1、国际经济贸易学院研究生课程班 固定收益证券 试题 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 securi
2、ty (or 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
3、 reference 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 r
4、eference 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 b
5、ecause 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 s
6、emiannually. 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
7、 given market 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,wh
8、ich investment 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 inve
9、stment is 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
10、 of the 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.Co
11、nsequently,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 t
12、o make a 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
13、.The yield 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 m
14、aturity is 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 inve
15、stor can 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 comp
16、are them based 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
17、when the manager 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.If These cash flows are spent when received,the manager would know exactly how muc
18、h money could 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 s
19、everal potential 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.Whe
20、n a total return 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 investigatin
21、g multiple scenarios 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
22、 case, which 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 the
23、re would be the 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 sp
24、ent when received,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 fo
25、r bonds A and B. Bond A Bond B Coupon 8% 9% Yield to maturity 8% 8% Maturity (years) 2 5 Par $100.00 $100.00 Price $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
26、 have an 8% coupon rate and an 8% 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 p
27、resent value of coupon bond formula, we get: 41111( 1 ) ( 1 0. 04 5 )$4 0 $1 43 .5 010. 04 5nrPC r The present value of the par or maturity value of $1,000 is: 4$ 1 , 0 0 0 $ 8 3 8 .5 6 1(1 ) (1 .0 4 5 )nM r Thus, the value of bond A with a yield of 9%, a coupon rate of 8%, and a maturity of 2 years
28、 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 $1,000 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 Thus, the value of bond A with
29、 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 $1,000 and a bond quote of $100 since a change of 100 basis points will make the yield and coupon rate t
30、he same, For example, inserting (b) Using duration, estimate the price of the bonds for a 100-basis-point increase in interest rates. Answer: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 calcu
31、lations required by the Macaulay procedure. The formula is: 211 ( 1 0 0 / )1( 1 ) ( 1 )nnC n C yy y yM o d ifi e d D u ra tionP Putting all applicable variables in terms of $100, we have C = $4, n = 4, y = 0.045, and P = $98.2062. Inserting these values, in the modified duration formula gives: 2 1 2
32、 4 51 ( 1 0 0 / ) $ 4 1 4 ( $ 1 0 0 $ 4 / 0 .0 4 5 )11( 1 ) ( 1 ) 0 .0 4 5 ( 1 .0 4 5 ) ( 1 .0 4 5 )9 8 .2 0 6 2nnC n C yy y yM o d ifi e d D u ra tion P ($1,975.3086420.161439 + $35.664491) / $98.2062 = ($318.89117 + $35.664491) / $98.2062 = $354.555664 / $98.2062 = 3.6103185 or about 3.61. Convert
33、ing to annual number by dividing by two gives a modified duration of 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: ( ) ( ) 1 . 8 0 5 1 5 9 (0 . 0 1 ) 0 . 0 1
34、 8 0 5 1 5dP M o d i f i e d D u r a t i o n d yP We can now solve for the new price of bond A as shown below: ( 1 ) ( 1 0 . 0 1 8 0 5 1 5 ) $ 1 , 0 0 0 $ 9 8 1 . 9 4 8dP PP This is slightly less than the actual price of $982.062. The difference is $982.062 $981.948 = $0.114. To estimate the price o
35、f 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 = 10, y = 0.045, and P = $100, we get: 2 1 2 1 0 1 11 ( 1 0 0 / ) $ 4 . 5 1
36、 1 0 ( $ 1 0 0 $ 4 . 5 / 0 . 0 4 5 )11( 1 ) ( 1 ) 0 . 0 4 5 ( 1 . 0 4 5 ) ( 1 . 0 4 5 )$100nnC n C yy y yM o d if ie d D u r a ti o n P ($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 dividin
37、g by two gives a modified duration of 3.956359 (before the increase in 100 basis points it was 3.994417). 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: ( ) ( ) 3 . 9 5 6 3 5 9 (0
38、. 0 1 ) 0 . 0 3 9 5 6 3 5dP M o d i f i e d D u r a t i o n d yP 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 of $1,000. The difference is $1,000 $999.382 = $0.618. (c) Using both duration and convexity measures, estimate
39、 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 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 pa
40、rt (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: ($982.062 y first approximating the dollar price change. With 100 basis points giving dy = 0.01 and a modif
41、ied duration computed in part (b) of 1.805159, we have: ( ) ( ) 1 . 8 0 5 1 5 9 (0 . 0 1 ) 0 . 0 1 8 0 5 1 5 9dP M o d i f i e d D u r a t i o n d yP This is slightly more negative than the actual percentage decrease in price ( 1 ) ( 1 0 . 0 1 8 0 5 1 5 9 ) $ 1 , 0 0 0 $ 9 8 1 . 9 4 8dP PP This is s
42、lightly less than the actual price of $982.062. 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) =22 3 2 1 21 2 1 2 ( 1 ) ( 100 / ) 11 ( 1 ) ( 1 ) ( 1 )n n nd P C C n n n
43、 C ydy P y y y y y P 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. 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
44、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 our convexity measure formula gives: convexity measure (half years) = 3 4 2 5 62$4 1 2( $4) 4 4( 5 ) ( 100 $4 / 0.04 5 ) 11 16.9 3250.04 5 ( 1.04 5 ) 0.04 5 ( 1.
45、04 5 ) ( 1.04 5 ) $98 .206 2y 22 1 6 . 9 3 2 5( ) 4 . 2 3 3 1 2 52C o n v e x i t y M e a s u r e i n m p e r i o d p e r y e a rT h e C o n v e x i t y M e a s u r e i n y e a r s m $1,000) = new price for bond A. We have: P r ( 1 ) ( 1 0 . 0 1 7 8 3 9 9 4 ) $ 1 , 0 0 0 (0 . 9 8 1 9 4 8 4 ) $ 1 , 0
46、 0 0 $ 9 8 2 . 1 6 0dPN e w i c e PP new price for bond A. We have: ( ) ( ) 3 . 0 5 6 3 5 9 (0 . 0 1 ) 0 . 0 3 9 5 6 3 5dP M o d i f i e d D u r a t i o n d yP This is slightly more negative than the actual percentage decrease in price of -3.896978%. The difference is (-3.896978%)-(-3.95635%)=0.0593
47、82% Using the -3.95635%just given by the duration measure, the new price for Bond B is: ( 1 ) ( 1 0 . 0 3 9 5 6 3 5 ) $ 1 , 0 4 0 . 5 5 $ 9 9 9 . 3 8 2dP PP This is slightly less than the actual price of $1,000. This difference is $1,000-$999.382=$0.618 We use the convexity measure to see if we can
48、account for the difference of 00594%. We have: 22 3 2 1 21 2 1 2 ( 1 ) ( 1 0 0 / ) 1( ) 1 ( 1 ) ( 1 ) ( 1 )n n nd P C Cn n n C yCon v e x it y M e a su re h a lf y e a rs d y P y y y y y P 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 equal. Prior to that, the price sold at P=$1,040.55. Expressing numbers in terms of a $10
