SOPHIA 2022
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Catégorie :Category: nCreator TI-Nspire
Auteur Author: rikkyriks
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 5.36 Ko KB
Mis en ligne Uploaded: 18/12/2024 - 19:13:30
Uploadeur Uploader: rikkyriks (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4420103
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 5.36 Ko KB
Mis en ligne Uploaded: 18/12/2024 - 19:13:30
Uploadeur Uploader: rikkyriks (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4420103
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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EXAMEN SOPHIA 2022 1. If the convexity of a 7-year zero coupon bond yielding 5% is 60, what would be its new price (not only % price change) if yields move up by 50BP? (2 points) Initial bond price = FV/(1+r)^t = 1 000 / (1 + 0.05)^7 = 710.7 % change in a bond price = -duration * YTM + ½* convexity * YTM² = -7*0.005+ ½*60*0.005² = -0.0343 = -3.43% New bond price = initial price * % change = 710.7 * (1-0.0343) = 686.32 2. Assume the following set of rates (3 points) Year Spot Rate 1 5% 2 6% 3 7% 4 6% a) What are the forward rates over each of the four years? b) What is the 2 year forward rate ending in 4 years ? a) (1+Spot n)^n = (1+Spot m)^m * (1*Forward m,n)^n-m =Fm,n = (1+Sn)^n / (1+Sm)^m -1 =F 1,2 =1.06²/1.05 1 =F 1,2 = 0.0701 F 2,3 = 1.07^3/1.06² -1 = 0.09 F 3,4 = 1.06^4/1.03^3 -1 = 0.0306 b) F2,4 = sqrt(1.06^4/1.06² - 1) = 0.06 3. Explain, mathematically (or not), whether it would make sense today to invest in a 10-year Treasury paper in USD rather than invest in a 5-year Treasury paper yielding 3,30% and reinvest the proceeds 5 years later in another 5-year paper. (3 points) It depends on the view you have of the interest rates : - If you think they will rise : invest in 5y paper and make profit from rising interest rates by reinvesting the proceeds 5y later - If you think they will decrease : invest in 10y paper to avoid a loss when reinvesting 5y later 4. Price (including accrued interest) the Exxon 6% USD denominated bond maturing March 31, 2034 yielding 4%. What is the total cost of buying 20 bonds assuming we are buying today December 5th 2022 (3 points) Exxon : corporate bonds denominated in $ à semi-annually coupon + ACT/360 If the bond is paid semi-annually, then you do : r/2, c/2, t*2 in the PV formulas. T= 2034-2023 + (31+30+5)/360 =11.18 AI = annual coupon * days/360 = 1000*0.06*66/360 =11 PV notional = FV/(1+r)^t = 1000/(1+0.04/2)^11*2 =646.87 PV coupons = c/r*(1 1/(1+r)^t) = (1000*0.06/2)/(0.04/2) * (1 1/(1+0.04/2)^11*2=568.38 Full bonds price = AI +PV notional + PV coupons = 11 + 646.87 + 568.38 =1 226.25 (to double check our result : coupon>yield, than price should be >par value, it is the case here) 5. A bond selling for par currently has a 9% yield. If the bond price increases to USD 101 when yields fall 10 basis points and the price falls to USD 98 when yields rise by 10 basis points, then what is this bond's modified duration? (2 points) Approximate modified duration = (V- - V+)/(2*V0*YTM) = (101-98)/(2*100*0.001)=15 6. A credit trader thinks rates will be declining starting next April and decides to use a 6-month FRA to trade on a notional of $200 million. What is his P&L if by next April, 6-month SOFR stands at 4%-4.25% ? The 4 x 10 FRA todays trades at 4.50%/4.75% (2 points) If the rates are declining and you to profit from it àgo short. 4x10 FRA bid/ask=4.50%/4.75% 6m SOFR bid/ask=4%/4.25% Strategy : - Short today at 4.50% - Long in 6 months at 4.25% P&L = Notional*spread*frequency of payment = 200 000 000 * (0.045-0.0425)*6/12 = $250 000 7. A bank entered a $1,000,000 one-year receive-fixed LIBOR-based interest rate swap, which is reset quarterly. The estimated spot rates are given in the following table. Last 6-month LIBOR is 1.70% (5 points) 90 days = 1.9% 180 days = 2.3% 270 days = 2.6% 360 days = 3% a. Determine the fixed rate of the swap. You need to calculate the forward or FRA rates first to solve this problem. b. What is the value of the swap to the party paying fixed? a) Lets first calculate the forward rate for each maturity: FRA 90days =0.019=1.90% FRA 180days = (R2*T2 -R1*T1)/(T2-T1) = (180-0.023-90*0.019)/(180-90)=0.027 =2.70% FRA 270days = (270*0.026 -180*0.023)/(270-180) =0.032 =3.2% FRA 360days = (360*0.03-270*0.026)/(360-270) = 0.0421 =4.21% The fixed rate of a swap is the average of the anticipated floating rates. Fixed rate = (0.019+0.027+0.032+0.0421)/4 =0.03 =3% b) The value of a swap at initiation is always zero. Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
EXAMEN SOPHIA 2022 1. If the convexity of a 7-year zero coupon bond yielding 5% is 60, what would be its new price (not only % price change) if yields move up by 50BP? (2 points) Initial bond price = FV/(1+r)^t = 1 000 / (1 + 0.05)^7 = 710.7 % change in a bond price = -duration * YTM + ½* convexity * YTM² = -7*0.005+ ½*60*0.005² = -0.0343 = -3.43% New bond price = initial price * % change = 710.7 * (1-0.0343) = 686.32 2. Assume the following set of rates (3 points) Year Spot Rate 1 5% 2 6% 3 7% 4 6% a) What are the forward rates over each of the four years? b) What is the 2 year forward rate ending in 4 years ? a) (1+Spot n)^n = (1+Spot m)^m * (1*Forward m,n)^n-m =Fm,n = (1+Sn)^n / (1+Sm)^m -1 =F 1,2 =1.06²/1.05 1 =F 1,2 = 0.0701 F 2,3 = 1.07^3/1.06² -1 = 0.09 F 3,4 = 1.06^4/1.03^3 -1 = 0.0306 b) F2,4 = sqrt(1.06^4/1.06² - 1) = 0.06 3. Explain, mathematically (or not), whether it would make sense today to invest in a 10-year Treasury paper in USD rather than invest in a 5-year Treasury paper yielding 3,30% and reinvest the proceeds 5 years later in another 5-year paper. (3 points) It depends on the view you have of the interest rates : - If you think they will rise : invest in 5y paper and make profit from rising interest rates by reinvesting the proceeds 5y later - If you think they will decrease : invest in 10y paper to avoid a loss when reinvesting 5y later 4. Price (including accrued interest) the Exxon 6% USD denominated bond maturing March 31, 2034 yielding 4%. What is the total cost of buying 20 bonds assuming we are buying today December 5th 2022 (3 points) Exxon : corporate bonds denominated in $ à semi-annually coupon + ACT/360 If the bond is paid semi-annually, then you do : r/2, c/2, t*2 in the PV formulas. T= 2034-2023 + (31+30+5)/360 =11.18 AI = annual coupon * days/360 = 1000*0.06*66/360 =11 PV notional = FV/(1+r)^t = 1000/(1+0.04/2)^11*2 =646.87 PV coupons = c/r*(1 1/(1+r)^t) = (1000*0.06/2)/(0.04/2) * (1 1/(1+0.04/2)^11*2=568.38 Full bonds price = AI +PV notional + PV coupons = 11 + 646.87 + 568.38 =1 226.25 (to double check our result : coupon>yield, than price should be >par value, it is the case here) 5. A bond selling for par currently has a 9% yield. If the bond price increases to USD 101 when yields fall 10 basis points and the price falls to USD 98 when yields rise by 10 basis points, then what is this bond's modified duration? (2 points) Approximate modified duration = (V- - V+)/(2*V0*YTM) = (101-98)/(2*100*0.001)=15 6. A credit trader thinks rates will be declining starting next April and decides to use a 6-month FRA to trade on a notional of $200 million. What is his P&L if by next April, 6-month SOFR stands at 4%-4.25% ? The 4 x 10 FRA todays trades at 4.50%/4.75% (2 points) If the rates are declining and you to profit from it àgo short. 4x10 FRA bid/ask=4.50%/4.75% 6m SOFR bid/ask=4%/4.25% Strategy : - Short today at 4.50% - Long in 6 months at 4.25% P&L = Notional*spread*frequency of payment = 200 000 000 * (0.045-0.0425)*6/12 = $250 000 7. A bank entered a $1,000,000 one-year receive-fixed LIBOR-based interest rate swap, which is reset quarterly. The estimated spot rates are given in the following table. Last 6-month LIBOR is 1.70% (5 points) 90 days = 1.9% 180 days = 2.3% 270 days = 2.6% 360 days = 3% a. Determine the fixed rate of the swap. You need to calculate the forward or FRA rates first to solve this problem. b. What is the value of the swap to the party paying fixed? a) Lets first calculate the forward rate for each maturity: FRA 90days =0.019=1.90% FRA 180days = (R2*T2 -R1*T1)/(T2-T1) = (180-0.023-90*0.019)/(180-90)=0.027 =2.70% FRA 270days = (270*0.026 -180*0.023)/(270-180) =0.032 =3.2% FRA 360days = (360*0.03-270*0.026)/(360-270) = 0.0421 =4.21% The fixed rate of a swap is the average of the anticipated floating rates. Fixed rate = (0.019+0.027+0.032+0.0421)/4 =0.03 =3% b) The value of a swap at initiation is always zero. Made with nCreator - tiplanet.org
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