Practice Problems on Term Structure

Transcription

1 Practice Problems on Term Structure 1- The yield curve and expectations hypothesis (30 points) Assume that the policy of the Fed is given by the Taylor rule that we studied in class, that is i t = 1.5 p t d t +1 where i t is the current Fed-funds interest rate, p t is the rate of inflation over the preceding four quarters and d t is the percentage deviation of output from the trend real GDP. As given in class the long-term interest rates are determined as averages of expected short term interest rates. The short rate, y t (1), is assumed to be the yield on a one year nominal US treasury bond. For n > 1 it follows y t (n)=[y t (1)+y t+1 (1) e +y t+2 (1) e + +y t+n-1 (1) e ]/n Note that all yields are in annual percentage terms and that n is measured in years. On January 1, 2000 the dollar prices of nominal U.S. treasury bonds (face value of 100 dollars) with maturity of one, two, and three years are q(1)=95.24$, q(2)=90.70$, and q(3)=86.38$ respectively. Further, on January 1, 2000 the dollar prices of Inflation Indexed U.S. treasury bonds (face value of 100 dollars) with maturity of one, two, and three years is Q(1)=97.09$, Q(2)=96.11$, and Q(3)=97.06$, respectively. Recall that an inflation indexed bond pays the face value multiplied by P t+n /P t where n is the maturity of the bond and P t+n is the CPI at date t+n. Also P t+n / P t = (1+pi t+1 ) * (1+pi t+2 ) * * (1+pi t+n ) In solving the problem assume that the above expression can be well approximated as P t+n / P t ~ 1+pi t+1 + pi t+2 + +pi t+n a) Compute the nominal yields on nominal US Treasury bonds on January 1, The annual percentage yields on the given nominal bonds are computed in the following way y t (n)=[(100 / q t (n)) (1/n) -1] * 100 Using this formula it follows

2 y 2000 (1)=[(100 / 95.24)-1] * 100 = 5.00% y 2000 (2)=[(100 / 90.70)^(1/2)-1] * 100 = 5.00% y 2000 (3)=[(100 / 86.38)^(1/3)-1] * 100 = 5.00% b. Compute the real yields on inflation indexed US Treasury bonds on January 1, The real yields are computed using the prices of inflation indexed Treasury bonds. r t (n)=[(100 / Q t (n)) (1/n) -1] * 100 r 2000 (1)=[(100 / 97.09)-1] * 100 = 3.00% r 2000 (2)=[(100 / 96.11)^(1/2)-1] * 100 = 2.00% r 2000 (3)=[(100 / 97.06)^(1/3)-1] * 100 = 1.00% c. Based on the above information and assuming that the expectations hypothesis determines the nominal yield curve at each period, compute the expected one year (short rate) nominal interest rate for 2001 and The expected short-term nominal yields are y 2000 (2)= [y 2000 (1) + y 2001 (1) e ]/ 2 From this we calculate y 2001 (1) e y 2001 (1) e =2* y 2000 (2) - y 2000 (1) = 5.00% By the same token we get y 2002 (1) e =3* y 2000 (3) - y 2000 (1) - y 2001 (1) e = 5.00% d. Based on the inflation indexed U.S. Treasury and the nominal U.S. treasury bond prices report the annual expected inflation for each of the following years: 2000, 2001, and 2002 in percentage terms. Using the information provided above we get. [P t+n / P t ] e = Q t (n) / q t (n) Q 2000 (1) / q t (1) = 1+pi 2000 e Solving for the expected inflation rates gives

3 Pi 2000 e = [(97.09 / 95.24)-1]* 100 = 1.94% Q 2000 (2) / q 2000 (2) = 1+pi 2000 e + pi 2001 e pi 2001 e = [(96.11 / 90.70)-1]* pi 2000 e = 4.02% Q 2000 (3) / q 2000 (3) = 1+ pi 2000 e + pi 2001 e + pi 2002 e pi 2002 e = [(97.06 / 86.38)-1]* pi 2000 e - pi 2001 e = 6.40% e. In answering this part assume that there is no difference between the Federal funds interest rate and the yield on a one-year nominal U.S. Treasury bond. This assumption implies that the Federal Reserve always sets the yield on the one-year nominal U.S. Treasury bill using the Taylor Rule. Using the solutions to parts c) and d) compute the expected value of d (departure of real GDP from trend) for years 2000, 2001 and 2002? From the Taylor rule it follows d t =2*(y t (1) e * pi t e ) d 2000 = 2*(y 2000 (1) 1-1.5* pi 2000 e ) = 2.18% d 2001 = 2*(y 2001 (1) e 1-1.5* pi 2001 e ) = -4.06% d 2002 = 2*( y 2002 (1) e 1-1.5* pi 2002 e ) = -11.2% 2- The yield curve and expectations hypothesis As given in class the long-term interest rates are determined as averages of expected short term interest rates. For n=1 the expectation hypothesis gives y t (1) is equal to the current short rate. For n > 1 it follows y t (n)=[y t (1)+y t+1 (1) e +y t+2 (1) e + + y t+n-1 (1) e + L(n)] / n where L(n) is equal to D*n 2. Assume through the problem that D=0.2% per annum. Note that all yields are in annual percentage terms and that n is measured in years. Report all yields beginning from n=1 and higher. Note that the term L(n) reports the risk premium on the bond. a. On January 1, 1998 the yields on the 1,2,5 and 10 year zero coupon Treasury bonds are 5.41%, 5.79%, 6.33% and 7.11% respectively. Plot the yield curve on January 1, 1998 using these data. This part is obvious.

4 b. Assume that the short term interest rate, that is y(1,t) is determined by the following law of motion y t+1 (1) = a + y t (1) + u t+1 Here u t+1 is a normally distributed random variable with mean zero and standard deviation of 1 percentage point. Assume that the value of a is equal to 0.01%. Also assume that on January 1, 1999 the (realized) short term interest rate, i.e. y(1,1999) is equal to 6.1%. Compute 2, 3, 5 and 10 year yields and report only the spread between the 10 year yield and the 2 year yield. To compute the long term interest rates first compute the expected short-term yields and then use the expectations theory. The expected short-term yields are given by the following formulas y t+1 (1) e =E[ a + y t+1 (1) + u t+1 ]=a+ y t (1) y t+2 (1) e =E[ a + y t+1 (1) + u t+2 ] Substitute y t+1 (1) with the law of motion stated above to get E[a+a+ y t (1) + u t+1 + u t+2 ]=2*a+ y t (1) or in general y t+k (1) e = a* k+ y t (1) Combining the above expression with the expectations theory and the expression for the risk premium gives y t (n)=[y t (1)+y t+1 (1) e +y t+2 (1) e + + y t+n-1 (1) e + L(n)] / n = =[ y t (1)+a+ y t (1)+a*2 + y t (1)+ + a* (n-1) + y t (1)+ D n 2 ]/ n For n > 1 this simplifies to =[ y t (1)+a+ y t (1)+a*2 + y t (1)+ + a* (n-1) + y t (1)+ D n 2 ]/ n y t (n) = y t (1)+ D*n + a*[ (n-1)]/n = y t (1) + D * n + a* (n-1) / 2 Finally, the difference between 10 year and 2 year yields is y 1999 (10) y 1999 (2)= y t (1) + D * 10 + a* (10-1) / 2-( y t (1)+ D * 2 + a* (2-1 / 2)) = 8D+4a

5 This equals 8D+4a=8 * 0.2% + 4 * 0.01% = 1.64% c. On January 1, 2000 the realized short term nominal interest rate was 6.6%. Compute 2, 3, 5 and 10-year yields. Report the difference between the five year yields on January 1, 2000 and January 1, (Note: this problem shows you the main source of movements in the yield curve is the short rate and that 90% of movements in the yield curve are of a parallel nature.) Given the above expression for the long term yields we have y 2000 (2) = y 2000 (1) + D*2 + a/2 y 2000 (3) = y 2000 (1) + D*3 + a y 2000 (5) = y 2000 (1) + D*5 + a*2 y 2000 (10) = y 2000 (1) + D*10 + a*9/2 y 1999 (5) y 2000 (5)= y 1999 (1) + D * 5 + a* (5-1)/ 2- (y 2000 (1) + D * 5 + a* (5-1)/ 2 = y 1999 (1) y 2000 (1) Which equals 0.5%. 3- Using the expectations hypothesis Explain the expectations hypothesis of the term structure of interest rates. The expectations hypothesis says that the current yield on a n maturity bond is the average of expected one period interest rates, e.g. i t (2) = [i t (1) + i t+1 e (1)]/2 On the next page is a graph of the yield curve on 12/13/94; here you see a shift in the yield curve between ``4 weeks ago'' and ``yesterday''. Using the expectations hypothesis, explain this shift in terms of the expectation of future short-term (one-period) interest rates. The shift from 4 weeks back to `yesterday' shows that the short term interest rates increased up to 3 years maturity and the long term yields dropped as compared to 4 weeks back. This implies that in the last 4 weeks the investors have raised their expectations regarding the one period interest rates up to the 3 years horizon and have revised their expectations downward regarding one period interest rates for the time period beyond three years

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7 4- Valuation of bonds On January 1, 2000 the prices on 1 and 2 year US Treasury bonds (assumed to be zero coupon bonds) are $94.16 and $88.17 respectively. The face value of both bonds is $100. On the same day the prices of inflation indexed Treasury bonds (IITB) of 1 and 2 year maturity have prices of $96.98 and $94.45 respectively. An inflation-indexed Treasury bond (IITB) is a bond that pays the face value ($100) multiplied by the gross inflation rate between January 1, 2000 and maturity of the bond. The gross inflation rate is simply P t+n /P t, where n is the number of years in the future, and P is the CPI index (consumer price index) in the US. Note that IITB's, protect the owner of the bond from inflation and pay off in inflation adjusted terms. Questions: a) What are the yields on the one-year and two-year US Treasury bonds? b) Based on the 1 and 2 year bond prices for the U.S. T-bonds and IITB's, what is the inflation expected (i.e., expected inflation) between January 1, 2000 and the end of year 1? c) What is the expected inflation between January 1, 2000 and the end of year 2? d) What is the one-year expected real interest rate in annual percentage terms. Solution Lets avoid the time subscript since date is always January a) The yield on the one- year Treasury bond is given by 1+i(1) = ($100 / q(1)) = $100 / $94.16 i(1) = (($100 / $94.16)-1) * 100 = 6.20% The yield on the two- year Treasury bond is given by (1+i(2))^2 = ($100 / q(2)) = ($100 / $88.17) i(2) = (($100 / $88.17)^0.5-1) * 100 = 6.50%

8 b) and c) Denote the prices of one and two year IITB's by Q(1) and Q(2) respectively. The links between IITB prices, expected inflation and yields are Q(1)=E[100(P t+1 /P t / (1 + i(1))] Q(2)=E[100(P t+1 /P t / (1 + i(2))^2)] Combining the above two expressions with the expressions for yields on non-indexed treasury securities, i.e., U.S. Treasury Bonds. q(1)=100 / (1 + i(1)) gives q(2)=100 / (1 + i(2))^2 Q(1) / q(1) = E[P t+1 /P t] = / which means that the expected rate on inflation in the first year is and ((96.98 / 94.16) - 1)* 100 = 2.99% Q(2)/ q(2) = E[P t+2 /P t ] = / which means that the expected average rate on inflation over the two years is ((94.45 / 88.17) - 1)* 100 = 7.12% d) The one-year expected real interest rate is or r(1) e = ((1+i(1))/(1+pi(1) e )-1) * 100 = (( )/( )-1) * 100 = 3.12% r(1) e = i(1) pi(1) e = 3.21% Both methods are acceptable.