SOPHIA 2020
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Catégorie :Category: nCreator TI-Nspire
Auteur Author: rikkyriks
Type : Classeur 3.0.1
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Mis en ligne Uploaded: 18/12/2024 - 19:25:10
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Shortlink : http://ti-pla.net/a4420163
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 6.07 Ko KB
Mis en ligne Uploaded: 18/12/2024 - 19:25:10
Uploadeur Uploader: rikkyriks (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4420163
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Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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EXAMEN SOPHIA 2020 3. Bank of America issued a USD 2% of March 31, 2026 yields 1%. You wish to purchase, today (exam date which is December 8th2020) $200,000 FV of this bond. What is your total cost (including accrued) in dollars. (4pts) Bond in $, so coupon paid semi-annually. Corporate bond so ACT/360 convention. If the bond is paid semi-annually, then you do : r/2, c/2, t*2 in the PV formulas. T = 2026-2021 + (31+30+8)/360 =5.2 Accrued Interest = annual coupon * days/360 = 200 000 *0.02 * 69/360 =766.67 PV notional = FV/(1+r)^t = 200 000 / (1+0.01/2)^2*5.2 =189,899.43. PV coupons = c/r * (1-1/(1+r)^t) = (200 000 *0.02/2)/(0.01/2) * (1 1/(1+0.01/2)^(2*5.2)) = 20,219.25 Full bonds price = AI + PV notional + PV coupons = 210 885.35 4. An investor has a 1-year, semiannual, 10% coupon bond priced at 102,5. If the 6- month spot rate on a bond-equivalent basis is 8%, calculate the 1-year theoretical spot rate ? (2 points) Lets calculate the first cash flow : PV1 = (100*0.1/2)/(1+0.08/2) = 4.81 Now lets estimate the second cash flow : PV2 = (100*0.1/2 +100)/(1+x/2)² But we know that the bonds price is equal to the sum of the present value of its cash flow. We can conclude that : Bonds price = PV1 + PV2 Ò 102.5 = 4.81 + (100*0.1/2 +100)/(1+x/2)² Ò x = 7.35% The one-year theorical spot rate is 7.35%. 5. A swap involves paying 3-month Libor and receiving 0,5% on a notional of $60 million. The swap has a remaining life of 7 months. What is the value of the swap ? We assume yield curve is flat and all necessary maturities yield 1%. Last 3-month Libor stands at 0,25%(4 points) FRA 0X3: 0,25% FRA 2X5 : 0,36% FRA 4X7 : 0,40% FRA 1x4: 0.32% FRA 3X6: 0,39% FRA 5 x 8: 0,44% Formula : =SUM [ notional * fixed rate * frequency of payment * exponential (- r *remaining days) ] Frequency of payment : for annual c =12/12, for semi annual c =6/12, for quarterly c= 3/12 for monthly c =1/12 Remaining days : for the 2nd exponential it is the remaining life of the swap (given, here 7/12), for the first exponential it is the difference between the remaining life and the frequency of payment (example here : = 7/12 3/12). Cash Flow from the fixed leg : CF1 = 60*0.05*3/12*exp(-0.01*(7-3)/12) + 60*0.05*3/12*exp(-0.01*7/12) =1 496 100 Cash flow from the floating leg : CF2 = 60*Last 3m Libor*3/12*exp(-0.01*(7-3)/12)+ 60*FRA 4x7*3/12*exp(-0.01*7/12) =97 100 Swap value from the point of view of the party receiving the fixed leg : Swap value = fixed leg floating leg = 1 496 100 97 100 =$ 1 399 000 of market value. 6. What is the total dollar cost of purchasing $10 million of a Treasury bill yielding 0,31% and maturing june 15th 2021. (apply adequate convention) (2 points) Today : 12/8/20 By definition : Treasury bill à ZC bond, in $, ACT/360. By definition, for discount instruments like Treasury bills, the yield is applied to the time remaining until maturity because that's when the instrument pays its face value. We count the day between the today date and the end date, and note the opposite (which is done usually with non ZC bonds). T = (23 +31+28+31+30+31+15)/360 =189/360 =0.525 AI = 0 because there is no coupon PV coupons = 0 because there is no coupon PV notional = FV/(1+r)^t = 10 000 000 / (1+0.0031)^0.525= $ 9,986,418.96 7. What would be the FX breakeven eurodollar rate of an euro denominated 3-year bond, purchased at 95, held to maturity, paying a 2% coupon annually if purchased by a American investor. Spot EUUSD1.2150. Volatility of the EURUSD is 9% and the risk free rate of the USD and EUR are respectively 1% and -0,1%% (3 points) Capital appreciation : FV-PV =100-95 =5 Coupon =100*0.02*3 =6 Yield = coupon/price=(5+6)/95=0.116 FX breakeven Eurodollar rate = spot * (1-yield) =1.215*(1-0.116)=1.0743 8. Consider the following Treasury spot rates expressed as bond-equivalent yields: Maturity Spot Rate 6 months 3.0% 1 year 3.5% 1.5 years 4.0% 2 years 4.5% If a Treasury note with two years remaining to maturity has a 5 % semiannual coupon and is priced at R 1,008. Show that this Treasury note is mispriced. (2 points) Semi-annual coupon = 1000*0.05/2 =25 Bonds price = 25/(1+0.03/2) + 25/(1+0.035/2)² + 25/(1+0.04/2)^3 + 1025/(1+0.045/2)^4 =1010.10. 1010.1`1008 then we can conclude that the Treasury note is mispriced. 9. Suppose at year 0, the 1-year interest rate is 5%. Assume that the market expectation for the next 3 years 1-year interest rate are: a. E(r1,2)= 7% b. E(r2,3)= 9% c. E(r3,4)= 10% What is the forward rate E(r1,3)? (2 points) (1+Spot n)^n = (1+Spot m)^m * (1*Forward m,n)^n-m Ò F1,3 = sqrt( 1.09^3/1.05 -1 ) = 0.4831 Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
EXAMEN SOPHIA 2020 3. Bank of America issued a USD 2% of March 31, 2026 yields 1%. You wish to purchase, today (exam date which is December 8th2020) $200,000 FV of this bond. What is your total cost (including accrued) in dollars. (4pts) Bond in $, so coupon paid semi-annually. Corporate bond so ACT/360 convention. If the bond is paid semi-annually, then you do : r/2, c/2, t*2 in the PV formulas. T = 2026-2021 + (31+30+8)/360 =5.2 Accrued Interest = annual coupon * days/360 = 200 000 *0.02 * 69/360 =766.67 PV notional = FV/(1+r)^t = 200 000 / (1+0.01/2)^2*5.2 =189,899.43. PV coupons = c/r * (1-1/(1+r)^t) = (200 000 *0.02/2)/(0.01/2) * (1 1/(1+0.01/2)^(2*5.2)) = 20,219.25 Full bonds price = AI + PV notional + PV coupons = 210 885.35 4. An investor has a 1-year, semiannual, 10% coupon bond priced at 102,5. If the 6- month spot rate on a bond-equivalent basis is 8%, calculate the 1-year theoretical spot rate ? (2 points) Lets calculate the first cash flow : PV1 = (100*0.1/2)/(1+0.08/2) = 4.81 Now lets estimate the second cash flow : PV2 = (100*0.1/2 +100)/(1+x/2)² But we know that the bonds price is equal to the sum of the present value of its cash flow. We can conclude that : Bonds price = PV1 + PV2 Ò 102.5 = 4.81 + (100*0.1/2 +100)/(1+x/2)² Ò x = 7.35% The one-year theorical spot rate is 7.35%. 5. A swap involves paying 3-month Libor and receiving 0,5% on a notional of $60 million. The swap has a remaining life of 7 months. What is the value of the swap ? We assume yield curve is flat and all necessary maturities yield 1%. Last 3-month Libor stands at 0,25%(4 points) FRA 0X3: 0,25% FRA 2X5 : 0,36% FRA 4X7 : 0,40% FRA 1x4: 0.32% FRA 3X6: 0,39% FRA 5 x 8: 0,44% Formula : =SUM [ notional * fixed rate * frequency of payment * exponential (- r *remaining days) ] Frequency of payment : for annual c =12/12, for semi annual c =6/12, for quarterly c= 3/12 for monthly c =1/12 Remaining days : for the 2nd exponential it is the remaining life of the swap (given, here 7/12), for the first exponential it is the difference between the remaining life and the frequency of payment (example here : = 7/12 3/12). Cash Flow from the fixed leg : CF1 = 60*0.05*3/12*exp(-0.01*(7-3)/12) + 60*0.05*3/12*exp(-0.01*7/12) =1 496 100 Cash flow from the floating leg : CF2 = 60*Last 3m Libor*3/12*exp(-0.01*(7-3)/12)+ 60*FRA 4x7*3/12*exp(-0.01*7/12) =97 100 Swap value from the point of view of the party receiving the fixed leg : Swap value = fixed leg floating leg = 1 496 100 97 100 =$ 1 399 000 of market value. 6. What is the total dollar cost of purchasing $10 million of a Treasury bill yielding 0,31% and maturing june 15th 2021. (apply adequate convention) (2 points) Today : 12/8/20 By definition : Treasury bill à ZC bond, in $, ACT/360. By definition, for discount instruments like Treasury bills, the yield is applied to the time remaining until maturity because that's when the instrument pays its face value. We count the day between the today date and the end date, and note the opposite (which is done usually with non ZC bonds). T = (23 +31+28+31+30+31+15)/360 =189/360 =0.525 AI = 0 because there is no coupon PV coupons = 0 because there is no coupon PV notional = FV/(1+r)^t = 10 000 000 / (1+0.0031)^0.525= $ 9,986,418.96 7. What would be the FX breakeven eurodollar rate of an euro denominated 3-year bond, purchased at 95, held to maturity, paying a 2% coupon annually if purchased by a American investor. Spot EUUSD1.2150. Volatility of the EURUSD is 9% and the risk free rate of the USD and EUR are respectively 1% and -0,1%% (3 points) Capital appreciation : FV-PV =100-95 =5 Coupon =100*0.02*3 =6 Yield = coupon/price=(5+6)/95=0.116 FX breakeven Eurodollar rate = spot * (1-yield) =1.215*(1-0.116)=1.0743 8. Consider the following Treasury spot rates expressed as bond-equivalent yields: Maturity Spot Rate 6 months 3.0% 1 year 3.5% 1.5 years 4.0% 2 years 4.5% If a Treasury note with two years remaining to maturity has a 5 % semiannual coupon and is priced at R 1,008. Show that this Treasury note is mispriced. (2 points) Semi-annual coupon = 1000*0.05/2 =25 Bonds price = 25/(1+0.03/2) + 25/(1+0.035/2)² + 25/(1+0.04/2)^3 + 1025/(1+0.045/2)^4 =1010.10. 1010.1`1008 then we can conclude that the Treasury note is mispriced. 9. Suppose at year 0, the 1-year interest rate is 5%. Assume that the market expectation for the next 3 years 1-year interest rate are: a. E(r1,2)= 7% b. E(r2,3)= 9% c. E(r3,4)= 10% What is the forward rate E(r1,3)? (2 points) (1+Spot n)^n = (1+Spot m)^m * (1*Forward m,n)^n-m Ò F1,3 = sqrt( 1.09^3/1.05 -1 ) = 0.4831 Made with nCreator - tiplanet.org
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