Below you can find a table presenting maturities and yields to maturity of a set
ID: 2614061 • Letter: B
Question
Below you can find a table presenting maturities and yields to maturity of a set of Treasury securities.
1) Based on the data from the table please perform bootstrapping and calculate the respective spot rates and construct the theoretical spot rate curve.
2) Please explain the shape of the curve referring to the expectations hypothesis, the liquidity preference hypothesis, and the segmented market hypothesis.
3) Based on the spot rates please calculate the forward rates and make conclusions regarding the market expectations of the changes in the interest rates. What would be your advice for the investors under such settings?
Explanation / Answer
(1) Let the 0.5-year spot rate be 2r0.5, the 1-year spot rate be 2r1, 1.5-year spot rate be 2r1.5 and so on.
Assuming that all bonds have a par value of $ 100, we have:
96.15 = 100 / (1+r0.5)
r0.5 = 0.04 or 4 % per half-year
92.19 = 100 / (1+r1)^(2)
r1 = 0.0415 or 4.15 % per half-year
For the 1.5 year coupon bond:
Annual Coupon Rate for 1,5 year bond = 8.5 % per annum or 4.25 % per half-year
Semi-Annual Coupon = 0.085 x 100 x 0.5 = $ 4.25
Therefore, 99.45 = 4.25 / 1.04 + 4.25 / (1.0415)^(2) + 104.25 / (1+r1.5)^(3)
r1.5 = 0.04465 or 4.465 % per half-year
For the 2 year coupon bond:
Annual Coupon Rate = 9 % per annum
Semi-Annual Coupon = 0.09 x 0.5 x 100 = $ 4,5
Therefore, 99.64 = 4.5 / 1.04 + 4.5 / (1.0415)^(2) + 4.5 / (1.04465)^(3) + 104.5 / (1+r2)^(4)
r2 = 0.046235 or 4.6235 % per half-year
A similar approach would give the half-year rates for the 2.5 year and 3 year bonds as well. These half year rates when doubled would give the theoretical spot rate curve as given below:
(2) The expectations hypothesis states that short-term and long-term interest rates are a result of the interest rate expectations of the market participants. In simpler terms it states that all short-term and long-term interest rates are equivalent to the geometric mean of the current and future interest rates. If current and future interest rates are increasing so would be their geometric mean and so would be the long-term interest rate(yield) curve.
The liquidity preference theory states that investors demand low yields on short-term bonds as such bonds have low exposure to interest rate volatility owing to their limited tenure. Consequently, long-term bond investors should require higher yields to compensate them for being exposed to interest rate volatility for longer durations. Hence, short-term bonds should have low-interest rates and long-term bonds should have relatively higher interest rates.
The segmented market hypothesis states that investors prefer bonds of specific maturities as per their needs and hence their selection of bonds is restricted to their preferred maturity segment. This results in an effective segmentation of the bond market and yields in the short-term and long-term bond markets are impacted by the demand and supply in each of these segments.
(3) The forward rate is the expected 6-month interest rate 6-months later. The same is calculated as given below:
r(t+0.5) = [(1+r1)^(2) / (1+r0.5)] - 1 = 0.043 or 4.3 % per half-year
Similarly, expected 6-month interest rate 1-year from now would be r(t+1) = [(1+r1.5)^(3) / (1+r1)^(2)] - 1 = [(1.04465)^(3) / (1.0415)^(2)] - 1 = 0.05097 or 5.1 % approximately
The remaining forward rates can be calculated as given above:
Maturity (in years) Theoretical Spot Rate (in %) 0.5 8 1 8.3 1.5 8.93 2 9.247 2.5 9.468 3 9.787Related Questions
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