) Ex) r = 10% - 5% = 5%

31. Real Interest Rates: 1975-1985 • Why would anyone accept a negative real rate of return?

32. Ex Ante. Vs. Ex Post • Ex Ante real interest rates are the rates investors expect based on anticipated inflation rates • Ex Post real interest rates are the rates investors actually receive after the fact. • The difference between the two depends on the accuracy of inflationary expectations

33. Inflation Expectations

34. Inflation Expectations and Real Returns • Inflation expectation tend to be quite persistent (i.e. investors don’t seem to update to new information). Therefore, real interest rates also have a high degree of persistence.