interest rates 2

8/3/2019 Interest Rates 2

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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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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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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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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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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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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

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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

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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.

interest rates 2

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