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Interest Rates: Key Concepts and Calculations

Explore the empirical properties of interest rates, including spot rates, forward rates, bond prices, and their relationship with inflation. Learn to calculate annualized returns and uncover the dynamics of interest rate movements.

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Interest Rates: Key Concepts and Calculations

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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.

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