Interest Rates Interest Rates

Empirical PropertiesEmpirical Properties

The Nominal Interest RateThe Nominal Interest Rate Suppose you take out a $1000 loan today. You Suppose you take out a $1000 loan today. You

agree to repay the loan with a $1050 payment in agree to repay the loan with a $1050 payment in one year. one year. Interest = Payment (Face Value) – Principal (Price)Interest = Payment (Face Value) – Principal (Price) Interest = $1,050 - $1,000 = $50Interest = $1,050 - $1,000 = $50

Interest Rate = (Interest/Principal)Interest Rate = (Interest/Principal) Interest Rate = ($50)/($1,000) = .05 (5%) Per YearInterest Rate = ($50)/($1,000) = .05 (5%) Per Year This is the one year spot rateThis is the one year spot rate

INTEREST RATES ALWAYS HAVE A TIME PERIOD INTEREST RATES ALWAYS HAVE A TIME PERIOD ASSOCIATED WITH THEM!!!ASSOCIATED WITH THEM!!!

AnnualizingAnnualizing Suppose that you invest $1 at a quarterly Suppose that you invest $1 at a quarterly

interest rate of 2%. What is your annual return?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

AnnualizingAnnualizing Suppose you earn a cumulative interest rate of 5% over Suppose you earn a cumulative interest rate of 5% over

a 4 year period. What is your annualized return?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 CurveThe Yield Curve

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Spot Rates are interest rates charged for loans Spot Rates are interest rates charged for loans contracted today: S(1), S(2), S(3), etc…contracted today: S(1), S(2), S(3), etc…

The Yield curve is a listing of current spot rates for The Yield curve is a listing of current spot rates for different maturities (on an annualized basis)different maturities (on an annualized basis)

Forward RatesForward Rates Forward rates are interest rates for contracts to be Forward rates are interest rates for contracts to be

written in the future. (F)written in the future. (F)

F(1,1) = Interest rate on 1 year loans contracted 1 F(1,1) = Interest rate on 1 year loans contracted 1 year from nowyear from now

F(1,2) = Interest rate on 2 yr loans contracted 1 F(1,2) = Interest rate on 2 yr loans contracted 1 yearyear

from now from now F(2,1) = interest rate on 1 year loans contracted 2 F(2,1) = interest rate on 1 year loans contracted 2

years from nowyears from nowS(1) = F(0,1)S(1) = F(0,1)

Forward rates are not explicitly stated, but are implied Forward rates are not explicitly stated, but are implied through observed spot ratesthrough observed spot rates

Calculating Forward RatesCalculating Forward Rates

The current annual yield on a 1 yr Treasury is 2.0% while The current annual yield on a 1 yr Treasury is 2.0% while a 2 yr Treasury pays an annual rate of 2.6%a 2 yr Treasury pays an annual rate of 2.6%

$1(1.02) = $1.02 ($1 invested for 1 year)$1(1.02) = $1.02 ($1 invested for 1 year) $1(1.026)(1.026) = $1.053 (invested for two years)$1(1.026)(1.026) = $1.053 (invested for two years)

($1.02)(1+F(1,1)) = $1.053($1.02)(1+F(1,1)) = $1.053

Therefore, the implied return from the 1Therefore, the implied return from the 1stst year to the year to the second is second is

$1.053/$1.02 = 1.032 = F(1,1) = 3.2%$1.053/$1.02 = 1.032 = F(1,1) = 3.2%

Calculating Forward RatesCalculating Forward Rates The current annual yield on a 2 yr Treasury is 2.6% while The current annual yield on a 2 yr Treasury is 2.6% while

a 3 yr Treasury pays an annual rate of 2.9%a 3 yr Treasury pays an annual rate of 2.9%

$1(1.026)(1.026) = $1.053 (invested for two years)$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(1.029)(1.029)(1.029) = $1.09 (invested for 3 years)

($1.053)(1+F(2,1)) = $1.09($1.053)(1+F(2,1)) = $1.09

Therefore, the implied return from the 2Therefore, the implied return from the 2ndnd year to the year to the third is third is $1.09/$1.053 = 1.035 = F(2,1) = 3.5%$1.09/$1.053 = 1.035 = F(2,1) = 3.5%

Spot Rates & Bond PricesSpot Rates & Bond Prices

Zero Coupon (Discount) Bonds are convenient Zero Coupon (Discount) Bonds are convenient because they only involve one payment.because they only involve one payment. Maturity date (Term)Maturity date (Term) Face Value (Assume $100)Face Value (Assume $100)

A 90 Day T-Bill is currently selling for $99.70 A 90 Day T-Bill is currently selling for $99.70 Yield (Yield to Maturity) = ($100 - $99.70)/$99.70 = .003 Yield (Yield to Maturity) = ($100 - $99.70)/$99.70 = .003

(.3%)(.3%) Annualized YTM = (1.003)^(365/90) = 1.012 (1.2%)Annualized YTM = (1.003)^(365/90) = 1.012 (1.2%)

Spot Rates & Bond PricesSpot Rates & Bond Prices

STRIPS (STRIPS (SSeparately eparately TTraded raded RRegistered egistered IInterest nterest and and PPrincipal) were created by the Treasury rincipal) were created by the Treasury department in 1985.department in 1985. Maturity date (Term)Maturity date (Term) Face Value (Assume $100)Face Value (Assume $100)

A 10 Yr. STRIP is selling for $63.69A 10 Yr. STRIP is selling for $63.69 YTM = ($100 - $63.69)/$63.69 = .5701 (57.01%)YTM = ($100 - $63.69)/$63.69 = .5701 (57.01%) Annual YTM = (1.5701)^(.1) = 1.0461 (4.61%)Annual YTM = (1.5701)^(.1) = 1.0461 (4.61%)

Forward Rates and Bond PricesForward Rates and Bond Prices

STRIP prices also imply forward rates…STRIP prices also imply forward rates…

An August 2015 STRIP is currently selling for $63.55 An August 2015 STRIP is currently selling for $63.55 while an August 2014 STRIP is selling for $68.07. while an August 2014 STRIP is selling for $68.07.

F(9,1) = $68.07/$63.55 = 1.07 = 7%F(9,1) = $68.07/$63.55 = 1.07 = 7%

Interest Rates & Bond PricesInterest Rates & Bond Prices

Consider a 1 year, $100 Consider a 1 year, $100 discount bond with a price discount bond with a price of $98.00of $98.00

i = i = ($100 – $98.00)($100 – $98.00) *100 =2% *100 =2%

$98.00$98.00

Now, consider the same 1 Now, consider the same 1 year, $100 discount bond year, $100 discount bond with a price of $94.00with a price of $94.00

i = i = ($100 – $94.00)($100 – $94.00) *100 = 6.4% *100 = 6.4%

$94.00$94.00

Higher bond prices are associated with Lower Returns!!

Interest Rates & Bond PricesInterest Rates & Bond Prices

What’s the difference between a bond What’s the difference between a bond price and an interest rate? price and an interest rate?

They are both They are both relative pricesrelative prices Interest Rate = Price of a current $ in terms of Interest Rate = Price of a current $ in terms of

foregone future dollars.foregone future dollars.Bond Price = Price of a Future $ in terms of Bond Price = Price of a Future $ in terms of

foregone current dollars foregone current dollars

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

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1 Year Treasury Rate1 Year Treasury Rate

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Interest Rates in the USInterest Rates in the US

TermTerm Federal Federal FundsFunds

1Yr TBill 1Yr TBill 5 Yr. 5 Yr. TBillTBill

10 Yr. 10 Yr. TBillTBill

MeanMean 5.885.88

Std. Dev.Std. Dev. 2.982.98

Corr (+1)Corr (+1) .988.988

Corr (+2)Corr (+2) .968.968

Corr (+3)Corr (+3) .949.949

Corr (+4)Corr (+4) .934.934

Interest Rates in the USInterest Rates in the US

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Interest Rates in the USInterest Rates in the US

TermTerm Federal Federal FundsFunds

1Yr 1Yr 5 Yr.5 Yr. 10 Yr. 10 Yr.

MeanMean 5.805.80 5.885.88 6.496.49 6.696.69

Std. Dev.Std. Dev. 3.393.39 2.982.98 2.752.75 2.682.68

Corr (+1)Corr (+1) .986.986 .988.988 .992.992 .994.994

Corr (+2)Corr (+2) .961.961 .968.968 .979.979 .985.985

Corr (+3)Corr (+3) .937.937 .949.949 .968.968 .976.976

Corr (+4)Corr (+4) .915.915 .934.934 .957.957 .969.969

CorrelationsCorrelations

  1YRTB 5YRTB 10YRTB FF

1YRTB 1  

5YRTB 0.966104 1  

10YRTB 0.934983 0.993211 1  

FF 0.973375 0.914724 0.879391 1

Interest RatesInterest Rates

Mean reverting (stationary)Mean reverting (stationary)Long term rates are less volatile than short Long term rates are less volatile than short

term ratesterm ratesLong term rates show more persistence Long term rates show more persistence

than short term ratesthan short term ratesHigh degree of persistenceHigh degree of persistenceHighly correlated with one another (long Highly correlated with one another (long

rates less correlated with shorter rates)rates less correlated with shorter rates)

Interest Rates & InflationInterest Rates & Inflation

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Interest Rates & InflationInterest Rates & Inflation

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Interest Rates & InflationInterest Rates & Inflation

Inflation rates are highly correlated with interest Inflation rates are highly correlated with interest rates (less so for longer term rates)rates (less so for longer term rates)

MEAN (Inflation Rate) 3.90

STDEV (Inflation Rate) 3.6746435

Corr(FF) 0.5899089

Corr(1YRTB) 0.5552795

Corr(5YRTB) 0.4879992

Corr(10YRTB) 0.4666077

Characteristics of Business Characteristics of Business CyclesCycles

All recessions/expansions “look similar”, that is, there All recessions/expansions “look similar”, that is, there seems to be consistent statistical relationships between seems to be consistent statistical relationships between GDP and the behavior of other economic variables.GDP and the behavior of other economic variables.

Correlation (procyclical, countercyclical)Correlation (procyclical, countercyclical)Timing (leading, coincident, lagging)Timing (leading, coincident, lagging)Relative VolatilityRelative Volatility

Interest Rates vs. GDPInterest Rates vs. GDP

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Nominal Interest Rates tend to be Procyclical and laggingNominal Interest Rates tend to be Procyclical and lagging

Interest Rates vs. MoneyInterest Rates vs. Money

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Interest rates tend to be negatively correlated with Interest rates tend to be negatively correlated with changes in money (in the short run)changes in money (in the short run)

Nominal vs. Real Interest RatesNominal vs. Real Interest Rates

A $1000 investment at a 10% annual interest rate will A $1000 investment at a 10% annual interest rate will pay out $1100 in one year. pay out $1100 in one year.

Nominal Return (i) = ($1100 - $1000)/$1000 = .10 (10%)Nominal Return (i) = ($1100 - $1000)/$1000 = .10 (10%)

or or

(1+i) = $1100/$1000 = 1.10(1+i) = $1100/$1000 = 1.10

Nominal vs. Real Interest RatesNominal vs. Real Interest Rates A $1000 investment at a 10% annual interest rate will A $1000 investment at a 10% annual interest rate will

pay out $1100 in one year. To get a real (inflation pay out $1100 in one year. To get a real (inflation adjusted) returns, we must divide by the price level adjusted) returns, we must divide by the price level (current and future)(current and future)

Real Return (r) = (($1100/P’) – ($1000/P))/($1000/P)Real Return (r) = (($1100/P’) – ($1000/P))/($1000/P)

oror

(1+r) = ($1100/$1000)/(P’/P) (1+r) = ($1100/$1000)/(P’/P)

(1+r) = (1+i) / (1+ inflation rate)(1+r) = (1+i) / (1+ inflation rate)

Nominal vs. Real Interest RatesNominal vs. Real Interest Rates

A $1000 investment at a 10% annual interest rate will A $1000 investment at a 10% annual interest rate will pay out $1100 in one year. To get a real (inflation pay out $1100 in one year. To get a real (inflation adjusted), we must divide by the price level (current and adjusted), we must divide by the price level (current and future). future).

Suppose that the inflation rate is equal to 5% annuallySuppose that the inflation rate is equal to 5% annually

Real Return (1+r ) = (1.10) / (1.05) = 1.048%Real Return (1+r ) = (1.10) / (1.05) = 1.048%

An Easy ApproximationAn Easy Approximation

We have the following: We have the following:

(1+i) = (1+r)(1+inflation)(1+i) = (1+r)(1+inflation)

(1+i) = 1 + r + inflation + r*inflation(1+i) = 1 + r + inflation + r*inflation

i = r + inflation. + r*inflation ( usually r*inf is small)i = r + inflation. + r*inflation ( usually r*inf is small)

Ex) r = 10% - 5% = 5%Ex) r = 10% - 5% = 5%

Real Interest Rates: 1975-1985Real Interest Rates: 1975-1985

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Why would anyone accept a negative real rate of return?Why would anyone accept a negative real rate of return?

Ex Ante. Vs. Ex PostEx Ante. Vs. Ex Post

Ex Ante real interest rates are the rates Ex Ante real interest rates are the rates investors expect based on anticipated investors expect based on anticipated inflation ratesinflation rates

Ex Post real interest rates are the rates Ex Post real interest rates are the rates investors actually receive after the fact.investors actually receive after the fact.

The difference between the two depends The difference between the two depends on the accuracy of inflationary on the accuracy of inflationary expectations expectations

Inflation ExpectationsInflation Expectations

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Inflation Expectations and Real Inflation Expectations and Real ReturnsReturns

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