Financial Engineering – IC302
Autumn Term 2020/1
Seminar 2: Simple Interest Rate Derivatives Answers
1. Suppose that an investment will pay you $110.25 in two years in exchange for investing $100 today. Calculate its annual percentage return based on:
(a) annual compounding
𝐹𝑉=𝑃𝑉%1+ 𝑟*!” → 𝑟=𝑚-.𝐹𝑉/!” −11=.110.25/#×% −1=5% 𝑚 𝑃𝑉 100
(b) semi-annual compounding
𝑟 = 𝑚 -.𝐹𝑉/!” − 11 = 2 -.110.25/%×% − 11 ≈ 4.94% 𝑃𝑉 100
(c) quarterly compounding
##
𝑟 = 𝑚 -.𝐹𝑉/!” − 11 = 4 -.110.25/&×% − 11 ≈ 4.91%
##
##
𝑃𝑉
(d) continuous compounding
100
𝑙𝑛 %110.25*
𝑙𝑛 %𝐹𝑉*
𝐹𝑉 = 𝑃𝑉𝑒'” → 𝑟 = 𝑃𝑉 𝑡2
→ 𝑟 = 100
≈ 4.88%
2. Consider a market that contains risk-free bonds with the following zero-coupon yields (all expressed with annual compounding):
(a) Does this market contain an arbitrage opportunity? Explain your answer.
Maturity (years)
1
2
3
Zero-coupon yield
3.09%
2.60%
1.55%
[Hint: Begin by calculating the discount factors.]
First calculate the discount factors implied by the zero-coupon yields:
𝐷# = 1 = 0.9700 1 + 0.0309
𝐷% = 1 = 0.9500 (1 + 0.0260)%
𝐷( = 1 = 0.9549 (1 + 0.0155)(
Notice that the 3-year discount factor is larger than the 2-year discount factor.
Suppose that zero-coupon bonds with face value 100 are traded for all three maturities. We short the 3-year zero-coupon bond. This gives us 95.49 today. Use 95.00 of this to go long the 2-year zero-coupon bond, leaving us 00.49 in cash. When the 2-year bond matures, it will pay us 100. We can hold this in cash and use it to repay the principal on the 3-year bond when it matures one year later. There are no net payments outstanding now or at future dates, we took no risk, and we have 00.49 in cash in our pocket. That is an arbitrage opportunity.
Note, however, that the no-arbitrage argument depends on our ability to hold cash imposing a lower bound of zero on nominal interest rates. In fact, interest rates are currently negative in some market, which suggests that this lower bound may not hold.
(b) What practical factors might limit the ability of market participants to eliminate arbitrage opportunities in bond markets?
The question is intended to encourage class discussion. Factors that might limit the ability of market participants to exploit arbitrage opportunities include:
• Transaction costs
• Liquidity
• Access to funding
• Regulatory capital requirement
• Mismatches in timing of cash flows (coupon payments, etc.
• Execution risk
3. Consider an investor who plans to make an investment with principal amount $10 million for a three-month period beginning on 16 December 2020. The investment will pay interest at a rate equal to 3-month USD LIBOR. She would like to use Eurodollar futures to lock in the interest rate on her investment. The December 2020 Eurodollar futures contract is currently trading at a price of 99.755.
(a) Should she go long or short Eurodollar futures?
She should go long Eurodollar futures. Her concern is that rates may fall between now and December, reducing the interest rate that she earns on her investment, so she needs her hedge to make money if rates go down. As we have seen, a long Eurodollar futures position makes money if the forward LIBOR rate for the period covered by the contract falls.
(b) How many futures contracts should she trade?
The simplest hedge would be to trade futures with notional amount equal to the principal of her investment. Each contract is on notional principal $1 million, so she should go long 10 contracts.
(c) Suppose that the investor implements the hedge you have just described, and the USD LIBOR fixing for the three-month period covered by the investment and the futures contract turns out to be 0.195%. Calculate the total profit or loss on her futures position and the interest rate that she will earn on her investment after taking into account that profit or loss.
If the three-month LIBOR for the period turns out to be 0.195%, then the final settlement price for the futures will be 99.805. The initial futures price was 99.755, so this is an increase of 5 basis points. The profit on her long position in 10 Eurodollar futures contracts will be:
𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑃𝑟𝑜𝑓𝑖𝑡
= 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 𝑐h𝑎𝑛𝑔𝑒 𝑖𝑛 𝑏𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡𝑠
× 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 × 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝐷𝑉01 = 5 × 10 × $25 = $1,250
The interest on her three-month investment will be:
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $10 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 0.00195 × 025 = $4,875
After taking into account the futures hedge, she will have earned:
𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝐻𝑒𝑑𝑔𝑖𝑛𝑔 𝑝𝑟𝑜𝑓𝑖𝑡 = = $4,875 + $1,250 = $6,125
This is equivalent to an an annual interest rate of 0.245%:
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $10 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 𝑅𝑎𝑡𝑒 × 0.25 = $6,125 → 𝑅𝑎𝑡𝑒 = 0.245%
(d) Suppose that the investor implements the hedge you have just described, and the USD LIBOR fixing for the three-month period covered by the investment and the futures contract turns out to be 0.295%. Calculate the total profit or loss on her futures position and the interest rate that she will earn on her investment after taking into account that profit or loss.
If the three-month LIBOR for the period turns out to be 0.295%, then the final settlement price for the futures will be 99.705. The initial futures price was 99.755, so this is a decrease of 5 basis points. The profit on her long position in 10 Eurodollar futures contracts will be:
𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑃𝑟𝑜𝑓𝑖𝑡
= 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 𝑐h𝑎𝑛𝑔𝑒 𝑖𝑛 𝑏𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡𝑠
× 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 × 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝐷𝑉01 = (−5 ) × 10 × $25 = −$1,250
The interest on her three-month investment will be:
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $10 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 0.00295 × 025 = $7,375
After taking into account the futures hedge, she will have earned:
𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝐻𝑒𝑑𝑔𝑖𝑛𝑔 𝑝𝑟𝑜𝑓𝑖𝑡 = = $7,375 − $1,250 = $6,125
This is equivalent to an an annual interest rate of 0.245%:
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $10 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 𝑅𝑎𝑡𝑒 × 0.25 = $6,125 → 𝑅𝑎𝑡𝑒 = 0.245%
(e) Has the investor succeeded in locking in the interest rate on her investment?
Yes. In each case, the net interest she earns on the investment (i.e. the amount she earns after taking into account the profit or loss on the futures hedge) is equivalent to the same annual rate, 0.245%. This is the forward rate implied by the price at which she entered into the long position in the December Eurodollar futures.
(f) What factors might lead hedges based on Eurodollar futures to perform less perfectly than in this simple example?
Even in this simple case, the hedge will not be perfect. The interest on the investment will be received at the end of the 3-month period, whereas the final futures settlement date is at the start of the period. In addition, the profit or loss on the futures will be received as a series of daily variation margin payments over the period she holds the futures position, rather than in a single payment at the end. As we
have seen, timing differences affect the relative value of the cash flows. More generally, the settlement periods of loans and investments may not correspond exactly to the quarterly settlement cycle of the futures, which will mean that futures hedges have to be unwound prior to their expiry dates. For example, if we hedge a 3- month loan or investment that starts on 1 December with a December futures contract, we are likely to unwind the futures hedge at the start of the loan or investment period. Since this is approximately two weeks before the futures expiry date, the 3-month LIBOR on that date (which will apply to the loan or investment) is likely to be different from the 3-month forward LIBOR for the period starting on 16 December (which will be reflected in the futures price). We cannot know in advance how large this difference will be, so it will introduce an element of variability or unpredictability in the performance of the hedge. This and other sources of basis risk are typical of futures hedges, since the standardized terms of the futures contract may not match exactly the risk that we are trying to hedge. Many market participants are willing to accept this basis risk, however, in exchange for access to a liquid instrument that allows them to construct a reasonably effective hedge for the market risk they face in their underlying position or business.
4. Suppose that you are an international bank that can obtain funding and deposit funds at LIBOR. For the purpose of this question, you may also assume that you discount derivative cash flows at LIBOR. Explain how you could use a FRA to lock in the interest rate for a floating-rate loan of $N million in which you receive money at date 𝑡# and repay the loan with interest at rate 𝐿”! at date 𝑡% , where 𝐿”! is the LIBOR rateatdate𝑡# fortheperiod𝛿=𝑡% −𝑡#.
[Hint: Be sure to take into account the fact that the FRA settlement will take place at date 𝑡#.]
You can lock in a fixed rate equal to the forward rate between the two dates by going long a FRA in which you pay the agreed FRA rate and receive LIBOR. Because the FRA settlement will take place at 𝑡#, you will also need to deposit these funds for the period between 𝑡# and 𝑡% if your strategy is to exactly replicate a fixed-rate loan.
Denoteby𝐿 theLIBORrateatdate𝑡 fortheperiod𝛿=𝑡 −𝑡 andby𝐹 the “! # %#”!×””
agreed fixed rate in the FRA for the same period. The cash flows on each date are as shown in the following table:
𝑡#
𝑡%
𝐺𝑜 𝑙𝑜𝑛𝑔 𝑡# × 𝑡% 𝐹𝑅𝐴
𝑁^𝐿 −𝐹 _𝛿 “! “!×””
1+𝐿”!𝛿
0
𝐷𝑒𝑝𝑜𝑠𝑖𝑡 𝐹𝑅𝐴 𝑠𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡 𝑎𝑚𝑜𝑢𝑛𝑡 𝑎𝑡 𝐿𝐼𝐵𝑂𝑅 𝑓𝑟𝑜𝑚 𝑡# 𝑡𝑜 𝑡%
−
𝑁^𝐿 −𝐹 _𝛿 “! “!×””
1+𝐿”!𝛿
𝑁^𝐿 −𝐹 _𝛿 “! “!×””
𝐿𝑜𝑎𝑛 𝑎𝑡 𝐿𝐼𝐵𝑂𝑅 𝑓𝑟𝑜𝑚 𝑡#𝑡𝑜 𝑡%
𝑁
−𝑁^1 + 𝐿”! 𝛿_
𝑆𝑢𝑚 (𝑟𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑛𝑔 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜)
𝑁
−𝑁^1 + 𝐹 𝛿_ “!×””
As the bottom line of the table shows, the portfolio replicates a fixed-rate loan.