對外經(jīng)濟貿(mào)易大學(xué)國際經(jīng)濟貿(mào)易學(xué)院固定收益證券部分答案

1、國際經(jīng)濟貿(mào)易學(xué)院研究生課程班固定收益證券試題1）Explain why you agree or disagree with the following statement: “The price of a floater will always trade at its par value.”Answer:I disagree with the statement: “The price of a floater will always trade at its par value.” First, the coupon rate of a floating-rate security (o

2、r floater) is equal to a reference rate plus some spread or margin. For example, the coupon rate of a floater can reset at the rate on a three-month Treasury bill (the reference rate) plus 50 basis points (the spread). Next, the price of a floater depends on two factors: (1) the spread over the refe

3、rence rate and (2) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price of a floater will trade close to its par value as long as (1) the spread above the refere

4、nce rate that the market requires is unchanged and (2) neither the cap nor the floor is reached. However, if the market requires a larger (smaller) spread, the price of a floater will trade below (above) par. If the coupon rate is restricted from changing to the reference rate plus the spread becaus

5、e of the cap, then the price of a floater will trade below par.2） A portfolio manager is considering buying two bonds. Bond A matures in three years and has a coupon rate of 10% payable semiannually. Bond B, of the same credit quality, matures in 10 years and has a coupon rate of 12% payable semiann

6、ually. Both bonds are priced at par.(a) Suppose that the portfolio manager plans to hold the bond that is purchased for three years. Which would be the best bond for the portfolio manager to purchase?Answer:The shorter term bond will pay a lower coupon rate but it will likely cost less for a given m

7、arket rate.Since the bonds are of equal risk in terms of creit quality (The maturity premium for the longer term bond should be greater),the question when comparing the two bond investments is:What investment will be expecte to give the highest cash flow per dollar invested?In other words,which inve

8、stment will be expected to give the highest effective annual rate of return.In general,holding the longer term bond should compensate the investor in the form of a maturity premium and a higher expected return.However,as seen in the discussion below,the actual realized return for either investment i

9、s not known with certainty. To begin with,an investor who purchases a bond can expect to receive a dollar return from(i)the periodic coupon interest payments made be the issuer,(ii)an capital gain when the bond matures,is called,or is sold;and (iii)interest income generated from reinvestment of the

10、periodic cash flows.The last component of the potential dollar return is referred to as reinvestment income.For a standard bond(our situation)that makes only coupon payments and no periodic principal payments prior to the maturity date,the interim cash flows are simply the coupon payments.Consequent

11、ly,for such bonds the reinvestment income is simply interest earned from reinvesting the coupon interest payments.For these bonds,the third component of the potential source of dollar return is referred to as the interest-on-interest components.If we are going to coupute a potential yield to make a

12、decision,we should be aware of the fact that any measure of a bonds potential yield should take into consideration each of the three components described above.The current yield considers only the coupon interest payments.No consideration is given to any capital gain or interest on interest.The yiel

13、d to maturity takes into account coupon interest and any capital gain.It also considers the interest-on-interest component.Additionally,implicit in the yield-to-maturity computation is the assumption that the coupon payments can be reinvested at the computed yield to maturity.The yield to maturity i

14、s a promised yield and will be realized only if the bond is held to maturity and the coupon interest payments are reinvested at the yield to maturity.If the bond is not held to maturity and the coupon payments are reinvested at the yield to maturity,then the actual yield realized by an investor can

15、be greater than or less than the yield to maturity.Given the facts that(i)one bond,if bought,will not be held to maturity,and(ii)the coupon interest payments will be reinvested at an unknown rate,we cannot determine which bond might give the highest actual realized rate.Thus,we cannot compare them b

16、ased upon this criterion.However,if the portfolio manager is risk inverse in the sense that she or he doesnt want to buy a longer term bond,which will likel have more variability in its return,then the manager might prefer the shorter term bond(bondA) of thres years.This bond also matures when the m

17、anager wants to cash in the bond.Thus,the manager would not have to worry about any potential capital loss in selling the longer term bond(bondB).The manager would know with certainty what the cash flows are.IfThese cash flows are spent when received,the manager would know exactly how much money cou

18、ld be spent at certain points in time.Finally,a manager can try to project the total return performance of a bond on the basis of the panned investment horizon and expectations concerning reinvestment rates and future market yields.This ermits the portfolio manager to evaluate thich of several poten

19、tial bonds considered for acquisition will perform best over the planned investment horizon.As we just rgued,this cannot be done using the yield to maturity as a measure of relative value.Using total return to assess performance over some investment horizon is called horizon analysis.When a total re

20、turn is calculated oven an investment horizon,it is referred to as a horizon return.The horizon analysis framwor enabled the portfolio manager to analyze the performance of a bond under different interest-rate scenarios for reinvestment rates and future market yields.Only by investigating multiple s

21、cenarios can the portfolio manager see how sensitive the bonds performance will be to each scenario.This can help the manager choose between the two bond choices.(b) Suppose that the portfolio manager plans to hold the bond that is purchased for six years instead of three years. In this case, which

22、would be the best bond for the portfolio manager to purchase?Answer:Similear to our discussion in part(a),we do not know which investment would give the highest actual relized return in six years when we consider reinvesting all cash flows.If the manager buys a three-year bond,then there would be th

23、e additional uncertainty of now knowing what three-year bond rates would be in three years.The purchase of the ten-year bond would be held longer than previously(six years compared to three years)and render coupon payments for a six-year period that are known.If these cash flows are spent when recei

24、ved,the manager will know exactly how much money could be spent at certain points in timeNot knowing which bond investment would give the highest realized return,the portfolio manager would choose the bond that fits the firms goals in terms of maturity.3） Answer the below questions for bonds A and B

25、.Bond ABond BCoupon8%9%Yield to maturity8%8%Maturity (years)25Par$100.00$100.00Price$100.00$104.055(a) Calculate the actual price of the bonds for a 100-basis-point increase in interest rates.Answer:For Bond A, we get a bond quote of $100 for our initial price if we have an 8% coupon rate and an 8%

26、yield. If we change the yield 100 basis point so the yield is 9%, then the value of the bond (P) is the present value of the coupon payments plus the present value of the par value. We have C = $40, y = 4.5%, n = 4, and M = $1,000. Inserting these numbers into our present value of coupon bond formul

27、a, we get:The present value of the par or maturity value of $1,000 is:Thus, the value of bond A with a yield of 9%, a coupon rate of 8%, and a maturity of 2 years is: P = $143.501 + $838.561 = $982.062. Thus, we get a bond quote of $98.2062. We already know that bond B will give a bond value of $1,0

28、00 and a bond quote of $100 since a change of 100 basis points will make the yield and coupon rate the same, For example, inserting Thus, the value of bond A with a yield of 9%, a coupon rate of 8%, and a maturity of 2 years is: P = $143.501 + $838.561 = $982.062. Thus, we get a bond quote of $98.20

29、62. We already know that bond B will give a bond value of $1,000 and a bond quote of $100 since a change of 100 basis points will make the yield and coupon rate the same, For example, inserting (b) Using duration, estimate the price of the bonds for a 100-basis-point increase in interest rates.Answe

30、r:To estimate the price of bond A, we begin by first computing the modified duration. We can use an alternative formula that does not require the extensive calculations required by the Macaulay procedure. The formula is: Putting all applicable variables in terms of $100, we have C = $4, n = 4, y = 0

31、.045, and P = $98.2062. Inserting these values, in the modified duration formula gives:($1,975.3086420.161439 + $35.664491) / $98.2062 = ($318.89117 + $35.664491) / $98.2062 = $354.555664 / $98.2062 = 3.6103185 or about 3.61. Converting to annual number by dividing by two gives a modified duration o

32、f 1.805159 (before the increase in 100 basis points it was 1.814948). We next solve for the change in price using the modified duration of 1.805159 and dy = 100 basis points = 0.01. We have:We can now solve for the new price of bond A as shown below:This is slightly less than the actual price of $98

33、2.062. The difference is $982.062 $981.948 = $0.114. To estimate the price of bond B, we follow the same procedure just shown for bond A. Using the alternative formula for modified duration that does not require the extensive calculations required by the Macaulay procedure and noting that C = $45, n

34、 = 10, y = 0.045, and P = $100, we get: ($791.27182 + $0) / $100 = 7.912718 or about 7.91 (before the increase in 100 basis points it was 7.988834 or about 7.99). Converting to an annual number by dividing by two gives a modified duration of 3.956359 (before the increase in 100 basis points it was 3

35、.994417). We will now estimate the price of bond B using the modified duration measure. With 100 basis points giving dy = 0.01 and an approximate duration of 3.956359, we have:Thus, the new price is(1 0.0395635)$1,040.55 = (0.9604364)$1,040.55 = $999.382. This is slightly less than the actual price

36、of $1,000. The difference is $1,000 $999.382 = $0.618.(c) Using both duration and convexity measures, estimate the price of the bonds for a 100-basis-point increase in interest rates.Answer:For bond A, we use the duration and convexity measures as given below. First, we use the duration measure. We

37、add 100 basis points and get a yield of 9%. We now have C = $40, y = 4.5%, n = 4, and M = $1,000. NOTE. In part (a) we computed the actual bond price and got P = $982.062. Prior to that, the price sold at par (P = $1,000) since the coupon rate and yield were then equal. The actual change in price is

38、: ($982.062 $1,000) = -$17.938 and the actual percentage change in price is: -$17.938 / $1,000 = -0.017938%. We will now estimate the price by first approximating the dollar price change. With 100 basis points giving dy = 0.01 and a modified duration computed in part (b) of 1.805159, we have:This is

39、 slightly more negative than the actual percentage decrease in price of -1.7938%. The difference is -1.7938% (-1.805159%) = -1.7938% + 1.805159% = 0.011359%. Using the -1.805159% just given by the duration measure, the new price for bond A is:This is slightly less than the actual price of $982.062.

40、The difference is $982.062 $981.948 = $0.114. Next, we use the convexity measure to see if we can account for the difference of 0.011359%. We have: convexity measure (half years) =For bond A, we add 100 basis points and get a yield of 9%. We now have C = $40, y = 4.5%, n = 4, and M = $1,000. NOTE. I

41、n part (a) we computed the actual bond price and got P = $982.062. Prior to that, the price sold at par (P = $1,000) since the coupon rate and yield were then equal. Expressing numbers in terms of a $100 bond quote, we have: C = $4, y = 0.045, n = 4, and P = $98.2062. Inserting these numbers into ou

42、r convexity measure formula gives: convexity measure (half years) = Adding the duration measure and the convexity measure, we get -1.805159% + 0.021166% = -1.783994%. Recall the actual change in price is: ($982.062 $1,000) = -$17.938 and the actual percentage change in price is: -$17.938 / $1,000 =

43、0.017938 or approximately -1.7938%. Using the -1.783994% resulting from both the duration and convexity measures, we can estimate the new price for bond A. We have: Adding the duration measure and the convexity measure, we get -1.805159% + 0.021166% = -1.783994%. Recall the actual change in price is

44、: ($982.062 $1,000) = -$17.938 and the actual percentage change in price is: -$17.938 / $1,000 = 0.017938 or approximately -1.7938%. Using the -1.783994% resulting from both the duration and convexity measures, we can estimate the new price for bond A. We have: This is slightly more negative than th

45、e actual percentage decrease in price of -3.896978%. The difference is (-3.896978%)-(-3.95635%)=0.059382%Using the -3.95635%just given by the duration measure, the new price for Bond B is:This is slightly less than the actual price of $1,000. This difference is $1,000-$999.382=$0.618We use the conve

46、xity measure to see if we can account for the difference of 00594%. We have:For Bond B, 100 basis points are added and get a yield of 9%. We now have C=$45, y=4.5%, n=10, and M=$1,000. Note in part (a), we computed the actual bond price and got P=$1,000 since the coupon rate and yield were then equa

47、l. Prior to that, the price sold at P=$1,040.55. Expressing numbers in terms of a $100 bond quote, we have C=$4.5, y-0.045, n=10 and P=$100. Inserting these numbers into our convexity measure formula gives:The convexity measure (in years)= Note. Dollar Convexity Measure=Convexity Measure (years) tim

48、es P=19.452564($100)=$1,945.2564.The percentage price change due to convexity is Inserting in the values, we get Thus, we have 0.097463% increase in price when we adjust for convexity measure.Adding the duration measure and convexity measure, we get -3.9563659%+0.097263% equals -3.859096%. Recall th

49、e actual change in price is ($1,000-$1,040.55)=-$40.55 and the actual new price is For Bond A. This is about the same as the actual price of $1,000. The difference is $1,000.394-$1,000=$0.394. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a d

50、ifference of -$0.618 to $0.394. (d) Comment on the accuracy of your results in parts b and c, and state why one approximation is closer to the actual price than the other.Answer:For bond A, the actual price is $982.062. When we use the duration measure, we get a bond price of $981.948 that is $0.114

51、 less than the actual price. When we use duration and convex measures together, we get a bond price of $982.160. This is slightly more than the actual price of $982.062. The difference is $982.160 $982.062 = $0.098. Thus, using the convexity measure along with the duration measure has narrowed the e

52、stimated price from a difference of -$0.114 to $0.0981. For bond B, the actual price is $1,000. When we use the duration measure, we get a bond price of $999.382 that is $0.618 less than the actual price. When we use duration and convex measures together, we get a bond price of $1,000.394. This is s

53、lightly more than the actual price of $1,000. The difference is $1,000.394 $1,000 = $0.394. Thus, using the convexity measure along with the duration measure has narrowed the estimated price from a difference of $0.618 to $0.394As we see, using the duration and convexity measures together is more ac

54、curate. The reason is that adding the convexity measure to our estimate enables us to include the second derivative that corrects for the convexity of the price-yield relationship. More details are offered below. Duration (modified or dollar) attempts to estimate a convex relationship with a straigh

55、t line (the tangent line). We can specify a mathematical relationship that provides a better approximation to the price change of the bond if the required yield changes. We do this by using the first two terms of a Taylor series to approximate the price change as follows: Dividing both sides of this

56、 equation by P to get the percentage price change gives us:The first term on the right-hand side of equation (1) is equation for the dollar price change based on dollar duration and is our approximation of the price change based on duration. In equation (2), the first term on the right-hand side is

57、the approximate percentage change in price based on modified duration. The second term in equations (1) and (2) includes the second derivative of the price function for computing the value of a bond. It is the second derivative that is used as a proxy measure to correct for the convexity of the pric

58、e-yield relationship. Market participants refer to the second derivative of bond price function as the dollar convexity measure of the bond. The second derivative divided by price is a measure of the percentage change in the price of the bond due to convexity and is referred to simply as the convexi

59、ty measure.(e) Without working through calculations, indicate whether the duration of the two bonds would be higher or lower if the yield to maturity is 10% rather than 8%.Answer: Like term to maturity and coupon rate, the yield to maturity is a factor that influences price volatility. Ceteris paribus, the hi