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Interest Rates
Empirical Properties
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The Nominal Interest Rate
l Suppose you take out a Rs.1000 loan today.You agree to repay the loan with a Rs.1050payment in one year.l Interest = Payment (Face Value) Principal (Price)
l Interest = Rs.1,050 - Rs.1,000 = Rs.50
l Interest Rate = (Interest/Principal)l Interest Rate = (Rs.50)/(Rs.1,000) = .05 (5%) Per Yearl This is the one year spot rate
l INTEREST RATES ALWAYS HAVE A TIME PERIODASSOCIATED WITH THEM!!!
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Annualizing
l Suppose that you invest Rs.1 at a quarterlyinterest rate of 2%. What is your annual return?
Rs.1 Rs.1.02 Rs.1.04 Rs.1.06 Rs.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
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Annualizing
l Suppose you earn a cumulative interest rate of 5% over a4 year period. What is your annualized return?
Rs.1 Rs.?? Rs.?? Rs.?? Rs.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 jIndiat divide by 4
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The Yield Curve
l Spot Rates are interest rates charged for loans contractedtoday: S(1), S(2), S(3), etc
l The Yield curve is a listing of current spot rates for
different maturities (on an annualized basis)
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Forward Rates
l Forward rates are interest rates for contracts to be writtenin the future. (F)
l F(1,1) = Interest rate on 1 year loans contracted 1year from now
l F(1,2) = Interest rate on 2 yr loans contracted 1year
l from nowl F(2,1) = interest rate on 1 year loans contracted 2
years from nowl S(1) = F(0,1)
l Forward rates are not explicitly stated, but are impliedthrough observed spot rates
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Calculating Forward Rates
l The current annual yield on a 1 yr Treasury is 2.0% whilea 2 yr Treasury pays an annual rate of 2.6%
l Rs.1(1.02) = Rs.1.02 (Rs.1 invested for 1 year)
l Rs.1(1.026)(1.026) = Rs.1.053 (invested for two years)
l (Rs.1.02)(1+F(1,1)) = Rs.1.053
l Therefore, the implied return from the 1st year to thesecond is
Rs.1.053/Rs.1.02 = 1.032 = F(1,1) = 3.2%
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Calculating Forward Rates
l The current annual yield on a 2 yr Treasury is 2.6% whilea 3 yr Treasury pays an annual rate of 2.9%
l Rs.1(1.026)(1.026) = Rs.1.053 (invested for two years)
l Rs.1(1.029)(1.029)(1.029) = Rs.1.09 (invested for 3years)
l (Rs.1.053)(1+F(2,1)) = Rs.1.09
l Therefore, the implied return from the 2nd year to thethird is
Rs.1.09/Rs.1.053 = 1.035 = F(2,1) = 3.5%
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Spot Rates & Bond Prices
l Zero Coupon (Discount) Bonds are convenientbecaIndiae they only involve one payment.l Maturity date (Term)
l
Face Value (Assume Rs.100)
l A 90 Day T-Bill is currently selling for Rs.99.70
l Yield (Yield to Maturity) = (Rs.100 - Rs.99.70)/Rs.99.70 = .
003 (.3%)l Annualized YTM = (1.003)^(365/90) = 1.012 (1.2%)
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Spot Rates & Bond Prices
l STRIPS (Separately Traded Registered Interestand Principal) were created by the Treasurydepartment in 1985.l
Maturity date (Term)l Face Value (Assume Rs.100)
l A 10 Yr. STRIP is selling for Rs.63.69
l YTM = (Rs.100 - Rs.63.69)/Rs.63.69 = .5701 (57.01%)l Annual YTM = (1.5701)^(.1) = 1.0461 (4.61%)
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Forward Rates and Bond Prices
l STRIP prices also imply forward rates
l An AugIndiat 2015 STRIP is currently selling for
Rs.63.55 while an AugIndiat 2014 STRIP is selling forRs.68.07.
l F(9,1) = Rs.68.07/Rs.63.55 = 1.07 = 7%
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Interest Rates & Bond Prices
l Consider a 1 year,Rs.100 discount bondwith a price of
Rs.98.00
i = (Rs.100 Rs.98.00) *100=2%
Rs.98.00
l Now, consider thesame 1 year, Rs.100discount bond with a
price of Rs.94.00
i = (Rs.100 Rs.94.00) *100 =6.4%
Rs.94.00
Higher bond prices are associated with Lower Returns!!
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Interest Rates & Bond Prices
l Whats the difference between a bondprice and an interest rate?
l They are both relative pricesl Interest Rate = Price of a current Rs. in terms
of foregone future dollars.
l Bond Price = Price of a Future Rs. in terms offoregone current dollars
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Interest Rates in the India (1984 2004)
0
2
4
6
8
10
12
14
1/1/84 1/1/89 1/1/94 1/1/99 1/1/04
1 YR TBILL
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1 Year Treasury Rate
0
24
6
8
1012
14
16
18
1/1/59 1/1/64 1/1/69 1/1/74 1/1/79 1/1/84 1/1/89 1/1/94 1/1/99 1/1/04
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Interest Rates in the India
Term FederalFunds
1Yr TBill 5 Yr. TBill 10 Yr. TBill
Mean 5.88
Std. Dev. 2.98
Corr (+1) .988
Corr (+2) .968
Corr (+3) .949
Corr (+4) .934
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Interest Rates in the India
02
4
6
810
12
14
16
1/1/84 1/1/89 1/1/94 1/1/99 1/1/04
1 YR 5 YR 10 YR Fed Funds
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Interest Rates in the India
Term FederalFunds
1Yr 5 Yr. 10 Yr.
Mean 5.80 5.88 6.49 6.69
Std. Dev. 3.39 2.98 2.75 2.68
Corr (+1) .986 .988 .992 .994
Corr (+2) .961 .968 .979 .985
Corr (+3) .937 .949 .968 .976
Corr (+4) .915 .934 .957 .969
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Correlations
1YRTB 5YRTB 10YRTB FF
1YRTB 1
5YRTB 0.966104 110YRTB 0.934983 0.993211 1
FF 0.973375 0.914724 0.879391 1
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Interest Rates
l Mean reverting (stationary)
l Long term rates are less volatile than shortterm rates
l Long term rates show more persistencethan short term rates
l High degree of persistence
l Highly correlated with one another (longrates less correlated with shorter rates)
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Interest Rates & Inflation
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Interest Rates & Inflation
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Interest Rates & Inflation
l Inflation rates are highly correlated with interestrates (less so for longer term rates)
MEAN (Inflation Rate) 3.90STDEV (Inflation Rate) 3.6746435
Corr(FF) 0.5899089
Corr(1YRTB) 0.5552795
Corr(5YRTB) 0.4879992
Corr(10YRTB) 0.4666077
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Characteristics of BIndiainessCycles
l All recessions/expansions look similar, that is, thereseems to be consistent statistical relationships betweenGDP and the behavior of other economic variables.
l Correlation (procyclical, countercyclical)
l Timing (leading, coincident, lagging)l Relative Volatility
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Interest Rates vs. GDP
l Nominal Interest Rates tend to be Procyclical and lagging
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Interest Rates vs. Money
l Interest rates tend to be negatively correlated withchanges in money (in the short run)
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Nominal vs. Real Interest Rates
l A Rs.1000 investment at a 10% annual interest rate willpay out Rs.1100 in one year.
l Nominal Return (i) = (Rs.1100 - Rs.1000)/Rs.1000 = .10
(10%)
or
(1+i) = Rs.1100/Rs.1000 = 1.10
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Nominal vs. Real Interest Rates
l A Rs.1000 investment at a 10% annual interest rate willpay out Rs.1100 in one year. To get a real (inflationadjIndiated) returns, we mIndiat divide by the price level(current and future)
l Real Return (r) = ((Rs.1100/P) (Rs.1000/P))/(Rs.1000/P)
or
(1+r) = (Rs.1100/Rs.1000)/(P/P)
(1+r) = (1+i) / (1+ inflation rate)
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Nominal vs. Real Interest Rates
l A Rs.1000 investment at a 10% annual interest rate willpay out Rs.1100 in one year. To get a real (inflationadjIndiated), we mIndiat divide by the price level (currentand future).
l Suppose that the inflation rate is equal to 5% annually
l Real Return (1+r ) = (1.10) / (1.05) = 1.048%
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An Easy Approximation
l We have the following:
(1+i) = (1+r)(1+inflation)
(1+i) = 1 + r + inflation + r*inflation
i = r + inflation. + r*inflation ( Indiaually r*inf is small)
Ex) r = 10% - 5% = 5%
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Real Interest Rates: 1975-1985
l Why would anyone accept a negative real rate of return?
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Ex Ante. Vs. Ex Post
l Ex Ante real interest rates are the ratesinvestors expect based on anticipatedinflation rates
l Ex Post real interest rates are the ratesinvestors actually receive after the fact.
l The difference between the two depends
on the accuracy of inflationaryexpectations
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Inflation Expectations
-10
-5
0
5
10
15
20
1/1/19
78
7/1/19
78
1/1/19
79
7/1/19
79
1/1/19
80
7/1/19
80
1/1/19
81
7/1/19
81
1/1/19
82
7/1/19
82
1/1/19
83
7/1/19
83
1/1/19
84
7/1/19
84
1/1/19
85
Expected
Actual
Real Rate
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Inflation Expectations and RealReturns
l Inflation expectation tend to be quitepersistent (i.e. investors dont seem toupdate to new information). Therefore,
real interest rates also have a high degreeof persistence.