FINA467 - Exam 2
A $100,000 interest rate swap has a remaining life of 10 months. Under the terms of the swap, six-month LIBOR is exchanged for 2.8% per annum (compounded semi-annually). Six-month LIBOR forward rates for all maturities are 3.4% (compounded semi-annually). The six-month LIBOR rate was 2.7% two months ago. The risk free rate is 4.5% (cont. comp) for all maturities. What is the value of the swap to the party paying floating?
-240.0 margin of error +/- 1 principal 100,000 remaining life 10 months fixed rate 2.8% (comp S/A) 6m LIBOR 2.7% (comp S/A) forward 6m 3.4% (comp S/A) r 4.5% (cont comp) time (m) / time (y) / LIBOR / fixed pmt / PV (fixed) / floating pmt / PV (floating) -2 / -0.1667 / 2.70% 4 / 0.3333 / 3.4% / 1400 / 1379.16 / 1350 / 1329.90 10 / 0.8333 / - / 1400 / 1348.47 / 1700 / 1637.43 - / - / - / TOTAL / 2727.63 / - / 2967.33 value of swap to party paying floating = pv(fixed) - pv(floating) value of swap = -239.70
A short forward contract that was negotiated some time ago will expire in three months and has a delivery price of $39. The current forward price for the three-month forward contract is $44. The three-month risk free interest rate is 8% (cont. comp). What is the value of the short forward contract today?
-4.9 margin of error +/- 0.01 Short forward contract T 0.25 years K 39 F0 44 r 8% f = (K-F0)*exp(-rT) f = -4.900993367
Companies A and B have been offered the following rates per annum on a $20M five-year loan. Company A requires a floating-rate loan. Company B requires a fixed rate loan. A bank will facilitate the swap and charge a 0.3% intermediation fee. Design a swap that will appear equally attractive to both parties (split any gains from the swap right down the middle). Fixed Rate Floating Rate Company A 4.3% LIBOR + 0.1% Company B 7.1% LIBOR + 0.8% A <-- X% -- F.I. <-- Y% -- B A --LIBOR--> F.I.--LIBOR--> B A diagram of the swap is shown above. What is the value of X (the fixed payment the financial intermediary makes to Company A) in the swap you designed?
0.051 margin of error +/- 0.0001 Fixed Rate Floating Rate Company A 4.30% LIBOR + 0.10% Company B 7.10% LIBOR + 0.80% B - A 2.80% 0.70% Diff-in-diff 2.10% intermediation fee 0.30% diff-in-diff - intermedation fee 1.80% above / 2 0.90% Both parties will reduce borrowing costs by 0.9%, meaning: Company A will borrow at LIBOR - 0.8% Company B will borrow at 3.4% Company A starts off the swap by borrowing where it has the comparative advantage, or the fixed rate of 4.3% As the diagram illustrates, Company A will then pay the financial intermediary LIBOR The resultant cash flows from the above two steps is LIBOR + 4.3% We know from above that the borrowing cost to A is LIBOR - 0.8% The cash flow received from A by the financial intermediary is 5.1%
An exchange rate is 0.500 and the one-year domestic and foreign risk-free interest rates are 8% and 8%, respectively (cont. comp). What is the one-year forward exchange rate?
0.5 margin of error +/- 0.001 S0 0.5 r 8% rf 8% T 1 F0 = S0*exp((r-rf)T) F0 = 0.5*exp((8%-8%)1) F0 = 0.5
Consider a 10-year bond with current price of $97.5 and a duration of 9.2 years. Suppose the yield on the bond is 6.1% per year with continuous compounding. What is the change in the price of the bond if the yield decreases by 0.06 percentage points?
0.54 margin of error +/- 0.01 maturity 10 Y current price 97.5 duration 9.2 Y yield 6.10% change yield -0.06% ΔB = -BDΔy ΔB = -97.5*9.2*-0.06% ΔB = 0.5382
What is the conversion factor for a bond with a 4.2% coupon (annual coupon paid semi-annually) with 10 years and 1 months to maturity?
0.87 margin of error +/- 0.01 coupon 4.20% maturity 10.08 discount rate for conversion factor is 6% (with S/A compounding) Price of 100 bond = PV(0.03,20,2.1,100) Price of 100 bond = ($86.61) Conversion factor = price of $100 bond / $100 Conversion factor = 0.87
Use put call parity to determine the size of the arbitrage profit at time T arising from the following situation. All options are European. So = $19.27 T = 3/12 (for both the call and the put) K = 20.00 (for both the call and the put) c = $2.70 p = $2.65 r = 10% (cont. comp. annual rate) Dividend = $1 in one month
1.31 margin of error +/- 0.01 S0 19.27 T 0.25 K 20 c 2.70 p 2.65 r 10% (cont comp) dividend 1 in one month D = PV(dividend) D = 1*exp(-0.1*1/12) D = 0.991701293 K*exp(-rT)+c+D = S0 + p Calculate LHS K*exp(-rT)+c+D = 20*exp(-0.1*0.25)+2.7+0.9917 K*exp(-rT)+c+D = 23.19789953 Calculate RHS S0+p = 19.27+2.65 S0+p = 21.92 There are two ways of getting the right answer. First is to take a shortcut. Second is to think through the arbitrage steps. Method 1: Use the shortcut. Arbitrage profit at time T = FV( absolute difference between LHS and RHS) abs(LHS - RHS) = abs(23.1978995 - 21.2144982) abs(LHS - RHS) = 1.277899533 FV(abs(LHS-RHS)) = abs(LHS-RHS) * exp(0.1*0.25) FV(abs(LHS-RHS)) = 1.310249714 Method 2: think through the arbitrage steps LHS > RHS, so we're going to buy put and stock and sell the call t=0 buy put -2.65 buy stock -19.27 sell call 2.70 total CF needed -19.22 amount we need to borrow 19.22 A $1 dividend is coming in 1 month, which we'll use to repay a portion of the loan amount needed to borrow over 1 month 0.991701293 amount needed to borrow for rest of time 18.23 t = 1/12 receive $1 dividend repay the portion of the loan that we borrowed for 1 month t = 3/12 we need to repay our loan by selling the stock. our loan has grown to 18.23*exp(0.1*3/12) = 18.68975029 if ST < K, we sell for K given our put if ST > K, we sell for K given our call profit = K - loan balance profit = 20 - 18.68975029 profit = 1.310249714 Both methods give us the right answer for the size of the arbitrage profit at time T
The price of a non-dividend paying stock is $48 and the strike price of a two year European call option is $41. The risk-free rate is 4% (continuously compounded). What is the lower bound of the call option's price?
10.15 margin of error +/- 0.01 S0 48 K 41 r 4% (cont. comp) If option is underpriced, we would buy the option and short the stock t=0 short stock, get 48 today buy call option today for c invest proceeds at the risk free rate, so it will grow to (48-c)*exp(0.04*2) t=2Y if ST < K, buy at ST if ST > K, buy at K (this is the more interesting scenario) Profit = (48-c)*exp(0.04*2) - 41 If no arbitrage, the profits are 0 (48-c)*exp(0.04*2) - 41 = 0 (48-c)*exp(0.04*2) = 41 (48-c) = 41*exp(-0.04*2) c = 48 - 41*exp(-0.04*2) c = 10.1522298 We know that the call option's price has to be at least 10.15 or there will be arbitrage profits
It is January 9, 2017. The price of a Treasury bond with a 6% coupon that matures on Oct 12, 2030, is quoted as 102-08. What is the cash price? (This bond pays coupons on Oct 12 and Apr 12. Excel might be helpful in calculating the difference in dates.)
103.72 margin of error +/- 0.01 previous coupon 10/12/2016 today 1/9/2017 next coupon 4/12/2017 coupon rate 6% semi-annual coupon 3 actual distance from last coupon 89 actual distance between coupons 182 quoted price 102.25 accrued interest = size of coupon * actual # days from last coupon / actual # days between coupons accrued interest = 1.467032967 cash price = quoted price + accrued interest cash price = 103.717033
A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $39.0 and the risk-free rate of interest is 5% per annum (cont. comp). What is the value of the forward contract 4 months later if the price of the stock is $42.3 and the risk-free rate interest rate is still 5%.
2.64 margin of error +/- 0.01 Initial contract length 1 years non-dividend paying stock S0 39 r 5% Elapsed time 0.333333333 years Remaining time 0.666666667 years S_t=4mo 42.3 Step 1: Calculate the initial forward rate at time = 0. This will be the delivery date of the contract (i.e. K) F0 = S0 * EXP(rT) F0 = 39 * EXP(0.05*1) F0 = K = 40.99957276 Step 2: Calculate the forward price at time = 4 months. F0 = S0 * EXP(rT) F0 = 42.3*exp(0.05*8/12) F0 = 43.7337633 Step 3: Calculate the value of the forward rate at time = 4 months. f = (F0 - K)*exp(-rT) f = (43.73376 - 40.99957)*exp(-0.05*0.666666) f = 2.644553115
The most recent futures settlement price is 93-06. What is the cost to deliver the following bond? Quoted Bond Price Conversion Factor 99.5 1.037
2.86 margin of error +/- 0.01 most recent settlement price 93.1875 quoted bond price 99.5 conversion factor 1.037 cost to deliver = quoted bond price - most recent settlement price * conversion factor cost to deliver = 99.5 - 93.1875*1.037 cost to deliver = 2.8645625
The stock price of google on May 13, 2015 is 531.90. Prices of call options are shown below. Strike Price Sept 2015 475 67.40 500 46.52 525 29.85 What is the time value of the call option with the strike price of $525?
22.95 margin of error +/- 0.01 option price 29.85 intrinsic value =max(0, 531.9-525) = 6.9 time value =29.85-6.9=22.95
The spot price of an investment asset is $27 and the risk-free rate for all maturities is 9% per annum (cont. comp). The asset provides an income of $3 at the end of the first year and at the end of the second year. What is the three-year forward price?
28.49 margin of error +/- 0.01 S0 27 r 9% T 3 income at end of first year 3 income at end of second year 3 F = (S0-I)*exp(rT) I = pv(future income payments) I = 3*exp(-0.09*1)+3*exp(-0.09*2) I = 5.24760419 F = 28.49486523
The below put option is European on a non-dividend paying stock. What is the minimum arbitrage profit at time T arising from the following prices? So = $19.56 T = 1Y K = $25 p = $0.57 r = 8% (cont. comp. annual rate)
3.19 margin of error +/- 0.01 S0 19.56 T 1 K 25 p 0.57 r 8% (cont. comp) If p is underpriced, we would do the following: t=0 buy put -0.57 buy stock -19.56 total CF -20.13 borrow 20.13 t=1 size of loan with interest = 20.13*exp(0.08*1) = 21.80656 if ST>K, we sell at ST and repay loan if ST<K, we exercise put and sell at K to repay the loan (this is the more interesting case) Profit = K - amount borrowed (after interest) Profit = K - 21.80656 Profit = 3.193431328
Consider a five-year bond with 9% annual coupon (paid semi-annually). Suppose the yield on the bond is 13% per year with continuous compounding. What is the duration of the bond (in years)?
4.04 margin of error +/- 0.01 maturity 5 annual coupon 9% paid S/A yield 13% time cf pv(cf) pv(cf)/Price * time 0.5 4.5 4.216803585 0.025031492 1 4.5 3.951429439 0.046912394 1.5 4.5 3.702755961 0.065940117 2 4.5 3.469732136 0.082387117 2.5 4.5 3.251373091 0.096502858 3 4.5 3.046755935 0.108515627 3.5 4.5 2.855015856 0.118634207 4 4.5 2.675342466 0.127049434 4.5 4.5 2.506976378 0.133935628 5 104.5 54.55378367 3.23838324 TOTAL 84.22996852 4.043292113
A currency swap has a remaining life of 15 months. It involves exchanging interest at 10.3% on £20,000 for interest at 5.6% on $30,000 once a year. The term structure of risk-free interest rates in the United Kingdom is flat at 7.3% and the term structure of risk-free interest rates in the US is flat at 3.7% (both with continuous compounding). The current exchange rate (dollars per pound sterling) is 1.46. What is the value of the swap (in dollars) to the party paying dollars?
439.0 margin of error +/- 1 GBP principal 20,000 GBP interest 10.30% rf 7.3% (comp comp) USD principal 30000 fUSD interest 5.6% r 3.7% (cont comp) S0 1.46 USD/GBP time (m) / time (y) / GBP pmt / PV (GBP pmt) / USD pmt / PV (USD pmt) 3 / 0.2500 / 2060 / 2022.75 / 1680 / 1664.53 15 / 1.2500 / 22060 / 20136.14 / 31680 / 30248.17 - / - / TOTAL / 22158.88 / - / 31912.70 Conv GBP->USD 32351.97 value of swap to party paying dollars = PV(pmts received in GBP) - pv(pmts made in USD) value of swap = 439.27
A bond fund manager is concerned about interest rate volatility over the next 3 months. The value of the bond portfolio is $8M and the duration of the portfolio is 6.8 years. To hedge interest rate volatility, the fund manager uses Treasury bond futures. The quoted price for the December Treasury bond futures contract is 93-03. The cheapest to deliver Treasury bond has a duration of 7.2 years. How many contracts should the fund manager short? (Enter as positive number, even though it's a short. Round to the nearest contract. Precision: 1 +/- 0.5)
81.0 margin of error +/- 0.5 Value of port 8,000,000 Duration of port 6.8 Y Duration of cheapest-to-deliver-bond 7.2 Y Price of cheapest-to-deliver-bond 93.09 Value of hedged bond (given 100k face value) 93,093.75 N* = h*(V_B/V_F) h* = duration of our portfolio / duration of cheapest-to-deliver bond N* = (6.8/7.2)*(8,000,00/93,093.75) N* = 81.16071762 N* = 81 after rounding
The spot price of oil is $82 per barrel and the cost of storing a barrel of oil for one year is $3.5, payable at the end of the year. The risk-free interest rate is 3% per annum (cont. comp). What is an upper bound for the one-year futures price of oil?
88.0 margin of error +/- 0.01 S0 82 oil storage cost 3.5 (payable at end of year) r 3% T 1 F0 <= (S0 + U)*exp(rT) U is the present value of storage costs U = 3.5*exp(-0.03*1) U = 3.396559367 F0 <= (S0 + U)*exp(rT) F0 <= (82 + 3.396559)*exp(0.03*1) F0 <= 87.99727178
An investor shorts 100 shares when the share price is $51 and closes out the position six months later when the share price is $40. The shares pay a dividend of $2 per share during the six months. The interest rate is zero. How much does the investor gain?
900.0 margin of error +/- 0.01 shares 100 S0 51 ST 40 dividend 2 T 0.5 years r 0% given 0 interest rates, we can ignore discounting in the following steps T=0 short 100 shares borrow 100 shares sell at 51 get 5100 0<T<0.5 owe 2/share to original investor corresponding cash flow is 200 T=0.5 rebuy 100 shares @ 40 per share for 4000 summation of cash flows 5100-200-4000 900
Which of the following describes an interest rate swap? A way of converting a liability from fixed to floating A portfolio of forward rate agreements An agreement to exchange interest at a fixed rate for interest at a floating rate All of the above
All of the above All are true. A and C are true b/c interest rate swaps are ways of transforming either liabilities/assets from fixed/floating to other. C is true b/c each specific cash flow can be thought of as a unique forward rate agreement.
When the time to maturity increases with all else remaining the same, which of the following is true? European options always increase in value The value of European options either stays the same or increases There is no effect on European option values European options may either increase or decrease in value
European options may either increase or decrease in value With American options, increasing the time to maturity will increase the price of an option. However, dividends complicate this relationship for European options. If the increase in time to maturity does not span a new dividend payment, the price of a European call option will increase. However, if an increase in time to maturity spans a new dividend payment, the price of a European call option may decrease. Hence, it is unclear which direction the European option's price will go.
When dividends increase with all else remaining the same, which of the following is true? Both calls and puts increase in value Both calls and puts decrease in value Calls increase in value while puts decrease in value Puts increase in value while calls decrease in value
Puts increase in value while calls decrease in value An increase in dividends will lower the stock price which will reduce the value of a call and increase the value of a put.
When the strike price increases with all else remaining the same, which of the following is true? Both calls and puts increase in value Both calls and puts decrease in value Calls increase in value while puts decrease in value Puts increase in value while calls decrease in value
Puts increase in value while calls decrease in value Value of put is increasing in strike price. Value of call is decreasing in strike price.
Which of the following is NOT true about duration? It equals the years-to-maturity for a zero-coupon bond It equals the weighted average of payment times for a bond, where weights are proportional to the present value of payments Equals the weighted average of individual bond durations for a portfolio, where weights are proportional to the present value of bond prices The prices of two bonds with the same duration change by the same percentage amount when interest rate move up by 2 percentage point (i.e. from 5% to 7%)
The prices of two bonds with the same duration change by the same percentage amount when interest rate move up by 2 percentage point (i.e. from 5% to 7%) Calculate the duration of a 30Y bond with 10% coupon (paid semi-annually). Then calculate the % error in bond price as a function of the change in yield. The duration formula (delta B = -B * D * delta y) works best for small changes in yields. For large changes in yield, the error is larger. At small changes in yield, two bonds with the same duration will change price by approximately the same amount. However, at large changes in yield, two bonds with the same duration will change price by different amounts. The technical reason is due to difference in curvature between the two bonds (the second derivative of bond price with respect to yield, where duration is the first-derivative). The simple explanation is that duration does not do well at large changes in yield. All other answers are true.
A company enters into an interest rate swap where it is paying fixed and receiving LIBOR. When interest rates increase, which of the following is true? The value of the swap to the company increases The value of the swap to the company decreases The value of the swap can either increase or decrease The value of the swap does not change providing the swap rate remains the same
The value of the swap to the company increases The company is paying fixed and receiving LIBOR. When interest rates increase, the floating rate (LIBOR) will increase, yet the fixed-rate remains the same. The result is an increase in the value of the swap.