Adv Investments Exam 1
A bond with annual coupon payments has a coupon rate of 8%, yield to maturity of 10%, and Macaulay's duration of 9 years. What is the bond's modified duration?
9/1.10 = 8.18
If a bond manager swaps a bond for one that is identical in terms of coupon rate, maturity, and credit quality but offers a higher yield to maturity, the swap is: a) A substitution swap b) An interest rate anticipation swap c) A tax swap d) An intermarket spread swap
A
When interest rates decline, the duration of a 30-year bond selling at a premium: a) Increases b) Decreases c) Remains the same d) Increases at first, then declines
A
A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, what is the predicted change in price based on the bond's duration?
Bond price decreases by $80.00, calculated as follows: 10 × 0.01 × 800 = 80.00
Which bond has the longest duration? a) 8 year maturity, 6% coupon b) 8 year maturity, 11% coupon c) 15 year maturity, 6% coupon d) 15 year maturity, 11% coupon
C
Consider the following $1000 par value zero-coupon bonds: Bond | Years to Maturity | YTM (%) A | 1 | 5% B | 2 | 6% C | 3 | 6.5% D | 3 | 7% According to the expectations hypothesis, what is the market's expectation of the yield curve one year from now? Specifically, what are the expected values of next year's yields on bonds with maturities 1 year from now? 2 years? 3 years?
If expectations theory holds, then the forward rate equals the short rate, and the one year interest rate three years from now would be one year bond = [(1+0.06)^2 / (1+0.05)] - 1 = 0.0701 = 7.01% two year bond = [(1+0.065)^3 / (1+0.05)]^1/2 - 1 = 0.0726 = 7.26% three year bond = [(1+0.07)&4 / (1+0.05)]^1/3 - 1 = 0.0768 = 7.68%
Long-term treasury bonds currently are selling at yields to maturity of nearly 6%. You expect interest rates to fall. The rest of the market thinks that they will remain unchanged over the coming year. In each question, choose the bond that will provide the higher holding-period return over the next year if you are correct. Briefly explain your answer. a) i. A Baa-rated bond with coupon rate 6% and time to maturity 20 years. ii. An Aaa-rated bond with coupon rate of 6% and time to maturity 20 years. b) i. An A-rated bond with coupon rate 3% and maturity 20 years, callable at 105. ii. An A-rated bond with coupon rate 6% and maturity 20 years, callable at 105. c) i. A 4% coupon noncallable T-bond with maturity 20 years and YTM = 6%. ii. A 7% coupon noncallable T-bond with maturity 20 years and YTM = 6%.
In each case, choose the longer-duration bond in order to benefit from a rate decrease. a) ii. The Aaa-rated bond has the lower yield to maturity and therefore the longer duration. b) i. The lower-coupon bond has the longer duration and greater de facto call protection. c) i. The lower coupon bond has the longer duration.
What is the difference between price risk and reinvestment risk?
Price risk: • Refers to the potential change in the value of a bond or bond portfolio due to changes in interest rates. • Occurs when market interest rates change and the value of an investor's bond holdings decreases. Reinvestment risk: • Refers to the potential loss of income or return on an investment due to changes in interest rates. • Occurs when an investor receives income or interest payments on a bond and has to reinvest those payments into a lower yielding security.
The 6-month Treasury bill spot rate is 4%, and the 1-year Treasury bill spot rate is 5%. What is the implied 6-month forward rate for six months from now?
The given rates are annual rates, but each period is a half-year. Therefore, the per period spot rates are 2.5% on one-year bonds and 2% on six-month bonds. The semiannual forward rate is obtained by solving for f in the following equation: 1 + f = 1.025^2/1.02 = 1.030 This means that the forward rate is 0.030 = 3.0% semiannually, or 6.0% annually.
A 9-year bond paying coupons annually has a yield of 10% and a duration of 7.194 years. If the market yield changes by 50 basis points, what is the percentage change in the bond's price?
The percentage change in the bond's price is: -(Duration/1 + y) x change in y = 7.194/1.1 x 0.005 = -0.0327 = -3.27% or a 3.2% decline
The table below shows, respectively, the characteristics of two annual-coupon bonds from the same issuer with the same priority in the event of default, as well as spot interest rates on zero-coupon bonds. Neither coupon bond's price is consistent with these spot rates. Using the information in these tables, recommend either bond A or bond B for purchase. Bond Characteristics: Bond A | Bond B Coupons | Annual | Annual Maturity | 3 years | 3 years Coupon rate | 10% | 6% Yield to maturity | 10.65% | 10.75% Price | 98.4 | 88.34 Spot Interest Rates: Term (years) | Spot rates (zero-coupon) 1 | 5% 2 | 8 3 | 11
The present value of each bond's payments can be derived by discounting each cash flow by the appropriate rate from the spot interest rate (i.e., the pure yield) curve: Bond A: PV = 10/1.05 + 10/1.08^2 + 110/1.11^3 = $98 53 Bond B: PV = 6/1.05 + 6/1.08^2 + 106/1.11^3 = 88.36 Bond A sells for $0.13 (i.e., 0.13% of par value) less than the present value of its stripped payments. Bond B sells for $0.02 less than the present value of its stripped payments. Bond A is more attractively priced.
Currently, the term structure is as follows: 1 year zero coupon bonds yield 7%, 2 year zero coupon bonds yield 8%, 3 year and longer maturity zero coupon bonds all yield 9%. You are choosing between 1, 2, and 3 year bonds all paying annual coupons of 8%. a) What is the price of each bond today? b) What will be the price of each bond in one year if the yield curve is flat at 9% at that time? c) What will be the rate of return on each bond?
The table below shows the holding period returns for each of the three bonds: Maturity | 1 year | 2 years | 3 years YTM at beginning of year | 7.00% | 8.00% | 9.00% Beginning of year prices | $1,009.35 | $1,000.00 | $974.69 Prices at year end (at 9% YTM) | $1,000.00 | $990.83 | $982.41 Capital gain | -$9.35 | -$9.17 | $7.72 Coupon | $80.00 | $80.00 | $80.00 1-year total $ return | $70.65 | $70.83 | $87.72 1-year total rate of return | 7.00% | 7.08% | 9.00% You should buy the 3-year bond because it provides a 9% holding-period return over the next year, which is greater than the return on either of the other bonds.
Patrick Wall is considering the purchase of one of the 2 bonds described in the following table. Wall realizes his decision will depend primarily on effective duration, and he believes that interest rates will decline by 50 basis points at all maturities over the next 6 months. Characteristic | CIC | PTR Market Price | 101.75 | 101.75 Maturity Date | June 1, 2030 | June 1, 2030 Call Date | Noncallable | June 1, 2025 Annual Coupon | 5.25% | 6.35% Interest Payment | Semiannual | Semiannual Effective Duration | 7.35 | 5.40 Yield to Maturity | 5.02% | 6.10% Credit Rating | A | A a) Calculate the percentage price change forecasted by the effective duration for both the CIC and the PTR bonds if interest rates decline by 50 basis points over the next 6 months. b) Calculate the 6-month horizon return (in percent) for each bond if the CIC bond price at the end of 6 months equals 105.55 and the PTR bond price equals 104.15. c) Wall is surprised by the fact that although interest rates fell by 50 basis points, the actual price change for the CIC bond was greater than the price change forecasted by the effective duration, whereas the actual price change for the PTR bond was less than the price change forecasted by effective duration. Explain why the actual price change would be greater for the CIC bond and less for the PTR bond.
a) % price change = (−Effective duration) × Change in YTM (%) CIC: (−7.35) × (−0.50%) = 3.675% PTR: (−5.40) × (−0.50%) = 2.700% b) Since we are asked to calculate horizon return over a period of only one coupon period, there is no reinvestment income. Horizon return = (Coupon Payment + Year-end Price - Initial Price) / Initial Price CIC = (26.25 + 1055.50 - 1017.50) / 1017.50 = 0.06314 = 6.314%\ PTR = (31.75 + 1041.50 - 1017.50) / 1017.50 = 0.05479 = 5.479% c) Notice that CIC is non-callable but PTR is callable. Therefore, CIC has positive convexity, while PTR has negative convexity. Thus, the convexity correction to the duration approximation will be positive for CIC and negative for PTR.
The following table shows the yields to maturity of zero-coupon Treasury securities: Term to Maturity (years) | Yield to Maturity (%) 1 | 3.5 2 | 4.5 3 | 5 4 | 5.5 5 | 6 6 | 6.5 a) Calculate the forward 1-year rate of interest for year 3. b) Describe the conditions under which the calculated forward rate would be an unbiased estimate of the 1-year spot rate of interest for that year. c) Assume that a few months earlier the forward 1-year rate of interest for that year had been significantly higher than what it is now. What factors could account for the decline in the forward rate?
a) (1+y4 )^4 = (1+ y3 )^3 (1 + f 4 ) (1.055)^4 = (1.05)^3 (1 + f 4 ) 1.2388 = 1.1576 (1 + f 4 ) ⇒ f 4 = 0.0701 = 7.01% b) The conditions would be those that underlie the expectations theory of the term structure: risk neutral market participants who are willing to substitute among maturities solely on the basis of yield differentials. This behavior would rule out liquidity or term premia relating to risk. c) Under the expectations hypothesis, lower implied forward rates would indicate lower expected future spot rates for the corresponding period. Since the lower expected future rates embodied in the term structure are nominal rates, either lower expected future real rates or lower expected future inflation rates would be consistent with the specified change in the observed (implied) forward rate.
The term structure for zero-coupon bonds is currently: Maturity (years) | YTM (%) 1 | 4% 2 | 5% 3 | 6% Next year at this time, you expect it to be: Maturity (years) | YTM (%) 1 | 5% 2 | 6% 3 | 7% a) What do you expect the rate of return to be over the coming 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? c) Is the market's expectation of the return on the 3-year bond greater or less than yours?
a) A 3-year zero coupon bond with face value $100 will sell today at a yield of 6% and a price of: $100/1.06^3 =$83.96 Next year, the bond will have a two-year maturity, and therefore a yield of 6% (from next year's forecasted yield curve). The price will be $89.00, resulting in a holding period return of 6%. b) The forward rates based on today's yield curve are as follows: Year | Forward Rate 2 | (1.05^2 /1.04) - 1 = 6.01% 3 | (1.06^3 /1.052 ) - 1 = 8.03% Using the forward rates, the forecast for the yield curve next year is: Maturity | YTM 1 | 6.01% 2 | (1.0601 × 1.0803)^1/2 - 1 = 7.02% The market forecast is for a higher YTM on 2-year bonds than your forecast. Thus, the market predicts a lower price and higher rate of return.
Rank the durations or effective durations of the following pairs of bonds: a) Bond A is a 6% coupon bond, with a 20-year time to maturity selling at par value. Bond B is a 6% coupon bond, with a 20-year time to maturity selling below par value. b) Bond A is a 20-year noncallable coupon bond with a coupon rate of 6%, selling at par. Bond B is a 20-year callable bond with a coupon rate of 7%, also selling at par.
a) Bond B has a higher yield to maturity than bond A since its coupon payments and maturity are equal to those of A, while its price is lower. (Perhaps the yield is higher because of differences in credit risk.) Therefore, the duration of Bond B must be shorter. b) Bond A has a lower yield and a lower coupon, both of which cause Bond A to have a longer duration than Bond B. Moreover, A cannot be called, so that its maturity is at least as long as that of B, which generally increases duration.
a) Assuming that the expectations hypothesis is valid, compute the expected price of the 4-year bond on card 46 at the end of the first year, the second year, the third year, and the fourth year. b) What is the rate of return of the bond in years 1, 2, 3, & 4? Conclude that the expected return equals the forward rate for each year.
a) Expected Price 1 = 792.16 2 = 1000/(1.05x1.06x1.07) = 839.69 3 = 1000/(1.06x1.07) = 881.69 4 = 1000/1.07 = 934.58 b) Expected RR 1 = (839.69/$792.16) - 1 = 6.00% 2 = ($881.68/$839.69) - 1 = 5.00% 3 = ($934.58/$881.68) - 1 = 6.00% 4 = ($1,000.00/$934.58) - 1 = 7.00%
Suppose that the prices of a zero-coupon bonds with various maturities are given in the following table. The face value of each bond is 1000. Maturity (years) | Price 1 | 925.93 2 | 853.39 3 | 782.92 4 | 715.00 5 | 650.00 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 part (b) for a 1-year forward loan beginning in year 4.
a) Maturity (years) | Price | YTM | Forward rate 1 | $925.93 | 8.00% | N/A 2 | $853.39 | 8.25% | 8.50% 3 | $782.92 | 8.50% | 9.00% 4 | $715.00 | 8.75% | 9.50% 5 | $650.00 | 9.00% | 10.00% b) For each 3-year zero issued today, use the proceeds to buy: $782.92/$715.00 = 1.095 four-year zeros Your cash flows are thus as follows: Time | Cash Flow 0 | $0 3 | -$1,000 | The 3-year zero issued at time 0 matures; the issuer pays out $1,000 face value 4 | +$1,095 | The 4-year zeros purchased at time 0 mature; receive face value This is a synthetic one-year loan originating at time 3. The rate on the synthetic loan is 0.095 = 9.5%, precisely the forward rate for year 4. c) For each 4-year zero issued today, use the proceeds to buy: $715.00/$650.00 = 1.100 five-year zeros Your cash flows are thus as follows: Time | Cash Flow 0 | $0 4 | -$1,000 | The 4-year zero issued at time 0 matures; the issuer pays out $1,000 face value 5 | +$1,100 | The 5-year zeros purchased at time 0 mature; receive face value This is a synthetic one-year loan originating at time 4. The rate on the synthetic loan is 0.100 = 10.0%, precisely the forward rate for year 5.
The yield to maturity on 1-year zero-coupon bonds is currently 7%; the YTM on 2-year 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 for which the bond will sell next year/ d) Recalculate your answer to part (c) if you believe in the liquidity preference theory and you believe that the liquidity premium is 1%.
a) P = $9/1.07 + $109/1.08^2 = $101.86 b) N = 2, FV = 100, PMT = 9, PV = -101.86 Solving for I, YTM = 7.958% c) The forward rate for next year, derived from the zero-coupon yield curve, is the solution for f 2 in the following equation: 1 + f2 = (1.08)^2/1.07 = 1.0901 -> f2 = 0.0901 = 9.01% Therefore, using an expected rate for next year of r2 = 9.01%, we find that the forecast bond price is: P = $109/1.0901 = $99.99 d) If the liquidity premium is 1% then the forecast interest rate is: E(r2) = f2 - liquidity premium = 9.01% - 1.00% = 8.01% The forecast of the bond price is: 109/1.0801 = $100 92
Pension funds pay lifetime annuities to recipients. If a firm will remain in business indefinitely, the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2 million per year to beneficiaries. The yield to maturity on all bonds is %16. a) If the duration of a 5-year maturity bonds with coupon rates of 12% (paid annually) is four years and the duration of 20-year maturity bonds with coupon rates of 6% (paid annually) is 11 years, how much of each of these coupon bonds (in market value) will you want to hold to both fully fund and immunize your obligation? b) What will be the par value of your holdings?
a) PV of obligation = $2 million/0.16 = $12.5 million Duration of obligation = 1.16/0.16 = 7.25 years Call w the weight on the 5-year maturity bond (which has duration of 4 years). Then: (w × 4) + [(1 - w) × 11] = 7.25 ⇒ w = 0.5357 Therefore: 0.5357 × $12.5 = $6.7 million in the 5-year bond and 0.4643 × $12.5 = $5.8 million in the 20-year bond. b) The price of the 20-year bond is: [$60 × Annuity factor (16%, 20)] + [$1,000 × PV factor (16%, 20)] = $407.12 Therefore, the bond sells for 0.4071 times its par value, and: Market value = Par value × 0.4071 $5.8 million = Par value × 0.4071 ⇒ Par value = $14.25 million Another way to see this is to note that each bond with par value $1,000 sells for $407.12. If total market value is $5.8 million, then you need to buy approximately 14,250 bonds, resulting in total par value of $14.25 million.
You will be paying 10,000 a year in tuition expenses at the end of the next 2 years. Bonds currently yield 8%. a) What are the present value and duration of your obligation? b) What maturity zero-coupon bond would immunize your obligation? c) Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your tuition obligation? d) What if rates fall immediately to 7%?
a) PV of the obligation = $10,000 × Annuity factor (8%, 2) = $17,832.65 Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 8%) | Weight Column (1) × Column (4) 1 | $10,000.00 | $9,259.259 | 0.51923 | 0.51923 2 | $10,000.00 | $8,573.388 | 0.48077 | 0.96154 Column Sums (starting from column 2) | $17,832.647 | 1.00000 | 1.48077 Duration = 1.4808 years b) A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be: $17,832.65 × 1.08^1.4808 = $19,985.26 c) If the interest rate increases to 9%, the zero-coupon bond would decrease in value to: $19,985.26/1.09^1.4808 = $17,590.92 The present value of the tuition obligation would decrease to: $17,591.11 The net position decreases in value by: $0.19 If the interest rate decreases to 7%, the zero-coupon bond would increase in value to: $19,985.26/1.07^1.4808 = $18,079.99 The present value of the tuition obligation would increase to: $18,080.18 The net position decreases in value by: $0.19 The reason the net position changes at all is that, as the interest rate changes, so does the duration of the stream of tuition payments.
Prices of zero-coupon bonds reveal the following pattern of forward rates: Year | Forward Rate 1 | 5% 2 | 7 3 | 8 In addition to the zero-coupon bond, investors may also purchase a 3-year bond making annual payments of $60 with par value $1000. 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 one year will be flat at 7%, what is your forecast for the expected rate of return on the coupon?
a) Solve for PV of $1 received at period end. Year 1 = $1/1.05 = $0.9524 Year 2 = $1/(1.05×1.07) = $0.8901 Year 3 = $1/(1.05×1.07×1.08) = $0.8241 Price = ($60 × 0.9524) + ($60 × 0.8901) + ($1,060 × 0.8241) = $984.14 b) To find the yield to maturity, solve for y in the following equation: $984.10 = [$60 × Annuity factor (y, 3)] + [$1,000 × PV factor (y, 3)] This can be solved using a financial calculator to show that y = 6.60% c) Period | Payment received at end of period: | Will grow by a factor of: | To a future value of 1 | $60.00 | 1.07 × 1.08 | $69.34 2 | $60.00 | 1.08 | $64.80 3 | $1,060.00 | 1.00 | $1,060.00 $69.34+$64.80+$1,060.00=$1,194.14 $984.10 × (1 + y realized) 3 = $1,194.14 1 + y realized = (1194.14/984.10)^1/3 = 1.0666 => y realized = 6.66% d) Next year, the price of the bond will be: [$60 × Annuity factor (7%, 2)] + [$1,000 × PV factor (7%, 2)] = $981.92 Therefore, there will be a capital loss equal to: $984.10 - $981.92 = $2.18 The holding period return is: (60 + -2.18)/$984.10 = 0.0588 = 5.88%
Below is a list of prices for zero-coupon bonds of various maturities: Maturity (Years) | Price of $1000 Par Bond (Zero-Coupon) 1 | $943.40 2 | 873.52 3 | 816.37 a) An 8.5% coupon $1000 par bond pays an annual coupon and will mature in 3 years. What should the yield to maturity on the bond be? b) If at the end of the first year the yield curve flattens out at 8%, what will be the 1-year holding period on the coupon bond?
a) The current bond price is: ($85 × 0.94340) + ($85 × 0.87352) + ($1,085 × 0.81637) = $1,040.20 This price implies a yield to maturity of 6.97%, as shown by the following: [$85 × Annuity factor (6.97%, 3)] + [$1,000 × PV factor (6.97%, 3)] = $1,040.17 b) If one year from now y = 8%, then the bond price will be: [$85 × Annuity factor (8%, 2)] + [$1,000 × PV factor (8%, 2)] = $1,008.92 The holding period rate of return is: [$85 + ($1,008.92 - $1,040.20)]/$1,040.20 = 0.0516 = 5.16%
My pension plan will pay me $10,000 once a year for a 10-year period. The first payment will come in exactly 5 years. The pension fund wants to immunize its position. a) What is the duration of its obligation to me? The current interest rate is 10% per year. b) If the plans uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond? c) What will be the face value of the holdings in each zero?
a) The duration of the annuity if it were to start in 1 year would be: Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 10%) | Weight Column (1) × Column (4) 1 | $10,000 | $9,090.909 | 0.14795 | 0.14795 2 | $10,000 | $8,264.463 | 0.13450 | 0.26900 3 | $10,000 | $7,513.148 | 0.12227 | 0.36682 4 | $10,000 | $6,830.135 | 0.11116 | 0.44463 5 | $10,000 | $6,209.213 | 0.10105 | 0.50526 6 | $10,000 | $5,644.739 | 0.09187 | 0.55119 7 | $10,000 | $5,131.581 | 0.08351 | 0.58460 8 | $10,000 | $4,665.074 | 0.07592 | 0.60738 9 | $10,000 | $4,240.976 | 0.06902 | 0.62118 10 | $10,000 | $3,855.433 | 0.06275 | 0.62745 Column Sums (starting from column 2) | $61,445.671 | 1.00000 | 4.72546 D = 4.7255 years Because the payment stream starts in five years, instead of one year, we add four years to the duration, so the duration is 8.7255 years. b) The present value of the deferred annuity is: [10,000 x Annuity factor (10%, 10)]/1.10^4 = $41,968 Call w the weight of the portfolio invested in the 5-year zero. Then: (w × 5) + [(1 - w) × 20] = 8.7255 ⇒ w = 0.7516 The investment in the 5-year zero is equal to: 0.7516 × $41,968 = $31,543 The investment in the 20-year zeros is equal to: 0.2484 × $41,968 = $10,423 These are the present or market values of each investment. The face values are equal to the respective future values of the investments. The face value of the 5-year zeros is: $31,543 × (1.10)^5 = $50,801 Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000, would be purchased. Similarly, the face value of the 20-year zeros is: $10,425 × (1.10)^20 = $70,123
You are managing a portfolio of $1 million. Your target duration is 10 years, and you can invest in 2 bonds, a zero-coupon bond with maturity of 5 years and a perpetuity, each yield 5%. a) How of (i) the zero-coupon bond and (ii) the perpetuity will you hold in your portfolio? b) How will these fractions change next year if target duration is now 9 years?
a) The duration of the perpetuity is: 1.05/0.05 = 21 years Call w the weight of the zero-coupon bond. Then: (w × 5) + [(1 - w) × 21] = 10 ⇒ w = 11/16 = 0.6875 Therefore, the portfolio weights would be as follows: 11/16 invested in the zero and 5/16 in the perpetuity. b) Next year, the zero-coupon bond will have a duration of 4 years and the perpetuity will still have a 21-year duration. To obtain the target duration of nine years, which is now the duration of the obligation, we again solve for w: (w × 4) + [(1 - w) × 21] = 9 ⇒ w = 12/17 = 0.7059 So, the proportion of the portfolio invested in the zero increases to 12/17 and the proportion invested in the perpetuity falls to 5/17.
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? b) What is the yield to maturity of the 2-year coupon bond? c) What is the forward rate for the second year? d) According to the expectations hypothesis, what are (i) the expected price of the coupon bond at the end of the first year and (ii) the expected holding-period return on the coupon bond over the first year?
a) The one-year zero-coupon bond has a yield to maturity of 6%, as shown below: $94.34 = 100/(1 + y1) => y1 = 0.060000 = 6% The yield on the two-year zero is 8.472%, as shown below: $84.99 = 100/(1 + y2)^2 => y2 = 0.08472 = 8.472% The price of the coupon bond is: 12/1.06 + 112/(1.08472)^2 = 106.51 Therefore: yield to maturity for the coupon bond = 8.333% [On a financial calculator, enter: n = 2; PV = -106.51; FV = 100; PMT = 12] b) f2 = (1 + y2)^2/(1 + y1) - 1 = (1.08472)^2/1.06 - 1 = 0.11 = 11% c) Expected price = 112/1.11 = 100.90 Expected holding period return = [$12 ($100 90. $106 51. )]/106.51 = 0.06 = 6% This holding period return is the same as the return on the one-year zero. d) If there is a liquidity premium, then: E(r2) < f 2 E(Price) = 112/1+ER2 > $100.90 E(HPR) > 6%
The yield to maturity (YTM) on a 1-year zero-coupon bonds is 5% and the YTM on 2-year zeros is 6%. The YTM on 2-year maturity coupon bonds with coupon rates of 12% (paid annually) is 5.8%. a) What arbitrage opportunity is available for an investment banking firm? b) What is the profit on the activity?
a) The price of the coupon bond, based on its yield to maturity, is: [$120 × Annuity factor (5.8%, 2)] + [$1,000 × PV factor (5.8%, 2)] = $1,113.99 If the coupons were stripped and sold separately as zeros, then, based on the yield to maturity of zeros with maturities of one and two years, respectively, the coupon payments could be sold separately for: 120/1.05 + 1120/1.06^2 = 1111.08 The arbitrage strategy is to buy zeros with face values of $120 and $1,120, and respective maturities of one year and two years, and simultaneously sell the coupon bond. The profit equals $2.91 on each bond.
You observe the following term structure: Bond | Effective Annual YTM 1-year zero-coupon bond | 6.1% 2-year zero-coupon bond | 6.2% 3-year zero-coupon bond | 6.3% 4-year zero-coupon bond | 6.4% a) If you believe that the term structure next year will be the same as today's calculate the return on (i) the 1-year zero and (ii) the 4-year zero. b) Which bond provides a greater expected 1-year return? c) Redo your answers to part (a) and (b) if you believed in the expectations hypothesis.
a) The return on the one-year zero-coupon bond will be 6.1%. The price of the 4-year zero today is: $1,000/1.0644 = $780.25 Next year, if the yield curve is unchanged, today's 4-year zero coupon bond will have a 3-year maturity, a YTM of 6.3%, and therefore the price will be: $1,000/1.0633 = $832.53 The resulting one-year rate of return will be: 6.70% Therefore, in this case, the longer-term bond is expected to provide the higher return because its YTM is expected to decline during the holding period. b) If you believe in the expectations hypothesis, you would not expect that the yield curve next year will be the same as today's curve. The upward slope in today's curve would be evidence that expected short rates are rising and that the yield curve will shift upward, reducing the holding period return on the four-year bond. Under the expectations hypothesis, all bonds have equal expected holding period returns. Therefore, you would predict that the HPR for the 4-year bond would be 6.1%, the same as for the 1-year bond.
An insurance company must make payments to a customer of $10 million in one year and $4 million in 5 years. The yield curve is flat at 10%. a) If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase/ b) What must be the face value and market value of that zero-coupon bond?
a) Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 10%) | Weight Column (1) × Column (4 1 | $10 million | $9.09 million | 0.7857 | 0.7857 5 | $4 million | $2.48 million | 0.2143 | 1.0715 Column Sums (starting from column 2) | $11.57 million | 1.000 | 1.8572 D = 1.8572 years = required maturity of zero coupon bond. b) The market value of the zero must be $11.57 million, the same as the market value of the obligations. Therefore, the face value must be: $11.57 million × (1.10)^1.8572 = $13.81 million
The current yield curve for default-free zero-coupon bonds is as follows: Maturity (years) | YTM (%) 1 | 10 2 | 11 3 | 12 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 yield to maturity on 1-year zero-coupon bonds next year? c) What about the yield on 2-year zeros? d) If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over the next year?
a) We obtain forward rates from the following table: Maturity | YTM | Forward Rate | Price (for parts c, d) 1 year | 10% | N/A | $1,000/1.10 = $909.09 2 years | 11% | (1.112 /1.10) - 1 = 12.01% | $1,000/1.11^2 = $811.62 3 years | 12% (1.123 /1.112 ) - 1 = 14.03% | $1,000/1.12^3 = $711.78 b) We obtain next year's prices and yields by discounting each zero's face value at the forward rates for next year that we derived in part (a): Maturity | Price | YTM 1 year | $1,000/1.1201 = $892.78 | 12.01% 2 years | $1,000/(1.1201 × 1.1403) = $782.93 | 13.02% c) Next year, the 2-year zero will be a 1-year zero, and will therefore sell at a price of: $1,000/1.1201 = $892.78 Similarly, the current 3-year zero will be a 2-year zero and will sell for: $782.93 Expected total rate of return: 2-year bond: 892.78/811.62 - 1 = 1.1 - 1 = 10% 3-year bond: 782.93/711.78 - 1 = 1.1 - 1 = 10% d) The current price of the bond should equal the value of each payment times the present value of $1 to be received at the "maturity" of that payment. The present value schedule can be taken directly from the prices of zero-coupon bonds calculated above. Current price = ($120 × 0.90909) + ($120 × 0.81162) + ($1,120 × 0.71178) = $109.0908 + $97.3944 + $797.1936 = $1,003.68 Similarly, the expected prices of zeros one year from now can be used to calculate the expected bond value at that time: Expected price 1 year from now = ($120 × 0.89278) + ($1,120 × 0.78293) = $107.1336 + $876.8816 = $984.02 Total expected rate of return = [$120 ($984.02. $ ,1,003.68. )]/$ ,1 003 68 = 0.1000 = 10%
The following is a list of prices for zero-coupon bonds of various maturities: Maturity | Price of Bond 1 | $943.40 2 | 898.47 3 | 847.62 4 | 792.16 a) Calculate the yield to maturity for a bond with a maturity of 1 year, 2 years, 3 years, & 4 years. b) Calculate the forward rate for the 2nd, 3rd, and 4th year.
a) YTM = (1000/Price)^1/n - 1 1 = ~6% 2 = ~5.5% 3 ~ ~5.67% 4 ~ ~6% b) 2. (1.055^2/1.06) - 1 = 5% 3. (1.0567^3/1.055^2) - 1 = 6% 4. (1.06^4/1.0567^3) - 1 = 7%
a) Find the duration of a 6% coupon bond making annual coupon payments if it has 3 years until maturity and has a yield to maturity of 6%. b) What is the duration if the yield to maturity is 10%? c&d) Repeat the problems above but now assume the coupons are paid semiannually.
a) YTM = 6% Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 6%) | Weight Column (1) × Column (4) 1 | $60.00 | $56.60 | 0.0566 | 0.0566 2 | $60.00 | $53.40 | 0.0534 | 0.1068 3 | $1,060.00 | $890.00 | 0.8900 | 2.6700 Column Sums (starting from column 2) | $1,000.00 | 1.0000 | 2.8334 Duration = 2.833 years b) YTM = 10% Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 10%) | Weight Column (1) × Column (4) 1 | $60.00 | $54.55 | 0.0606 | 0.0606 2 | $60.00 | $49.59 | 0.0551 | 0.1102 3 | $1,060.00 | $796.39 | 0.8844 | 2.6532 Column Sums (starting from column 2) | $900.53 | 1.0000 | 2.8240 Duration = 2.824 years, which is less than the duration at the YTM of 6%. c) For a semiannual 6% coupon bond selling at par, we use the following parameters: coupon = 3% per half-year period, y = 3%, T = 6 semiannual periods. Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 3%) | Weight Column (1) × Column (4) 1 | $3.00 | $2.913 | 0.02913 | 0.02913 2 | $3.00 | $2.828 | 0.02828 | 0.05656 3 | $3.00 | $2.745 | 0.02745 | 0.08236 4 | $3.00 | $2.665 | 0.02665 | 0.10662 5 | $3.00 | $2.588 | 0.02588 | 0.12939 6 | $103.00 | $86.261 | 0.86261 | 5.17565 Column Sums (starting from column 2) | $100.000 | 1.00000 | 5.57971 D = 5.5797 half-year periods = 2.7899 years d) If the bond's yield is 10%, use a semiannual yield of 5%, and semiannual coupon of 3%: Time until Payment (years) | Cash Flow | PV of CF (Discount rate = 5%) | Weight Column (1) × Column (4) 1 | $3.00 | $2.857 | 0.03180 | 0.03180 2 | $3.00 | $2.721 | 0.03029 | 0.06057 3 | $3.00 | $2.592 | 0.02884 | 0.08653 4 | $3.00 | $2.468 | 0.02747 | 0.10988 5 | $3.00 | $2.351 | 0.02616 | 0.13081 6 | $103.00 | $76.860 | 0.85544 | 5.13265 Column Sums (starting from column 2) | $89.849 | 1.00000 | 5.55223 D = 5.5522 half-year periods = 2.7761 years
As part of your debt analysis of debt issued by Monticello Corporation, you are asked to evaluate two of its bond issues, shown in the following table. Second Column is Bond A (Callable) Third Column is Bond B (Noncallable) Maturity | 2030 | 2030 Coupon | 11.50% | 7.25% Current Price | 125.75 | 100 Yield to Maturity | 7.7% | 7.25% Modified Duration to Maturity | 6.2 | 6.8 Call Date | 2024 | -- Call Price | 105 | -- Yield to Call | 5.10% | -- Modified Duration to Call | 3.1 | -- a) Using the duration and yield information in the table above, compare the price and yield behavior of the two bonds under each of the following 2 scenarios: i. Strong economic recovery with rising inflation expectations. II. Economic recession with reduced inflation expectations. b) Using the information in the table, calculate the projected price change for bond B if its yield to maturity falls by 75 basis points. c) Describe the shortcoming of analyzing bond A strictly to call or to maturity.
a) i. Strong economic recovery with rising inflation expectations. Interest rates and bond yields will most likely rise, and the prices of both bonds will fall. The probability that the callable bond will be called would decrease, and the callable bond will behave more like the non-callable bond. (Note that they have similar durations when priced to maturity). The slightly lower duration of the callable bond will result in somewhat better performance in the high interest rate scenario. ii. Economic recession with reduced inflation expectations. Interest rates and bond yields will most likely fall. The callable bond is likely to be called. The relevant duration calculation for the callable bond is now modified duration to call. Price appreciation is limited as indicated by the lower duration. The non-callable bond, on the other hand, continues to have the same modified duration and hence has greater price appreciation. b) Projected price change = (modified duration) × (change in YTM) = (-6.80) × (-0.75%) = 5.1% Therefore, the price will increase to approximately $105.10 from its current level of $100. c) For Bond A, the callable bond, bond life and therefore bond cash flows are uncertain. If one ignores the call feature and analyzes the bond on a "to maturity" basis, all calculations for yield and duration are distorted. Durations are too long and yields are too high. On the other hand, if one treats the premium bond selling above the call price on a "to call" basis, the duration is unrealistically short and yields too low. The most effective approach is to use an option valuation approach. The callable bond can be decomposed into two separate securities: a non-callable bond and an option: Price of callable bond = Price of non-callable bond - price of option Since the call option always has some positive value, the price of the callable bond is always less than the price of the non-callable security.
Which one of the following statements about the term structure of interest rates is true? a) The expectations hypothesis predicts a flat yield curve if anticipated future short-term rates exceed current short term rates. b) The expectations hypothesis contends that the long-term spot rate is equal to the anticipated short-term rate. c) The liquidity premium theory indicates that, all else being equal, longer maturity bonds will have lower yields. d) The liquidity preference theory contends that lenders prefer to buy securities at the short end of the yield curve.
d) The liquidity preference theory contends that lenders prefer to buy securities at the short end of the yield curve.
A 6% coupon bond with semiannual coupons has a convexity (in years) of 120, sells for 80% of par, and is priced at a yield to maturity of 8%. If the YTM increases to 9.5%, what is the predicted contribution of convexity to the percentage change in price due to convexity?
½ × 120 × (0.015)2 = 0.0135 = 1.35%
Will a bond price rise more for a given % rate decrease, or fall more for the same % increase?
• A bond price will typically rise more for a given percentage decrease in interest rates than it will fall for the same percentage increase in interest rates. • This is because interest rate risk is non-linear and asymmetrical, meaning that the relationship between bond prices and interest rates is not proportional. • When interest rates decrease, the yield of a bond decreases, which increases the price of the bond. • When interest rates increase, the yield of a bond increases, which decreases the price of the bond. • The magnitude of the impact of changes in interest rates on bond prices is not proportional, and the impact can be more significant for a given percentage decrease in interest rates than for the same percentage increase.
How is a coupon paying bond like a portfolio of ZCB's?
• A coupon paying bond can be seen as a combination of zero-coupon bonds (ZCBs) of different maturities. • The coupon payments on the bond can be thought of as the return on investing in a series of ZCBs at different times. • Each ZCB has its own maturity date, and its value is based on its face value and the current market rate for bonds of that maturity. • By combining these ZCBs, the coupon paying bond effectively replicates the cash flows of a portfolio of ZCBs with different maturities.
What is a forward rate?
• A rate at which an investor can invest funds for a specific future period of time and receive a specific rate of return. • A rate for a future time period, rather than the present. • Determined based on the current spot rates and expectations for future interest rate changes. • Often used as a benchmark for pricing financial instruments, such as interest rate derivatives.
What is a spot interest rate?
• A rate at which an investor can invest funds for a specific period of time and receive a specific rate of return. • A rate for a particular point in time, rather than an average rate over a period of time. • Often used to determine the value of a financial instrument, such as a bond.
What is a short rate?
• A short rate is the rate for a very short-term investment, typically one day or one week. • It is often used as a benchmark for pricing short-term financial instruments. • It is considered to be the most basic building block in modeling the term structure of interest rates.
What are pure yield pickup swaps?
• A type of financial instrument that offers higher yield compared to traditional fixed-income securities. • Involves swapping lower-yielding for higher-yielding fixed income securities. • Goal: enhance return of fixed-income portfolio without increasing risk. • Used by institutional investors and large hedge funds. • Can generate additional income but also involves risks such as credit, market, and liquidity risk.
What is a yield curve?
• A yield curve is a graph that shows the relationship between the yield of a bond and its maturity. • It is used to predict future interest rate changes, assess the risk of bonds, and determine the relative value of bonds with different maturities. • The shape of the yield curve provides information about market expectations for future interest rates and current market conditions. • The yield curve is a key tool for analyzing market expectations and conditions and is regularly monitored by investors and analysts.
How does active bond management seek to maximize portfolio profits?
• Active bond management aims to generate returns that exceed a benchmark or market index. • It involves making investment decisions such as buying, selling, and holding different types of bonds to achieve this objective. • The bond manager may seek to take advantage of market inefficiencies, identify undervalued securities, or use interest rate and credit analysis to make informed investment decisions. • The manager may also make decisions about the types of bonds, the duration and credit quality of the bonds, and the geographical and sector diversification of the portfolio. • The goal of active bond management is to achieve a higher return on investment compared to a passive strategy, while also considering the level of risk involved.
Which shapes are considered "normal" and "inverted?"
• An upward sloping shape is considered "normal." This shape reflects higher yields for longer maturity bonds, reflecting higher expected interest rates in the future. • A downward sloping shape is considered "inverted." This shape reflects lower yields for longer maturity bonds, reflecting lower expected interest rates in the future or a flight to safety.
What is a bond index? What is a bond index strategy?
• Bond index: A bond index is a benchmark that tracks the performance of a specific segment of the fixed income market. • It is composed of a set of bonds with specific characteristics, such as maturity, credit quality, and sector. • Bond index strategy: A bond index strategy refers to an investment strategy that seeks to replicate the performance of a bond index by investing in a portfolio of bonds that closely matches the composition of the benchmark index. The goal is to achieve a return that closely tracks the performance of the bond market.
What is bond portfolio immunization?
• Bond portfolio immunization is a strategy used in fixed income management to reduce the interest rate risk in a bond portfolio. • The goal of this strategy is to create a bond portfolio with a duration that matches the investment horizon of the portfolio. In this way, the bond portfolio is protected from changes in market interest rates because the price change of the bonds due to changes in interest rates will offset each other. This helps to ensure that the value of the portfolio remains stable over the investment horizon, regardless of changes in interest rates. • To achieve this, the bond portfolio manager may need to make active adjustments to the portfolio, such as rebalancing, or replacing bonds as they approach maturity.
How are bond prices and market yields related?
• Bond prices and market yields are inversely related. • When the price of a bond increases, its yield decreases. • When the price of a bond decreases, its yield increases. • The relationship between bond prices and yields is affected by market interest rates, credit risk, and time remaining until maturity.
How is a bond portfolio immunized?
• Determine the target portfolio duration: The target portfolio duration is the desired sensitivity of the portfolio to changes in interest rates, taking into account the portfolio's investment objectives and constraints. • Choose the right mix of bonds: The bonds selected should have durations that match the target portfolio duration as closely as possible. The bonds can be Treasury bonds, corporate bonds, or other types of fixed-income securities. • Rebalance the portfolio as needed: The portfolio should be regularly rebalanced to maintain the desired duration, taking into account any changes in market conditions, interest rates, and individual bond prices. • Monitor the portfolio: It is important to continuously monitor the portfolio and make changes as needed to ensure that it remains immunized against interest rate risk.
How is duration related to high vs low coupon rate (ceteris paribus)?
• Duration generally increases with coupon rate (ceteris paribus) • A high coupon rate bond has more cash flows (coupon payments) to reinvest, so the duration is typically longer • The reinvestment of coupon payments and bond's cash flows are longer in duration, making the bond more sensitive to interest rate changes • A low coupon rate bond has fewer cash flows (coupon payments), so the duration is typically shorter
Why does duration only approximate the percent change in price for a given percent change in yield?
• Duration is a linear approximation of the price sensitivity of a bond to changes in interest rates. • It assumes a small change in interest rates and a corresponding small change in the bond's price. • It only provides an estimate of price change and not an exact prediction. • Factors such as credit risk, optionality, and the shape of the yield curve can have a significant impact on a bond's price. • In practice, the relationship between bond price and interest rates can be much more complex than what is captured by duration alone.
What is duration in finance?
• Duration is a measure of a bond's sensitivity to changes in interest rates. • It reflects the average time, in years, that an investor will receive the bond's cash flows. • The longer the duration, the more sensitive the bond's price is to changes in interest rates. • Duration is calculated by weighting the time period of each cash flow by the size of the cash flow and the present value of the cash flow. • A bond's duration can be used to estimate the change in its price given a change in interest rates. • The formula for duration is: Duration = sum of present value of cash flows / present value of bond.
Is duration higher or lower when YTM is lower?
• Duration is lower when YTM is lower.
How is duration used to measure interest rate risk?
• Duration is used to measure the sensitivity of the bond's price to changes in interest rates. • A bond with a longer duration will have a greater sensitivity to changes in interest rates compared to a bond with a shorter duration. • A bond's duration is a useful tool for investors to understand the interest rate risk associated with a bond investment. • The duration of a bond can be used to estimate the percentage change in the price of a bond given a 1% change in interest rates. • Investors can use duration to compare the interest rate risk of bonds with different maturities and coupon rates. • The duration of a portfolio of bonds can also be calculated to estimate the portfolio's overall interest rate sensitivity.
What are the implications of various shapes of the yield curve under each of the theories?
• Expectations Theory: • Upward sloping (normal) shape: Expectations for higher future interest rates, potentially indicating economic growth and increased inflation. • Downward sloping (inverted) shape: Expectations for lower future interest rates, potentially indicating economic slowdown and decreased inflation. • Flat shape: Uncertainty about future interest rate changes, potentially indicating economic uncertainty. • Liquidity Preference Theory: • Normal: upward slope, healthy economy with growth/inflation expectations • Inverted: downward slope, signals upcoming recession • Flat: short/long-term rates similar, neutral outlook • Steep: significant difference in short/long-term rates, strong economy expectations
What are the two theories that may explain the term structure of interest rates?
• Expectations Theory: This theory suggests that the term structure of interest rates reflects market expectations for future interest rates. Under this theory, longer-term rates are influenced by expectations for future short-term rates. • Liquidity Preference: The liquidity preference theory by Keynes argues that people demand money not just for transactions but also for the ease of accessing cash. The theory suggests that interest rates are determined by the supply and demand for money, with higher interest rates reducing demand and lower interest rates increasing demand for money. • Long-term spot rates include risk premium to compensate investors for interest rate risk
What are rate anticipation swaps?
• Financial instrument that allows investors to speculate on changes in interest rates. • Involves swapping a fixed rate for a floating rate. • Used to benefit from expected changes in interest rates. • Primarily used by institutional investors and large hedge funds. • Can generate profits but also involves risks such as market, credit, and liquidity risk.
How is duration related to length of time to maturity for discount bonds?
• For discount bonds, duration is longer than the time to maturity • This is because a discount bond has a negative yield, which means its cash flows are expected to come later in its life than for a bond with a positive yield • As a result, the price sensitivity to changes in market yields is greater for discount bonds than for bonds selling at or above par value • Therefore, the duration of a discount bond increases with the length of time to maturity, whereas for par and premium bonds, duration decreases with the length of time to maturity.
Which is more sensitive to a given change in market yield: low-yield bonds or high-yield bonds?
• High-yield bonds typically have longer durations and thus have more exposure to changes in interest rates. • The longer duration means that these bonds have more weight on their longer-term cash flows, making them more sensitive to changes in interest rates. • As a result, a change in interest rates can have a greater impact on the price of high-yield bonds compared to low-yield bonds. • When interest rates rise, the price of high-yield bonds will fall more than the price of low-yield bonds. Conversely, when interest rates fall, the price of high-yield bonds will rise more than the price of low-yield bonds. • The additional sensitivity to interest rate changes is why high-yield bonds offer higher yields compared to low-yield bonds. • The higher yield compensates investors for the additional risk associated with these bonds.
What is the risk difference between holding short-term securities and planning to "roll-over" and holding long-term securities and planning to sell prior to maturity?
• Holding short-term securities and planning to "roll-over" involves interest rate risk. • Holding long-term securities and planning to sell prior to maturity involves credit risk. • Long-term securities generally have lower interest rate risk compared to short-term securities.
What is meant by interest rate risk being non-linear and asymmetrical?
• Interest rate risk is considered to be non-linear and asymmetrical because the relationship between bond prices and interest rates is not proportional. • The effect of changes in interest rates on bond prices can vary depending on the magnitude and direction of the change. • A small increase in interest rates may have a larger impact on the price of a long-term bond than a short-term bond. • A decrease in interest rates may have a smaller impact on the price of a long-term bond than an increase of the same magnitude would have had.
What is interest rate risk?
• Interest rate risk refers to the potential for losses due to changes in interest rates. • It affects fixed-income securities such as bonds and other fixed-rate investments. • When interest rates rise, the value of fixed-income securities decreases. • When interest rates fall, the value of fixed-income securities increases. • Interest rate risk is an important factor to consider when investing in fixed-income securities and can impact the return on investment.
What are intermarket spread swaps?
• Intermarket spread swaps are financial instruments used to hedge or speculate on the price differences between two related markets. • They involve swapping the price movements between two different financial assets, such as commodities, bonds, or currencies. • The goal is to take advantage of price differences between the two markets and profit from the change in the spread between them. • These swaps are typically used by institutional investors and large hedge funds, who have the resources and expertise to manage the complex financial transactions involved. • Intermarket spread swaps can be highly speculative and carry a high level of risk, and they are generally not suitable for individual investors or those with limited experience in complex financial instruments.
What should a bond portfolio manager do about anticipated changes in interest rates using portfolio duration to maximize returns?
• Portfolio managers may use bond portfolio duration to prepare for anticipated changes in interest rates. • Duration measures the sensitivity of a bond's price to changes in interest rates. • If rates are expected to rise, the manager may reduce the duration of the portfolio to minimize price losses from rising rates. • If rates are expected to fall, the manager may increase the duration of the portfolio to benefit from potential price gains. • The manager may also use other strategies such as bond yield curve positioning or bond sector allocation to prepare for changes in interest rates. • The goal is to maximize returns while managing interest rate risk.
What are substitution swaps?
• Substitution swaps are financial instruments used in bond portfolio management • They involve replacing one bond with another in a portfolio • The goal is usually to change the duration, credit quality, or coupon rate of the portfolio • Substitution swaps are used as a way to change the risk-return profile of a bond portfolio without selling the original bonds and incurring transaction costs • The new bonds are often chosen to better match the investor's objectives and the prevailing market conditions.
Does the change in price sensitivity to change in maturity increase or decrease with increase in time to maturity?
• The change in price sensitivity to a change in maturity generally increases with an increase in time to maturity. • As the time until maturity of a bond increases, the bond becomes increasingly sensitive to changes in interest rates. • A rise in interest rates will lead to a larger decline in the price of a longer-term bond. • A fall in interest rates will lead to a larger increase in the price of a longer-term bond.
How is duration related to length of time to maturity for par and premium bonds?
• The duration of a par bond and a premium bond is positively related to the length of time to maturity. • As the time to maturity increases, the duration also increases, meaning that the bond's price sensitivity to changes in interest rates increases. • The effect of maturity on duration is greater for premium bonds compared to par bonds. • This is because premium bonds have higher coupons, which results in a larger portion of the bond's cash flows being received sooner.
Is the duration of a perpetuity infinite? Why or why not?
• The duration of a perpetuity can be considered infinite • A perpetuity is a bond with no maturity date and pays a constant coupon payment forever • The present value of the infinite stream of cash flows can be calculated as the coupon payment divided by the yield-to-maturity • The longer the duration, the more sensitive the bond price is to changes in interest rates • The infinite stream of cash flows in a perpetuity makes the duration effectively infinite, since changes in interest rates will continue to affect the present value of the cash flows indefinitely.
Is interest rate sensitivity lower for high coupon rate bonds or low coupon rate bonds?
• The interest rate sensitivity of a bond is generally lower for high coupon rate bonds compared to low coupon rate bonds. • High coupon rate bonds have a greater portion of their return coming from the coupon payments, which are fixed. • Low coupon rate bonds have a smaller portion of their return coming from the coupon payments, which are fixed. • As a result, high coupon rate bonds are typically less sensitive to changes in interest rates than low coupon rate bonds.
How is price volatility related to length of time until maturity?
• The longer the time until maturity of a bond, the higher its price volatility. • Changes in interest rates have a greater impact on the future cash flows of a longer-term bond than on the cash flows of a shorter-term bond. • If interest rates rise, the price of a longer-term bond will fall more than the price of a shorter-term bond. • If interest rates fall, the price of a longer-term bond will rise more than the price of a shorter-term bond.
Is price sensitivity greater for discount bonds or premium bonds (ceteris paribus)?
• The price sensitivity of a bond is generally greater for discount bonds compared to premium bonds, ceteris paribus. Discount bonds have a lower market price relative to their face value. • Premium bonds have a higher market price relative to their face value. • As a result, discount bonds are typically more sensitive to changes in interest rates than premium bonds.
What is the risk premium under the liquidity preference theory and how is it related to forward rates and expected future short rates?
• The risk premium under the Liquidity Preference Theory is the additional return demanded for holding a security with higher risk. • The risk premium is related to the difference between the expected future short-term rate and the current forward rate. • If the expected future short-term rate is higher than the current forward rate, the risk premium is positive. • If the expected future short-term rate is lower than the current forward rate, the risk premium is negative.
What is the Term Structure of Interest Rates?
• The term structure of interest rates refers to the relationship between the yield of a bond and its maturity. • It is represented by a graph of yields for bonds with different maturities. • The shape of the term structure provides information about market expectations for future interest rates and current market conditions. It can change over time as market conditions and expectations change. • The term structure is a crucial indicator for investors and analysts to monitor changes in the broader financial markets.
Why do we typically use U.S. Treasury securities to construct a yield curve?
• U.S. Treasury securities are widely considered safe and liquid. • They are issued by the U.S. government with low credit risk. • They have a wide range of maturities and are actively traded in large volumes. • Their yields are widely followed and used as a benchmark.
How are forward rates and future spot rates related under the expectations theory?
• Under the Expectations Theory, forward rates are based on expectations for future spot rates. • If expectations for future interest rates are upward, forward rates will be higher than the current spot rate. • If expectations for future interest rates are downward, forward rates will be lower than the current spot rate. • Forward rates and future spot rates should be equal under efficient and rational expectations.
How are forward rates and future spot rates related under the liquidity preference theory?
• Under the Liquidity Preference Theory, forward rates reflect a premium required for holding a longer-term investment. • The premium, or "liquidity premium," compensates investors for bearing the risk of a longer investment period. • The liquidity premium can vary based on various factors, such as monetary policy and financial market risk. • The relationship between forward rates and future spot rates is less predictable under the Liquidity Preference Theory.
Which type of investor dominates the market under the liquidity preference theory?
• Under the Liquidity Preference Theory, investors who value liquidity and safety dominate the market for short-term securities. • Investors seeking higher returns and willing to accept longer-term and higher risk investments dominate the market for long-term securities.
What various shapes may a yield curve have?
• Upward sloping (normal): Yields increase as maturity increases, reflecting higher expected interest rates in the future. • Downward sloping (inverted): Yields decrease as maturity increases, reflecting lower expected interest rates in the future or a flight to safety. • Flat: Yields are roughly the same across all maturities, reflecting uncertainty about future interest rate changes. • Humped: Yields are higher for intermediate maturities and lower for both shorter and longer maturities, reflecting a mix of expectations for future interest rates.
What is the duration of a ZCB?
• Zero coupon bond (ZCB) is a bond that pays no coupon (interest) payments and is sold at a discount to its face value • The duration of a ZCB can be calculated as the time it takes for the present value of all its future cash flows (the face value at maturity) to equal its current market price. • ZCBs have longer durations than coupon-paying bonds with similar maturities and face values due to their lack of periodic coupon payments. • The longer the duration of a bond, the more sensitive it is to changes in interest rates.
Are coupon bonds or ZCB's more sensitive to interest rates?
• Zero coupon bonds (ZCBs) are generally more sensitive to changes in interest rates than coupon bonds. • ZCBs do not have any periodic coupon payments to partially offset changes in market yields. • Coupon bonds receive periodic coupon payments, which can partially offset changes in market yields. • As a result, coupon bonds are typically less sensitive to changes in interest rates than ZCBs.
