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## Interest Rates

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**Interest Rates**Empirical Properties**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!!!**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**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**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)**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**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%**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%**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%)**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%)**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%**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!!**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**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)**Interest Rates & Inflation**• Inflation rates are highly correlated with interest rates (less so for longer term rates)**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**Interest Rates vs. GDP**• Nominal Interest Rates tend to be Procyclical and lagging**Interest Rates vs. Money**• Interest rates tend to be negatively correlated with changes in money (in the short run)**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**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)**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%**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%**Real Interest Rates: 1975-1985**• Why would anyone accept a negative real rate of return?**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**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.