[Solution] IPA Week 9 - Chapter 15 PDF

Preview

Summary

Lecturer/Tutor: Peiming. These documents will be useful for Semester 2 2018 and Semester 1 2019 students undertaking this paper - These are what got me A overall in this paper. ...

Description

Week 9 Lecture 9 – Chapter 15: The Term Structure of Interest Rate

Problem Sets

Question 2: Under the expectations hypothesis, if the yield curve is upward-sloping, the market must expect an increase in short-term interest rates. True/false/uncertain? Why?

Expectation theory, according to expectation theory future short term interest rates can be forecasted with the help of long term interest rates. It is also known as Unbiased Expectations Theory. This theory suggests that an investor will earn same amount of interest if invests his money in one-year bond in current year and then in another one-year bond next year as compared to investing in a single two-year bond in the current year. According to the expectations theory, long term interest rates are equal to the average of short-term rates expected to prevail.

Here, the given statement is True, because bond has no risk premium associated with it which is related to its price. The single reason due to which long-term yields are higher than the short-term yields which may result in Yield curve being upward, the investor wants to economic growth and based on their forecasted inflation they prefer long term securities over short term securities. Thus, the statement is true.

Question 6: Assuming the pure expectations theory is correct, an upward-sloping yield curve implies: A. Interest rates are expected to increase in the future. B. Longer-term bonds are riskier than short-term bonds. C. Interest rates are expected to decline in the future.

According to the expectations theory, long term interest rates are equal to the average of short-term rates expected to prevail.

According to the expectations theory, those yield curves which are upward sloping leads to short-term rates to rise or vice versa. The yield curve slopes upward because short-term rates are lower than long-term rates. Hence, an upward-sloping yield curve implies interest rates are expected to increase in the future.

Question 7: The following is a list of prices for zero-coupon bonds of various maturities. Calculate the yields to maturity of each bond and the implied sequence of forward rates.

Yield to Maturity (YTM) can be defined as the annual rate of return that will be earned if the bond is purchased today at the current market price and is held by the investor till maturity. Thus, YTM is the average rate of return that will be earned on a bond if it is bought now and held till maturity. It shows an effective

annual return from a security expressed as a percentage of the current market price of the security.

The following given information will be used for the computation of YTM of oneyear maturity, 2-years maturity, 3-years maturity, and 4-years maturity zero coupon bond.

Compute YTM of 1-year maturity zero coupon bond as follows:

Compute YTM of 2-year maturity zero coupon bond as follows:

Compute YTM of 3-year maturity zero coupon bond as follows:

Compute YTM of 4-year maturity zero coupon bond as follows:

The following information will be used for the computation of implied forward rate in year-2, year-3, and year-4.

Compute implied forward rate for 2nd year as follows:

Compute implied forward rate for 3rd year as follows:

Compute implied forward rate for 4th year as follows:

Question 9: Consider the following $1,000 par value zero-coupon bonds:

According to the expectations hypothesis, what is the expected 1-year interest rate 3 years from now?

According to the expectations theory, long term interest rates are equal to the average of short-term rates expected to prevail.

The expected 1-year interest rate after 3 years will be calculated as follows:

To find the expected 1-year interest rate, divide the higher YTM by the YTM with 3yrs to maturity and then deduct 1.

According to the expectations hypothesis, the forward rate must be equal to the short rate.

Question 10: The term structure for zero-coupon bonds is currently:

Next year at this time, you expect it to be:

a. What do you expect the rate of return to be over the coming year on a 3-year zero-coupon bond?

b. Under the expectations theory, what yields to maturity does the market expect to observe on 1- and 2-year zeros at the end of the year? Is the market’s expectation of the return on the 3-year bond greater or less than yours?

Zero coupon bonds are types of bonds which are issued at discount and redeemed at premium. There is no coupon rate on these kinds of bonds. Thus, the difference in the issue price and redeemed price is the interest which the bond holder receives.

Yield to Maturity (YTM) 

Can be defined as the annual rate of return that will be earned if the bond is purchased at the current market price and the investor holds it till maturity.

Part A A 3-year zero coupon bonds which have the face value of $100 and YTM of 6% and the price will be:

Where FV is face value, YTM is yield to maturity and PV is present value.

On substituting the values,

Hence, the face value of 3 year Zero coupon bond is Part B

The forward rates based on today’s yield curve are as follows:

Yield curve for the next year by using the forward rates will be as follows:

The market forecast support the higher YTM on 2–year bonds. Therefore, the market forecast an inferior price and superior rate of return.

Question 11: The yield to maturity on 1-year zero-coupon bonds is currently 7%; the YTM on 2year zeros is 8%. The Treasury plans to issue a 2-year maturity coupon bond,

paying coupons once per year with a coupon rate of 9%. The face value of the bond is $100.

A. At what price will the bond sell? B. What will the yield to maturity on the bond be? C. If the expectations theory of the yield curve is correct, what is the market expectation of the price that the bond will sell for next year? D. Recalculate your answer to ( c ) if you believe in the liquidity preference theory and you believe that the liquidity premium is 1%. Bonds are the debt instrument issued by the government or corporate to raise money from the market under the borrowing agreement. Under the agreement, the issuer has to pay periodic interest payments to the bond holder on the specific date. This rate of interest rate is called the coupon rate.

Yield to Maturity (YTM) 

Can be define as the annual rate of return that will be earned if the bond is purchased today at the current market price and is held by the investor till maturity.

(a) Calculate the Current selling price of the bond: YTM on 1-year zero coupon bond is 7% YTM on 2-year zero coupon bond is 8% Selling price of the bond will be calculated as follows:

(b) Calculate the Yield to Maturity (YTM) of the bond: YTM of the bond will be calculated as follows: Let be the YTM is Y so:

By using the financial calculator: We can solve for Yield to Maturity using a financial calculator by doing the following:

(c) Calculate the market expectation of the price: The next year forward rate will be calculated with the help of zero-coupon yield curve, is the solution for f 2 in the following equation:

So by following the expected rate of return for next year be 9.01%, the forward bond price would be:

Hence, the market expectation of the bond price is $99.99

Part D If the liquidity premium is 1% so the forecast interest rate is:

And the forecast of the bond price is:

Therefore, if the liquidity premium is 1%, the forecasted interest rate is 8.01% and current bond price is $100.92.

Question 13: Prices of zero-coupon bonds reveal the following pattern of forward rates:

In addition to the zero-coupon bond, investors also may purchase a 3-year bond making annual payments of $60 with par value $1,000.

A. What is the price of the coupon bond? B. What is the yield to maturity of the coupon bond? C. Under the expectations hypothesis, what is the expected realized compound yield of the coupon bond? D. If you forecast that the yield curve in 1 year will be flat at 7%, what is your forecast for the expected rate of return on the coupon bond for the 1-year holding period?

Bonds are the debt instrument issued by the government or corporate to raise money from the market under the borrowing agreement. Under the agreement, the issuer has to pay periodic interest payments to the bond holder on the specific date. This rate of interest rate is called the coupon rate.

Yield to Maturity (YTM) 

Is a rate of return expected on a bond that held by someone until its maturity is known as yield to maturity. It is fundamentally IRR (Internal Rate of Return) on the bond as it equalizes the present value of bond's future cash flows to current price of the bond.

(a) Calculate the price of the coupon bond: Price of the bond will be calculated with the help of given table:

Now the price will be calculated as follows:

Therefore, the Current price of the coupon bond is $984.10 (b) Calculate the Yield to Maturity of the coupon bond: YTM of the bond will be calculated as follows: Let be the YTM is Y so:

By using the financial calculator: We can solve for Yield to Maturity using a financial calculator by doing the following:

(c) Calculate the expected realized compound yield of the coupon bond: Expected realized compound yield of the coupon bond will be calculated as follows:

Now the equation will be as follows:

(d) Calculate the forecasted expected rate of return on the coupon bond: Forecast for the expected rate of return on the coupon bond for the 1-year holding period will be calculated as follows: Calculate the bond price, if the YTM of the bond is 7%:

Therefore, the Holding-period return on the coupon bond will be:

Question 14: You observe the following term structure:

A. If you believe that the term structure next year will be the same as today’s, will the 1-year or the 4-year zeros provide a greater expected 1year return? B. What if you believe in the expectations hypothesis? Bonds are the debt instrument issued by the government or corporate to raise money from the market under the borrowing agreement. Under the agreement, the issuer has to pay periodic interest payments to the bond holder on the specific date. This rate of interest rate is called the coupon rate.

Yield to Maturity (YTM) 

Can be define as the annual rate of return that will be earned if the bond is purchased today at the current market price and is held by the investor till maturity.

Part A The return on the one-year zero coupon bond is 6.1%. The 4-year zero-coupon bond yield is 6.4%. Calculate the price for the 4-year zero-coupon bond as follows:

The term structure next year remains the same. So the 4-year zero coupon bond will have 3-year maturity and the YTM will be 6.3%. Calculate the price for 3-year maturity zero coupon bond as follows:

Calculate the one-year rate of return as follows:

Therefore, the one-year rate of return considering 3-year and 4-year periods is 6.7%.

The return on the one-year zero coupon bonds is 6.1% and the one-year rate of return considering 3-year and 4-year periods is 6.7%. This states that long-term bond provides higher return as the YTM is expected to reduce during the holding period.

Part B Considering the expectations hypothesis, the yield curve for the next year should not be expected to be similar as today’s curve. The upward slope in today's curve will show that the predictable small rates are increasing. Therefore the yield curve will shift upward with dipping the holding period return on the fouryear bond.

Question 15: The yield to maturity (YTM) on 1-year zero-coupon bonds is 5% and the YTM on 2-year zeros is 6%. The yield to maturity on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is 5.8%.

What arbitrage opportunity is available for an investment banking firm? What is the profit on the activity?

Bonds are the debt instrument issued by the government or corporate to raise money from the market under the borrowing agreement. Under the agreement, the issuer make periodic interest payments to the bond holder on the specific date. This rate of interest rate is called the coupon rate.

Yield to Maturity (YTM) 

Can be defined as the annual rate of return that will be earned if the bond is purchased today at the current market price and is held by the investor till maturity.

Arbitrage 

Is a risk less transactions, which take benefit of mispricing securities in different market, without incurring any loss. The benefit occurred from mispricing termed as arbitrage profit.

Compute the price of coupon bond as follows:

The arbitrage opportunity can be framed if the coupon of year-1 and the face value plus coupon of year-2 can be sold separately like zeros. Then the yield to maturity of zeros with maturities of one and two years will be the yield of coupon payment in year-1 and the coupon payment plus face value in year-2 respectively. These could be sold separately now for the combined value computed below.

The strategy of arbitrage will be to buy zeros having the face value of $120 and $1,120 with the 1-year and 2-year maturity respectively. Simultaneously, sell the coupon bond of 2-years.

Compute the arbitrage profit from the strategy as follows:

Question 16: Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $94.34, while a 2-year zero sells at $84.99. You are considering the purchase of a 2-year-maturity bond making annual coupon payments. The face value of the bond is $100, and the coupon rate is 12% per year.

A. What is the yield to maturity of the 2-year zero? The 2-year coupon bond? B. What is the forward rate for the second year?

C. If the expectations hypothesis is accepted, what are (1) the expected price of the coupon bond at the end of the first year and (2) the expected holding-period return on the coupon bond over the first year? D. Will the expected rate of return be higher or lower if you accept the liquidity preference hypothesis?

Zero coupon bonds are those bonds which do not pay any payment in terms of interest during the whole life of bond and these bonds are sold at a deep discount from its face value or we can say that investor bought these bonds at a price below than the face value.

Yield to Maturity (YTM) 

Can be define as the annual rate of return that will be earned if the bond is purchased today at the current market price and is held by the investor till maturity.

(a) Calculate the yield to maturity of the 2-year zero and the 2-year coupon bond: YTM of a 1 year zero coupon bonds will be as follows:

And the YTM of a 2 year zero coupon bonds will be as follows:

So the price of the coupon bond is:

(b) Calculate the forward rate for the second year: The forward rate for the second year will be as follows:

(c)

Calculate the expected price of the coupon bond at the end of the first year:

Calculate the expected holding-period return on the coupon bond over the first year:

Part D

Conclusion: 

If the yields on long-term bonds are greater than the expected return then the investors in long-term bonds for interest rate risk bearing. The bonds having different maturities may have different yields. If the yield curve is showing upward slope then the liquidity premium are high (negative) and if the yield curve is showing downward slope then the liquidity premium is positive (low).

Question 17: The current yield curve for default-free zero-coupon bonds is as follows:

A. What are the implied 1-year forward rates? B. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will be the pure yield curve (that is, the yields to maturity on 1- and 2-year zero coupon bonds) next year? C. If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over the next year? What if you purchase a 3-year zerocoupon bond? (Hint: Compute the current and expected future prices.) Ignore taxes. D. What should be the current price of a 3-year maturity bond with a 12% coupon rate paid annually? If you purchased it at that price, what would your total expected rate of return be over the next year (coupon plus price change)? Ignore taxes. Bonds are the debt instrument issued by the government or corporate to raise money from the market under the borrowing agreement. Under the agreement,

the issuer has to pay periodic interest payments to the bond holder on the specific date. This rate of interest rate is called the coupon rate.

Yield to Maturity (YTM) 

Can be define as the annual rate of return that will be earned if the bond is purchased today at the current market price and is held by the investor till maturity.

(a) Calculate the 1-year implied forward rate: Implied 1-year forward rates will calculate as follows:

Part B The YTM on 1- and 2-year zero coupon bonds can be calculated by discounting each zero’s face value at the forward rates for next year that we resulting in part:

Part C The expected total rate of return over the next year will be a 1-year zero and selling price will be as follows:

Likewise, the current 3-year zero will be a 2-year zero and will sell for: $782.93

And the expected total rate of return:

Part D Current price of a 3-year maturity bond with a 12% coupon rate will equal to the value of each payment times the present value of $1 to be received at the “maturity” of that payment. Therefore, the current price will be as follows:

Similarly, the expected price 1 year from now will be a follows:

And the total expected rate of return will be:

Question 18: Suppose that the prices of zero-coupon bonds with various maturities are given in the following table. The face value of each bond is $1,000.

A. Calculate the forward rate of interest for each year. B. How could you construct a 1-year forward loan beginning in year 3? Confirm that the rate on that loan equals the forward rate. C. Repeat (b) for a 1-year forward loan beginning in year 4. (a) Calculate the forward rate of interest for each year: In order to calculate the forward rate, first calculate the YTM:

Therefore, the formula to calculate YTM is:

So, in the next step is to calculate the Forward rate: In order to calculate the forward rate, the formula is as under:

Where,

Hence, by applying the above formula for calculating YTM is followed by forward rate is mention below:

Therefore, as per the said question, the formula to calculate YTM and Forward rate is being applied to calculate the same up to maturity period of 5 years.

As, you can see that in the initial year yield to maturity is been calculated till to the end, but in the same year the forward rate is calculated from next year to the end.

It is because while calculating forward rate it takes into account the preceding year YTM rate which is not applicable in the case.

Finally, to compute the forward rate, first, calculate the YTM up to maturity period. Similarly, in the next step is to calculate the forward rate with help of YTM.