Financial Mathematics Problems
Financial Mathematics Problems MATH 3615
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Date Created: 09/17/15
University of Connecticut Math 3615 Financial Mathematics Problems Fall 2008 Summary Module 14 McDonald Chapter 8 Lombardi text Unit 10 WAP Review of Forward Contracts and Forward Rate Agreements In Module 9 we were introduced to forward contracts which involve the sale of an asset where the price is determined currently but the payment of that price and the transfer of the asset occur at a specified future date or in the case of a prepaid forward contract the price is determined and paid currently and the asset is delivered at the future date The buyer s payoff for a forward contract which equals hisher profit is the spot price on the delivery date less the agreed to forward price In Module 13 we were introduced to forward rate agreements FRAs which are essentially forward contracts for interest rates The agreement relates to the cost of interest during a future period e g the 3 month period beginning 6 months from now The forward interest rate for that period times the notional principal amount is analogous to a forward price for the agreement The actual interest rate spot rate at the beginning of the future period times the notional principal amount is analogous to the spot price for settling the contract The payoff for an F RA equals the spot rate minus the forward interest rate specified in the agreement times the notional principal amount1 Definition of Swaps A swap contract amounts to a series of forward contracts or forward rate agreements combined into a single contract A commodity swap xes the price that one party will pay to the other party at a series of future dates in return for a series of deliveries of the commodity or equivalently in return for payment of the spot price of the commodity on each of those dates An interest rate swap xes the interest rate that Party 1 will pay to Party 2 at a series of future dates in return for payments from Party 2 equal to the actual spot rates on those dates The interest rates are multiplied by a notional principal amount to determine the dollar amount that will be paid A forward contract may be regarded as a swap involving only a single payment Conversely a swap can be regarded as a series of forward contracts combined with borrowing andor lending money This borrowinglending aspect is discussed in the next section 1 Since interest is due at the end of the period 9 months after the initial date for our example and the payoff is made at the beginning of the period 6 months after the initial date the calculated difference in interest payments must be discounted for the length of the period 3 months This discounting is done at the spot rate on the date of settlement the 3month spot rate in this case This is explained in later sections of this summary Commodity Swaps Suppose that a jewelry manufacturer needs to purchase 100 ounces of gold one year from now and another 100 ounces in two years and that the forward price for gold is 750 per ounce for delivery in one year and 800 for delivery in two years The manufacturer could enter into two long forward contracts for these two dates at these two prices and eliminate the risk of price variation Alternatively a single swap agreement could be used to fix a single price for the purchases on these two dates Because a swap is a fair exchange the present value of the level amounts to be paid must equal the present value of the amounts that would have been paid based on the current forward prices For instance if the swap price in the above example is 7741535 then this equality can be expressed as 750 P01 800P0 2 7741535 P01 P0 2 where P0n is the price at time 0 for an nyear zerocoupon bond with a maturity value of 1 If we know the 1 year and 2 year spot interest rates 500 and 600 in this case we can calculate P01 and P02 and use them to determine the level swap price 7741535 In this example there is a single level swap price to be paid at times 1 and 2 However any combination of prices that has the same present value as the current forward prices could be used in a swap contract Letting P1 and P2 represent the prices to be paid this can be expressed as 750P01800P0 2 P1 P01P2 P0 2 P1 and P2 could be 750 and 800 or 7741535 and 7741535 or any other pair of prices that satisfy this equation A swap agreement typically does not involve paying the swap price in return for delivery of the commodity on each delivery date Instead a swap agreement usually involves paying the swap price in return for receiving a payment equal to the spot price for the commodity on that date The fixed payer receives the spot price and is therefore able to purchase the commodity which is selling at that price Thus the payoff for a swap agreement on each payment date is the difference between the spot price and the swap price The fixed payer receives and the variable payer pays an amount equal to the spot price less the swap price If the fixed payer then purchases the commodity the net cost of the purchase equals the swap price Swap price 2 Spot price 7 Spot price 7 Swap price net cost of purchase pmt to purchase gold 7 pmt rec39d from swap contract This formula describes the payments for a single contract ie the payments per ounce of gold If the swap agreement involves 100 ounces of gold then the values in the formula are multiplied by the notional amount of the swap 100 ounces 3 There is a timing difference between the payments under a swap agreement and the payments under a series of forward contracts In the above example the fixed payer under the swap agreement would pay and the variable payer would receive the swap price of 7741535 on each of the two dates Under a pair of forward contracts the forward prices of 750 and 800 would have been paid on the first and second dates respectively Relative to the forward prices on the date of the contract the fixed payer lends to the variable payer an amount of 241535 2 7741535 7 750 on the first date by paying that much more than the forward rate then the vaiiable payer repays the loan by accepting 7741535 on the second date which is less than the forward price by 258465 2 800 7 7741535 The implicit interest rate earned on this loan is the implied forward rate based on the interest rates used to determine the swap price2 The loan has the effect of making the two payments under the swap agreement equal to each other at 7741535 The swap agreement is thus equivalent to two forward agreements at 750 and 800 each for 100 ounces of gold plus a forward rate agreement at the one year forward one year interest rate of 701 for a notional principal of 241535 2 100 X 7741535 7 750 Note that if the swap agreement had more than two payment dates additional FRAs would be required in order to offset the interest rate risk for all of the interest periods If the counterparty to the manufacturer s gold swap is a securities dealer the dealer may also enter into an agreement with a seller of gold and thus create a back to back transaction or matched book transaction In that case the dealer is subject to each paity 5 credit risk but is not exposed to price risk Alternatively the dealer can eliminate its price risk by hedging the swap contract through the use of multiple forward contracts In that case as discussed in the preceding paragraph the dealer must also enter into forward rate agreements in order to hedge the interest rate exposure resulting from the timing difference between the forward contracts payments 750 and 800 and the level payments of 7741535 under the swap agreement When a swap agreement is created the payments to be made by each party have equal value In other words the present value of the fixed payments equals the expected present value of the variable payments expected because the actual variable payments are not known at the time the contract is made Therefore the value of the contract is zero because the present value of the amounts that each party expects to receive equals the present value of the payments that party expects to pay As time passes however the fixed payments remain constant while gold prices and interest rates change So over time the swap contract develops a nonzero value Even if prices and interest rates do not change the swap agreement will have a non zero value once the first payment has been made If the parties agree to terminate the contract prior to the final payment one party must pay to the other an amount equal to the value of the contract as of the date of termination This value is equal to the difference in expected present value as of that date between the future payments to be made by one party and the future payments to be made by the other paity3 Z In this case the rate earned is 701 2258465241535 7 1 This is the lyearforward 1year rate based on the 1year spot rate 500 and the 2year spot rate 600 3 The payoff would equal the difference in present values between the swap contract s fixed prices and the market s current forward prices with the interest calculation being based on current spot rates Interest Rate Swaps An interest rate swap is a means of converting a series of future interest payments that vary with changes in interest rates to a series of fixed level payments or vice versa At the time the swap agreement is entered into the present value of the xed payments is equal to the expected present value of the variable payments In terms of a formula RP0tiP0titiilti where the fixed interest rate r0 tHt the estimate at t0 of the variable interest rate for the period from tH to t ie the forward rate as of time 0 for the period between rm and ti this equals P0t1P0t 7 1 P0t the price for a zero coupon bond maturing for 1 in t years ie the ti year discount factor based on the ti year spot rate For problem solving purposes it is critical to understand that the present value of the variable payments the right hand member in the above equation is equal to 1 P0 tn 4 The left hand member is a level annuity immediate for n periods with payments equal to the swap contract s fixed rate R We can solve for R using the following formula 1 P0 imam i1 R This is a very useful formula which can also be rearranged as follows P0tnRZI0t 1 391 This formula states that a payment of 1 that will be received n years from now plus a payment of R at the end of each of the next n years has a total present value of 1 But this is simply the formula for the payments under a par n year bond with a face amount of 1 R represents the annual coupon payment the summation is an annuity factor applied to the coupon rate to produce the present value of the coupon payments and P0tn is the present value of the bond s redemption payment its maturity value From this we see that R is the coupon rate for an 11year bond that is selling at par This can be demonstrated by converting the forward rates r0 in the righthand member into bond prices and simplifying the result To understand it consider that l is the present value of l payable now and POt is the present value of l payable in t years The difference 1 POt represents the value of the interest that can be earned during those t years if the payment of l is received now rather than t years from now So the right member of the equation equals the interest earned by an investment of 1 between time 0 and time t The lefthand member is an annuity of a level amount R on each payment date Ordinarily we would think of this annuity as being R a where ag is based on a single level interest rate But conceptually each of the n payments in a has a present value equal to P0 ll and drill is the sum of those payments 5 An interest rate swap acts much like the gold price swap described earlier When initiated the swap has a value of zero ie the fixed rate payments and the variable rate payments have equal present values But as interest rates change the changes affect both the amount of each variable payment that will be made and the interest rate used to determine its present value As a result of these effects the value of the swap increases andor decreases over time as interest rates change When interest rates rise the variable rate payer must make larger payments and the value to the fixed rate payer increases When interest rates fall the value to the variable rate payer increases Even in the absence of changes in interest rates as soon as the first swap payment occurs the swap will have a value The swap will have a positive value for the party who paid too much e g the fixed rate payer if the fixed rate exceeds the variable rate on the first payment date and a negative value for the party who received the net payment If the yield curve is normal ie if interest rates are higher for longer term investments then the level payments made by the fixed rate payer are higher than the variable rates in the early periods and lower than the variable rates in the later periods In effect the fixed rate payer is lending money to the variable rate payer The rate for this loan if interest rates do not change is the appropriate forward rate for that period determined as of the date the swap is initiated As with the commodity swap an interest rate swap is equivalent to entering into a series of forward contracts forward rate agreements and also undertaking some borrowing and lending Calculating the Amount of a Swap Payment Under commodity swaps the spot price for the commodity is determined as of each settlement date and the swap payment Spot price 7 Swap price is made on that same date Consequently there is no interest adjustment in the payment calculation The amount of the payment is simply the difference between the current spot price and the swap price specified in the agreement The situation is more complicated for interest rate swaps and also for FRAs The variable rate is determined as of the beginning of each interest rate period But it is a rate that applies over a period of time and interest is payable at the end of the period Since the swap payment between the parties is typically made at the beginning of the period the calculated interest difference must be discounted from the end of the period to the beginning of the period to re ect the time value of money ie to recognize that the receiving party is able to earn interest on the payment during the current interest period The amount of the payment is the difference between the spot rate for the period and the xed swap rate times the notional principal amount discounted for one period using the current spot rate Payment6 spot rate 7 swap rate X notional principal 1spot rate7 6 This is the net amount paid by the variable payer to the fixed payer If it is a negative amount then the fixed payer pays the absolute value of that amount to the variable payer 7 Note Each of the rates in this formula is an effective rate per period If the agreement is based on periods other than one year then the rate must be adjusted to re ect that period A Numerical Example Consider a 3 year interest rate swap based on the following interest rate data erm n rate 2 per 1000 of maturity value 95238 89845 84200 rate An exam problem would typically show the bond prices but not the spot rates or forward rates The additional information is provided here for greater clarity but you need to know how to calculate spot rates and forward rates if given only the bond prices We know that the fixed rate for this swap R will be the coupon rate for a par 3 year bond based on these interest rates We also know that the formula for R in this case is 1 P0 t3 R 3 Z P0 ti i1 Using the values in the above table we have R 1 842 952388984584200 The level interest rate for the swap is thus 58678 05867 If the notional principal for this swap is 100000 then we can calculate the difference between the variable rate and fixed rate payments for the first year as follows Variable rate payment 2 05000 X 100000 2 5000 Fixed rate payment 2 05 867 X 100000 2 5867 Difference in interest payments 2 867 This difference is determined at the beginning of the first year at i0 but it represents the difference in interest that is earned on the 100000 notional principal throughout the first year and that is payable at the end of the year It is customary to determine the amount of the difference and make the appropriate payment at the beginning of each year The calculated difference must therefore be discounted for one period s interest and this is done using the actual spot rate in effect for the period 5 in this case Thus the net payment under the swap at time 0 is 867 105 82571 3 This may seem like a high rate relative to the spot rates of 5 55 and 59 However if we recognize that the level rate is a weighted average of the forward rates which range from 5 to 67 a value of 5867 appears much more reasonable 7 Because we calculated the excess of the variable spot rate over the fixed rate and it is negative the absolute value 82571 is an amount owed by the fixed rate payer to the variable rate payer Similarly at the beginning of the second year assuming that the actual 1 vear spot rate at 11 equals 06002 the forward rate for that period as of i0 the swap payment is determined as follows Variable rate payment 2 06002 X 100000 2 6002 Fixed rate payment 2 05 867 X 100000 2 5867 Difference in interest payments 2 135 Present value at beginning of period 135 106002 12736 Because the result is positive 12736 is an amount to be paid by the variable rate payer to the fixed rate payer at i1 For the third year following the same process and again assuming that the one year spot rate is the same as the original forward rate for the third year we would calculate an amount of 78534 6705 5867 106705 meaning that the variable rate payer owes the fixed rate payer 78534 at i2 to re ect the excess of the actual variable rate the spot rate for the third year of the swap over the contractual fixed interest rate These three swap payments 82571 12736 and 78534 have a present value of zero at i09 This is what we would expect in the case of a fair swap Now consider the effect of changes in interest rates after the date of the agreement The first payment in this swap can be calculated at the time the swap agreement is initiated i0 since the actual spot rate for the first period 5 is already known However the second and third payments can not be calculated at i0 and will be affected by subsequent interest rate changes Suppose that at t1 the 1 year spot rate for the second year of the agreement is 58 rather than 6002 Then the second payment would be calculated as follows Variable rate payment 2 05 800 X 100000 2 5800 Fixed rate payment 2 05 867 X 100000 2 5867 Difference in interest payments 2 67 67 1 05 800 6332 The fixed rate payer would owe 6332 to the variable rate payer at t1 Similarly if the third year spot rate at 12 is 65 rather than 6705 the third payment at t2 would be 59437 2 6500 5867 1065 That is the variable rate payer would owe 59437 to the fixed rate payer 9 The present value of the payments is not exactly 0 due to rounding The exact swap rate should actually be 5867424 To perform this calculation keep in mind that the proper discount factors to apply are 1000 P01 and P02 This is because the payments are made at times t0 1 and 2 so they should be discounted for 0 1 and 2 years not 1 2 and 3 years 8 In this case the present value of the three actual payments is not 0 The actual spot rates declined relative to the forward rates in effect when the agreement was made which is to the advantage of the variable rate payer The present value of the payments is negative indicating that the variable rate payer has received value The fixed rate payer has a net cost a loss from the contract On the other hand the fixed rate payer has likely used the swap agreement to hedge its variable interest rate payments and has gained the advantage of knowing in advance the net cost of the three interest payments Value of the Swap Contract When this interest rate swap was initiated its value to either party was zero because the present value of each party s expected payments was equal to the present value of the other party s expected payments PVfixed payments 2 PVvariable payments After the first payment and before any subsequent interest rate changes and before the passage of time the value of the contract to the fixed rate payer is as follows 100 000 06002 89845 06705 84200 05 867 89845 84200 82610 Except for rounding this is equal to the amount that the fixed rate payer paid to the variable rate payer on the date of the contract Because the swap agreement is a fair contract the present value of expected future payments to the fixed rate payer is equal to the amount the fixed rate payer has paid on the contract s inception date At the end of the first year the value of the swap agreement to the fixed rate payer has increased with interest However if interest rates have changed the change in rates will also affect the value of the agreement Assume that at the end of the first year of this swap current interest rates are as follows erm 94518 88750 of rate Based on these rates the payments for the second and third years at i1 and i2 will be 6332 and 59437 as calculated previously The present value of these payments as of i1 is 633294518 59437 88750 46765 This is the fair price that the variable rate payer should pay the fixed rate payer if they agree to cancel the swap agreement just before the payment at i2 with market interest rates as shown in the above table The value of the swap agreement to the fixed rate payer has decreased from 82610 at i0 to 46765 at i1 even though no payments have been made during this period and even though the effect of interest during the first year would be expected to increase the value of the contract The contract s value to the xed rate payer decreased as a result of the decline in interest rates
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