Financial Engineering
IC302
2. Simple Interest Rate Derivatives
David Oakes
–
Autumn Term 2020/1
Interest Rates and Yield Curves
2
Interest Rates
• Interest rates are prices that relate present values to future values.
• Investing present value $100 at annual interest rate 2% for one year gives future value equal to:
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 × 1 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒 = $100× 1 + 0.02 = $102
• Discounting $100 for one year at annual rate 2% gives present value of:
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = $100 = $98.04 1 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒 1 + 0.02
3
Compound Interest
• When interest is paid more than once during life of
investment, we must account for compound interest.
• Investing $100 for two years at 2% per year paid annually
gives future value of:
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 × 1 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒 ! = $100× 1 + 0.02 ! = $104.04
• Present value of $100 discounted for five years at 2% per year is:
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = $100 = $90.57 1+𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑎𝑡𝑒 ” 1+0.02 #
4
Compounding Frequency
• Some financial instruments make interest payments more than once each year. We must compound based on the coupon interest paid in each coupon period.
• $100 invested for five years at annual rate 2%, with interest paid twice each year, has future value:
$×”
𝐹𝑢𝑡𝑢𝑟𝑒𝑉𝑎𝑙𝑢𝑒=𝑃𝑟𝑒𝑠𝑒𝑛𝑡𝑉𝑎𝑙𝑢𝑒× 1+𝐴𝑛𝑛𝑢𝑎𝑙𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑎𝑡𝑒
0.02 !×# 𝑚 = $100× 1 + 2 = $110.46
where m is the compounding frequency per year.
5
Compounding Frequency
•
The present value of $100 to be received 5 years from now when discounted at an annual rate of 5% per year that is compounded semi-annually is:
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = $100 = $78.12 1 + 𝐴𝑛𝑛𝑢𝑎𝑙 𝑅𝑎𝑡𝑒 $×” 1 + 0.05 !×#
𝑚2
6
Nominal and Effective Rates
• Promised annual interest rate is the nominal rate.
• Calculated return per year taking into account the compounding frequency is the effective annual rate (also known as the annual percentage rate).
1 + 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝐴𝑛𝑛𝑢𝑎𝑙 𝑅𝑎𝑡𝑒 = 1 + 𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑎𝑡𝑒
$ 𝑚
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑎𝑡𝑒 $
→ 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝐴𝑛𝑛𝑢𝑎𝑙𝑅𝑎𝑡𝑒= 1+ 𝑚 −1
7
Nominal and Effective Rates
• The effective annual rate that corresponds to a nominal annual rate of 3% compounded four times per year is:
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑎𝑡𝑒 $ 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝐴𝑛𝑛𝑢𝑎𝑙 𝑅𝑎𝑡𝑒 = 1 + 𝑚 − 1
0.03 &
= 1+ 4 −1=3.0339%
8
Continuous Compounding
• The limit as the compounding frequency m tends to
infinity is known as continuous compounding.
• Present and future values with continuous compounding
can be calculated using the exponential function: lim 1+𝑥$≡𝑒𝑥𝑝𝑥≡𝑒)
$→( 𝑚 where 𝑒 ≈ 2.71828.
• This function is built into most calculators.
9
Continuous Compounding
• Future value under continuous compounding at rate r for t years is:
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 × 𝑒*”
• Present value under continuous compounding is: 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 × 𝑒+*”
• So present value of $100 discounted for 5 years at a continuously compounded rate of 3% per year is:
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 × 𝑒+*”= $100𝑒+,.,. # = $86.07
10
Continuous Compounding
• Continuously compounded returns are time consistent.
• They add up over sub-periods and preserve their distributional properties (e.g. if daily continuously compounded returns are normally distributed, then annual returns will also be normally distributed).
• This makes continuously compounded returns very convenient to use in quantitative applications, and we frequently employ them in financial modelling.
11
Yield Curve
• A yield curve is a plot of interest rates that apply over different maturities in a particular market.
• A yield curve summarizes current pricing conditions in the market to which it applies.
• Yield curve movements (e.g. an increase or decrease in rates or a flattening or steepening of the curve) imply changes in the interest rates at which future cash flows will be discounted and therefore changes in value for financial instruments priced relative to that curve.
12
Government Yield Curves
Refinitiv Eikon
13
Zero-Coupon Yields
• In principle, investors may discount future cash flows from a given source (e.g. the Government of Canada) to be received on different future dates at different rates.
• We call these term-specific interest rates zero-coupon yields because they represent the return on an investment that pays a single cash flow at a future date.
• For consistency, we might expect investors to use the same set of zero-coupon yields to value different sets of cash flows or financial instruments from the same source or with the same credit quality and liquidity.
14
Estimating Zero-Coupon Yields
• Most of the interest rates that we observe directly in the market are not zero-coupon yields.
• Estimating the zero-coupon yields for a particular market requires the application of mathematical and statistical techniques to a contemporaneous sample of prices for financial instruments that are homogeneous in credit quality and sufficiently liquid for prices to be reliable.
• In practice, the only markets that typically meet these criteria are government bonds and interest rate swaps.
15
Building Benchmark Yield Curves
• We build zero-coupon yield curves for these benchmarks.
• In simple cases, this can be done through bootstrapping.
• In the next lecture, we will use bootstrapping to build zero-coupon yield curves from fixed-for-floating interest rate swap rates and overnight index swap (OIS) rates.
• These are used to value swaps and other derivatives.
16
US Treasury Zero Curve
Refinitiv Eikon
17
USD OIS Zero Curve
Refinitiv Eikon
18
Discount Factors
• The pricing information contained in zero-coupon yields
can also be represented using discount factors.
• The t-year discount factor is calculated as:
1 1+𝑟” ”
where rt is the t-year zero-coupon yield.
• Knowing the discount factors is equivalent to knowing the zero-coupon yields, and vice versa. They contain the same information about market prices.
𝐷” =
19
Discount Factors
•
Consider the following zero-coupon yields:
t
1
2
3
rt
2%
3%
4%
𝐷/ =
1 =0.9804 𝐷! = 1 =0.9426 𝐷. = 1+0.02 1+0.03 !
1 =0.8890 1+0.04 .
•
The corresponding discount factors are:
• Discount factors are decreasing with maturity.
20
Forward Rates
21
Forward Rates
• A forward rate is an interest rate agreed today for borrowing or lending between two future dates.
• Forward rates must be consistent with zero-coupon yields or discount factors within any given market.
• Suppose that we invest $100 for two years at a two-year rate r2 of 3% per year. In two years’ time we will have:
𝐹𝑉! = 𝑃𝑉, 1+𝑟! ! = $100 1+0.03 ! = $106.09
22
Implied Forward Rate
• Now suppose that we invest $100 for one year at rate r1 = 2% and commit to roll over the proceeds for a second year at a 1×2-year forward rate f1x2.
• In two years we will have:
𝐹𝑉!∗ = 𝑃𝑉, 1 + 𝑟/ 1 + 𝑓/×! = $100 1 + 0.02 1 + 𝑓/×!
• If neither strategy involves any risk, then no- arbitrage requires that 𝐹𝑉P = 𝐹𝑉P∗ .
• The implied or fair-value forward rate is the forward rate for which this will be the case.
23
Implied Forward Rate
Spot 1 year
2 years
24
Implied Forward Rate r2
Spot 1 year
2 years
25
Implied Forward Rate
r1
r2
Spot
1 year
2 years
26
Implied Forward Rate
r2
r1
Spot
1 year
2 years
f1x2
27
Implied Forward Rate
• Setting the future values for both strategies equal gives: 𝐹𝑉! = 𝐹𝑉!∗
𝑃𝑉, 1+𝑟! ! =𝑃𝑉, 1+𝑟/ 1+𝑓/×!
1 + 𝑟! ! 1 + 0.03 !
→ 𝑓/×! = 1+𝑟/ −1= 1+0.02 −1=0.0401
• The fair-value or implied forward rate that satisfied the no-arbitrage condition is approximately 4.01%.
• Forward rates contain the same pricing information as zero-coupon yields or discount factors.
28
Engineering a Forward Loan
29
Engineering a Forward Loan
• We can use this idea to engineer a forward loan from cash deposit rates, which are a kind of zero-coupon yield.
• Suppose that we agree today (date t0) to borrow $100 at date t1, repaying the loan with interest at date t2. We agree today the interest rate on the loan, denoted f1x2.
+100
𝑡,
• Interest on the loan is paid on a money market basis with year fraction 𝛿RP = 𝑡P − 𝑡R ⁄360 .
Forward Loan 𝑡/ from t1 to t2
−100 1+𝑓!×#𝛿!#
𝑡!
30
Engineering a Forward Loan • Split the loan cash flows into two parts:
𝑡, 𝑡, 𝑡,
+100
+100
𝑡/ 𝑡/ 𝑡/
Forward Loan
from t1 to t2 −100 1+𝑓!×#𝛿!#
𝑡!
𝑡!
−100 1+𝑓!×#𝛿!#
31
Engineering a Forward Loan
• Transform the new cash flows into tradable contracts by
adding and subtracting compensating cash flows at t0:
𝑡,
+100
+100
𝑡/ 𝑡/ 𝑡/
Forward Loan
from t1 to t2 −100 1+𝑓!×#𝛿!#
−𝐶$! 𝐶$!
𝑡,
−100 1+𝑓!×#𝛿!#
𝑡!
𝑡,
𝑡!
32
Engineering a Forward Loan
• These new cash flows can be interpreted as a deposit and
a loan, both with present value 𝐶S! : +100
Forward Loan
from t1 to t2 −100 1+𝑓!×#𝛿!#
Deposit from t0 to t1
Loan
from t0 to t2
− 𝐶$! + 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡,
𝑡/ 𝑡/ 𝑡/
𝑡!
+100
−𝐶$! 𝐶$!
𝑡,
𝑡,
𝑡!
33
Engineering a Forward Loan
• If there is no credit risk and there are no arbitrage
opportunities, then the amount we can borrow at t0 is: 𝐶” =1001+𝑓/×!𝛿/!
% 1 + 𝑟!𝛿,!
• We immediately reinvest this to obtain: 𝐶”% 1+𝑟/𝛿,/ =100
• Adding vertically, we recover the cash flows for the forward loan, confirming that the loan plus the deposit constitute a synthetic portfolio for the forward loan.
34
Engineering a Forward Loan
• No-arbitrage requires that the cash flows on the forward
loan and the synthetic at time t2 be identical: 1001+𝑓/×!𝛿/! =𝐶”% 1+𝑟!𝛿,!
• Substituting for 𝐶S! from the expression for the return on the deposit (see the previous slide) gives:
100 1+𝑓/×!𝛿/!
→ 1+𝑓/×!𝛿/! = 1+𝑟!𝛿,! 1 + 𝑟/𝛿,/
=100 1+𝑟!𝛿,! 1 + 𝑟/𝛿,/
→ 𝑓/×!𝛿/! = 1+𝑟!𝛿,! −1 1 + 𝑟/𝛿,/
• This is essentially the implied forward rate we saw earlier. 35
Contractual Equation
• Contractual equation for forward loan:
+
• The synthetic replicates the forward loan and is a starting point for understanding forward interest rates.
• In practice, the deposit rates referenced by financial instruments (e.g. LIBOR and Euribor) often include credit risk premiums, which complicates forward pricing.
• We show how to solve this problem in the next lecture.
36
Forward Loan from t1 to t2
=
Loan from t0 to t2
Deposit from t0 to t1
Derivatives on Forward Rates
• In fact, forward loans are not liquid and are rarely traded.
• Instead, market participants who want to hedge or ‘lock in’ interest rates for forward periods rely on liquid derivative instruments that create exposure to forward rates.
• In the remainder of the lecture, we look at two of these derivative structures: interest rate futures and forward rate agreements (FRAs).
37
Interest Rate Futures
38
Interest Rate Futures
• Interest rate futures are exchange-traded, centrally cleared contracts with standardized features in which
counterparties make forward commitments with respect to debt instruments or interest rates.
• Short-term interest rate futures create forward exposure to short-term interest rates (e.g. 3-month USD LIBOR).
• Bond futures create forward exposure to bond prices (e.g. US Treasuries or UK Gilts).
• We consider only short-term interest rate futures here.
39
Short-Term Interest Rate Futures
• Eurodollar futures (CME, ICE Futures Europe)
• Euribor futures (ICE Futures Europe, Eurex, CME)
• Each contract makes payoffs linked to a specific money market interest rate for the three-month period that starts when the futures contract expires.
Dec 2020 Mar 2021 Jun 2021 Sep 2021 Dec 2021
• Fixed quarterly cycle of delivery months plus contracts that expire in nearby serial months to add flexibility.
40
Eurodollar Futures
41
Implied Forward Rate
• Futures price is 100 minus implied forward rate: 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑃𝑟𝑖𝑐𝑒 = 100 − 𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑅𝑎𝑡𝑒
• A quote of 99.730 for the December 2020 Eurodollar futures contract implies a 3-month US dollar LIBOR rate of 0.270% for the period that starts on 16 December 2020:
𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑃𝑟𝑖𝑐𝑒 = 100 − 𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑅𝑎𝑡𝑒
→ 𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑅𝑎𝑡𝑒 = 100 − 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑃𝑟𝑖𝑐𝑒 = 100 − 99.730 = 0.270
42
Futures Profit and Loss
• Suppose December 2020 futures price goes up to 99.740.
• This implies a forward LIBOR of 0.260% for the same 3- month period beginning on 16 December 2020.
• Futures price has increased by one basis point.
• Contract specifies daily settlement of $12.50 per contract for each one-half of one basis point, so the long counterparty makes a mark-to-market profit of $25.00.
43
Futures Profit and Loss
• This will be received as variation margin.
• Short counterparty makes a loss of the same size.
• If the price had fallen to 99.720, this would imply a a
forward LIBOR of 0.280% for the same period.
• Long counterparty would make a mark-to-market loss of $25.00, to be paid as variation margin.
• Short counterparty would make a profit of the same size. 44
Eurodollar Futures DV01
• Long counterparty makes money if the underlying forward rate falls and loses money if it increases.
• Contract specifies fixed payout of $25.00 for each one- basis-point movement in contract price or forward rate.
• DV01 of Eurodollar futures is therefore $25.00 per contract and does not change as interest rates change.
• Other short-term interest rate futures contracts (e.g. Euribor futures) work in the same way.
45
Futures and Forward Loans
• We went long one December Eurodollar futures at 99.730.
• When futures price increased to 99.740, we made $25.
• This equals the profit from agreeing to lend $1 million for three months starting in December at 0.270% and then later offsetting this position by borrowing the same amount for the same period at 0.260%.
• Long position in futures is like forward lending.
• Short position in futures is like forward borrowing.
46
Hedging with Futures
• Suppose that a company plans to borrow $1 million for three months beginning on 16 December 2020.
• Interest on loan will be 3-month LIBOR rate at that date.
• This rate is not known now, so they face interest rate risk.
• They can hedge this risk by going short Eurodollar futures.
• Suppose they go short one December contract at 99.730.
47
Hedging with Futures
• • •
•
On 16 December, the futures expires and the loan begins.
Suppose that 3-month USD LIBOR on that date is 0.300%.
Interest on the 3-month loan will be approximately:
𝐿𝑜𝑎𝑛 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $1 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 0.003 × 0.25 = $750
Final settlement price of December futures will be:
𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑃𝑟𝑖𝑐𝑒 = 100 − 𝑈𝑆𝐷 𝐿𝐼𝐵𝑂𝑅 𝑟𝑎𝑡𝑒 = 100 − 0.300 = 99.700
48
Hedging with Futures
• Futures price fell by 3 basis points, from 99.730 to 99.700.
• Profit or loss on futures position is:
𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑜𝑓𝑖𝑡 𝑜𝑟 𝑙𝑜𝑠𝑠
= 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 𝑐h𝑎𝑛𝑔𝑒 𝑖𝑛 𝑏𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡𝑠 × 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 × 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝐷𝑉01 =−3× −1 ×$25=$75
• Setting the hedging profit against the interest on the loan give them an effective interest cost of:
𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝐶𝑜𝑠𝑡 = 𝐿𝑜𝑎𝑛 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 − 𝐻𝑒𝑑𝑔𝑖𝑛𝑔 𝑃𝑟𝑜𝑓𝑖𝑡 = $750 − $75 = $675
49
Hedging with Futures
• This is equivalent to paying interest on the loan at 0.270%:
𝐿𝑜𝑎𝑛 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $1 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 𝑅𝑎𝑡𝑒 × 0.25 = $675 → 𝑅𝑎𝑡𝑒 = 0.270%
• But this is equal to the forward rate implied by the price at which they entered the December futures (99.730).
• Similar calculations show that, whatever value LIBOR settles at on 16 December, the company pays the same effective rate after taking into account its hedging profit.
• Going short futures allows them to lock in their borrowing rate for the 3-month period beginning in December.
50
Futures and Derivatives Markets
• Most futures hedges, of course, are not as exact as this.
• Various sources of basis risk affect their performance.
• Nonetheless, interest rate futures offer exposure to forward rates similar to that of a forward loan in a liquid instrument with fewer balance sheet implications and an effective mechanism for managing counterparty risk.
• Futures are widely used to hedge other interest rate derivatives (e.g. swaps), and futures-implied forward rates play an important role in derivatives pricing.
51
Forward Rate Agreements
Forward Rate Agreements (FRAs)
• A forward rate agreement (FRA) is an over-the-counter derivative that fixes an interest rate for a future period.
• FRAs typically reference money market interest rates such as LIBOR or Euribor but may also reference other rates.
• FRAs can be used for many of the same purposes as short- term interest rate futures.
• For example, the company in our earlier example could have hedged using a FRA rather than Eurodollar futures.
FRAs and Interest Rate Risk
• The long counterparty in a FRA pays an agreed fixed rate and receives the floating rate (e.g. 3-month USD LIBOR) on the notional principal amount for the forward period.
• A long position in a FRA therefore makes a profit if the underlying interest rate increases.
• This is exactly the opposite of the exposure in futures
• In terms of interest rate exposure, a long position in a FRA is like a short position in a futures contract.
54
Contractual Equation
• The FRA involves an exchange of fixed cash flow (at the agreed FRA rate) and a floating cash flow (e.g. at LIBOR).
• It can be replicated by a forward fixed rate loan and a forward floating rate deposit.
• The contractual equation for a FRA is therefore: =+
FRA
Pay fixed, receive floating for forward period t1 to t2
Fixed-Rate Loan from t1 to t2
Floating-Rate Deposit from t1 to t2
55
FRA Settlement
• Suppose that we go long a FRA on 3-month USD LIBOR covering the 3-month period from December 2020.
• The agreed fixed rate is 0.270% and the notional principal amount is $1 million.
• If the LIBOR fixing at the start of the forward period turns out to be 0.300%, the payoff to our long position will be:
𝐹𝑅𝐴𝑃𝑎𝑦𝑜𝑓𝑓=𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙× 𝐿𝐼𝐵𝑂𝑅−𝐴𝑔𝑟𝑒𝑒𝑑𝐹𝑅𝐴𝑅𝑎𝑡𝑒 ×𝑌𝑒𝑎𝑟𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 = $1 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 × 0.003 − 0.0027 × 0.25 = $75
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FRA Settlement
• Notice that this is the same as the payoff we saw earlier for the short futures position in the same market scenario.
• The FRA is structured like a lending agreement, in which the interest is paid in arrears (i.e. in March 2021).
• In practice, however, FRAs are usually settled at the start of the forward period to which they apply, based on the present value of the settlement amount on that date.
• The FRA settlement in our example would therefore be the present value in December of $75 to be paid in March.
57
FRA Settlement • We can write this as:
𝐹𝑅𝐴𝑆𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡=𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙× 𝐿𝐼𝐵𝑂𝑅−𝐴𝑔𝑟𝑒𝑒𝑑𝐹𝑅𝐴𝑅𝑎𝑡𝑒 ×𝛿 1+𝑟𝛿
where 𝛿 is the year fraction for the forward period and 𝑟 is the interest rate used to discount the FRA payoff.
• This makes the DV01 of the FRA slightly less than than the DV01 of a Eurodollar futures on the same notional.
• In current market practice, the rate used to discount the FRA payoff will be the risk-free zero rate for the period covered by the FRA (see Hull, ch. 4).
58
FRA Settlement
• Prior to the financial crisis of 2007-9, derivative cash flows were typically discounted at LIBOR rates.
• In this framework, the FRA payoff can be written: 𝐹𝑅𝐴𝑆𝑒𝑡𝑡𝑙𝑒𝑚𝑒𝑛𝑡=𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙× 𝐿𝐼𝐵𝑂𝑅−𝐴𝑔𝑟𝑒𝑒𝑑𝐹𝑅𝐴𝑅𝑎𝑡𝑒 ×𝛿
1 + 𝐿𝐼𝐵𝑂𝑅 𝛿
where 𝐿𝐼𝐵𝑂𝑅 is the 3-month LIBOR fixing at the start of the FRA period (this is the approach taken in Kosowski and Neftci, ch. 3).
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FRAs and Interest Rate Risk
• The fixed rate in a newly initiated FRA is usually the market forward rate for the period covered by the FRA.
• This makes the initial market value of the FRA zero.
• Over time, changes in the market forward rate will result
in changes in value for a FRA with a given fixed rate.
• FRAs can therefore be used to hedge interest rate risk and express views on interest rates in much the same way as interest rate futures contracts.
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FRA Mark-to-Market Value
• The mark-to-market value of a FRA can be found by calculating its payoff as if the LIBOR rate for the period
were equal to the current forward rate and then discounting this payoff to the present (see Hull, ch. 4).
• In current market practice, this discounting will be at the risk-free zero rate for the end date of the forward period.
• We will look more closely at valuation in the next lecture.
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FRA Convexity
• A change in interest rates will change both the payoff to the FRA at the end of the forward period and the rate at which that payoff is discounted to the present.
• This means that the DV01 of a FRA is not constant.
• Futures are linear instruments with fixed DV01s, but FRAs
are convex instruments with DV01s that vary with rates.
• We consider implications of this in the next section.
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Convexity Adjustment
FRAs, Futures, and Convexity
• Suppose that we go short 100 Eurodollar futures contracts (equivalent to $100 million notional) and short the DV01-
equivalent number of USD LIBOR FRAs (a slightly larger notional amount because of the smaller DV01 of the FRA).
• What will happen as interest rates change?
• An increase in the forward rate will result in a MTM loss
on the short FRA and a MTM profit on the short futures.
• A decrease in the forward rate will result in a MTM profit on the short FRA and a MTM loss on the short futures.
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FRAs, Futures, and Convexity
• Since the DV01s of the positions are matched, the net change in MTM value will be approximately zero.
• As rates increase, however, the loss on the short FRA will be discounted at higher rates, reducing its present value.
• As rates fall, the profit on the short FRA will be discounted at lower rates, increasing its present value.
• The payoff to the futures is not discounted, so it remains the same for each one-basis-point change in rates.
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FRAs, Futures, and Convexity
• As a result, our strategy will make a small profit as rates change, which will be increasing with the change in rates.
• The profit arises because the discounting of the FRA payoff makes it convex, while the futures is linear.
• Equivalently, the DV01 of the FRA changes as rates change, but the DV01 of the futures does not.
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FRAs, Futures, and Convexity
Interest Rate
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Convexity Bias
• To correct for this, the futures must trade at a slightly lower price, so that our strategy (which has zero risk and zero cost) does not generate an arbitrage profit.
• But a lower futures price will imply a higher forward rate.
• Forward rates implied by short-term interest rate futures
prices will be higher than true forward rates.
• They must therefore be adjusted downward to correct for convexity bias.
68
More Convexity Bias
• There is a second source of convexity bias in futures- implied forward rates, due to daily settlement of futures.
• Suppose that we are short the futures.
• If rates go up, we receive variation margin each day, which
can be invested at the now higher interest rate.
• If rates go down, we must pay variation margin each day, which can be borrowed at the now lower interest rate.
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More Convexity Bias
• This gives an advantage to the short futures position relative to a long FRA without daily settlement.
• Once again, the futures should trade at a lower price, but a lower price means a higher futures-implied forward rate.
• The futures-implied forward rate will be higher than a true forward rate and must be adjusted downward.
70
Convexity Adjustment
• This second source of convexity bias has become less important in recent years, as non-cleared derivatives have
become subject to collateralization and margin requirements much like those on cleared derivatives.
• Together, these sources of convexity bias mean that futures-implied forward rates are not true forward rates.
• They must be adjusted downward if they are to be used in derivatives valuation and other applications.
• How should this be done?
71
Convexity Adjustment
• One popular convexity adjustment for futures-implied forward rates is based on the Ho-Lee interest rate model:
𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑅𝑎𝑡𝑒 = 𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝑅𝑎𝑡𝑒 − 12 𝜎!𝑇/𝑇!
𝜎 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐h𝑎𝑛𝑔𝑒 𝑖𝑛 𝑠h𝑜𝑟𝑡 𝑡𝑒𝑟𝑚 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑜𝑣𝑒𝑟 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟
𝑇/ = 𝑡𝑖𝑚𝑒 𝑡𝑜 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑜𝑓 𝑓𝑢𝑡𝑢𝑟𝑒𝑠 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠
𝑇! = 𝑡𝑖𝑚𝑒 𝑡𝑜 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑜𝑓 𝑡h𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑡h𝑒 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡
• Adjustment is applied to continuously compounded rates. 72
References Main reading:
• Hull, ch. 4 and 6
• Kosowski and Neftci, ch. 3
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