Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Interest Rate Basics

Ferdinando M. Ametrano
October 11, 2018
1.3k

Interest Rate Basics

First lesson of the Interest Derivatives course of Milano-Bicocca

https://www.ametrano.net/ird/

Ferdinando M. Ametrano

October 11, 2018
Tweet

Transcript

  1. Discounting and Forwarding Curves
    Models and Implementation
    Interest Rate Basics
    Ferdinando M. Ametrano
    https://www.ametrano.net/about/

    View full-size slide

  2. Outline
    1. Interest Rates and Conventions
    2. Deposits
    3. Forward Rate Agreements
    4. Interest Rate Futures
    5. Bonds
    6. Interest Rate Swaps
    7. Indexes
    8. Bibliography
    2/97

    View full-size slide

  3. Interest Rate Basics
    1. Interest Rates and Conventions

    View full-size slide

  4. Basic common sense observation
    Cash availability
    is a privilege
    To give up this
    privilege there
    must be a
    compensation
    This
    compensation is
    paid
    by the borrower
    to the lender
    Inflation
    Time
    value of
    money
    Current
    risk-free rate
    Liquidity
    preference
    Borrower
    credit
    worthiness
    4/97

    View full-size slide

  5. Interest rate
    Cost of borrowing is expressed as interest, a percentage of the notional (or nominal or principal
    amount)
    If you invest an amount at the end of the investment you receive
    +
    where is the interest, or equivalently
    × ℎ(, )
    where
    ▪ ℎ , ≥ 1
    ▪ is the loan tenor
    ▪ is the interest rate
    5/97

    View full-size slide

  6. Capitalization and Discounting (1/2)
    , Capitalization factor
    ▪ The future value (in T) of one unit of currency paid today
    ▪ How many units of currency received in are equivalent to 1
    unit of currency received today?

    ,
    Discount factor
    ▪ The present value (today) of one unit of currency paid at time T
    ▪ How many units of currency received today are equivalent to 1
    unit of currency received in ?
    toda
    y
    T
    1 ℎ(, )
    toda
    y
    T
    1
    1
    ℎ(, )
    6

    View full-size slide

  7. Capitalization and Discounting (2/2)
    How has to be measured?
    ✓ Day Count Convention
    ✓ Business Day Convention
    How many types for ?
    ✓ Zero Rates
    ✓ Forward Rates
    ✓ Instantaneous Forward Rate
    How many forms for ?
    ✓ Simple Interest
    ✓ Compounded Interest
    ✓ Continuous Comp. Interest
    h ,
    1 2
    3
    7/97

    View full-size slide

  8. Conventions
    The tenor
    ▪ is expressed as year fraction
    ▪ is defined by two dates 1 e 2
    ▪ has to be measured according to a day count
    8/97
    Input Convention Output
    1
    , 2 DayCount Convention (1
    , 2
    )
    1
    , Business Day Convention
    + Calendar
    2

    View full-size slide

  9. Day Count ISDA Convention
    Defines
    ▪ the period of time to which the interest rate applies
    ▪ the period of time used to calculate accrued interest (relevant when the instrument is bought of sold)
    ▪ Money market
    ◦ Actual/360
    ◦ Actual/365 (fixed) - Actual/365 (actual)
    ◦ Actual/Actual
    ▪ Bond basis (corporate bond and derivatives)
    ◦ 30/360 (30 days month, 360 days year)
    ▪ Treasury
    ◦ Actual/Actual
    9/97

    View full-size slide

  10. Day Count Convention - The February Effect
    How many days of interest are earned between February 28, 2013 and March 1, 2013 when
    ▪ day count is Actual/Actual in period?
    ▪ day count is 30/360?
    10/97

    View full-size slide

  11. Business Day Convention
    Used to determine what business day should be used for a payment when the given date falls on a bank holiday
    ▪ Following [Preceding]
    Choose the first business day after [before] the given holiday.
    ▪ Modified Following [Preceding]
    Choose the first business day after [before] the given holiday, unless it belongs to a different month. If the first business day is
    in a different month, choose the first business day before [after] the holiday.
    ▪ End of Month
    When the start date of a period is on the final business day of a particular calendar month, the end date is on the final business
    day of the end month (not necessarily the corresponding date in the end month).
    11/97

    View full-size slide

  12. Simple Interest
    In the simplest case we have
    = × ×
    + = × (1 + × )
    Capitalization factor
    ℎ(, ) = 1 + R ×
    Discount factor
    1
    ℎ ,
    =
    1
    1 + ×
    12/97

    View full-size slide

  13. Simple Interest - Examples
    ▪ Bond 8% Actual/Actual in period
    ◦ 4% is earned between coupon payment dates. Accruals on an Actual basis. When coupons are paid on March 1 and Sept 1,
    how much interest is earned between March 1 and April 1?
    ▪ Bond 8% 30/360
    ◦ Assumes 30 days per month and 360 days per year. When coupons are paid on March 1 and Sept 1, how much interest is
    earned between March 1 and April 1?
    ▪ T-Bill 8% Actual/360
    ◦ 8% is earned in 360 days. Accrual calculated by dividing the actual number of days in the period by 360. How much interest is
    earned between March 1 and April 1?
    13/97

    View full-size slide

  14. When we compound times per year for (1
    , 2
    ) years at rate an amount grows to
    × 1 +



    Usually the interest is payed
    ▪ At the end of the period for tenor less than 1Y
    ▪ Annualy for longer tenors
    Capitalization factor
    ℎ , = 1 +



    Discount factor
    1
    ℎ ,
    = 1 +



    Compounded Interest
    14/97

    View full-size slide

  15. Compounding frequency
    Interest rate payment frequency is very relevant: a montly rate has higher value than the same rate payed annually.
    Example
    = €100, = 10%, DayCount = 30/360
    Compounding frequency Value of €100 in one year at
    10%
    Annual (m=1) 110.00
    Semiannual (m=2) 110.25
    Quarterly (m=4) 110.38
    Bimonthly (m=6) 110.43
    Monthly (m=12) 110.47
    Biweekly (m=26) 110.50
    Weekly (m=52) 110.51
    Daily (m=365) 110.52
    15/97

    View full-size slide

  16. Continuous Compounding
    In the limit as we compound more and more frequently we obtain continuously compounded interest rates. grows to
    lim
    →+∞
    × 1 +



    = ×
    when invested at a continuously compounded rate for time (1
    , 2
    )
    Capitalization factor
    ℎ , =
    Discount factor
    1
    ℎ ,
    = −
    16/97

    View full-size slide

  17. Spot and forward rates
    ▪ A zero (or spot) rate for maturity
    o is the rate of interest earned on an investment that starts accruing interests from today (= spot date) until
    o provides a payoff only at time (no intermediate coupons)
    ▪ A forward rate
    o is the rate of interest earned on an investment that starts accruing at a future date (> spot date).
    o is the future zero rate implied by today’s term structure of interest rates
    Example
    ▪ 6M spot rate
    ▪ 3x6 forward rate: the 3M zero rate starting in 3M
    17/97

    View full-size slide

  18. Spot and forward rates - Example
    Once upon a time...
    Two spot rates imply a forward rate
    A. Borrow 1,000,000 at 4% for 2Y. Have to pay back
    1,000,000 × (1 + 4% × 2) = 1,080,000
    1. Lend 1,000,000 at 3% per 1Y. Gain:
    1,000,000 × (1 + 3% × 1) = 1,030,000
    2. Then invest 1,030,000 for another 1Y
    1,030,000 × (1 + × 1)
    For [A] to be equivalent to [1+2] , the 12x24 rate must be
    =
    1,080,000
    1,030,000
    − 1 = 4.85%
    18/97

    View full-size slide

  19. Spot and forward rates - Formula (1/2)
    Suppose that the zero rates for time periods 1 and 2 are 1 and 2 with both rates continuously compounded
    The forward rate for the period between times 1 and 2 satisfies
    1(0,1)(1,2) = 2(0,2)
    So we have
    =
    2
    (0, 2
    ) − 1
    (0, 1
    )
    (1
    , 2
    )
    19/97

    View full-size slide

  20. Spot and forward rates - Formula (2/2)
    The last formula is only approximately true when rates are not expressed with continuous compounding.
    For example, with simple rates we have
    1 + 1
    1
    1 + 1
    , 2
    = 1 + 2
    2
    =
    1
    (1
    , 2
    )
    1 + 2
    2
    1 + 1
    1
    − 1 =
    2
    2
    − 1
    1
    (1
    , 2
    ) 1 + 1
    1
    or
    =
    1
    (1
    , 2
    )
    1
    2
    − 1
    where 1 , 2 are the discount factors corresponding to rates 1 , 2
    20/97

    View full-size slide

  21. Spot and forward rates - Application of the Formula
    Year (n) Zero rate for n-year investment
    (% per annum)
    1Y rate forward n years
    (% per annum)
    1 3.0
    2 4.0 5.0
    3 4.6 5.8
    4 5.0 6.2
    5 5.5 7.5
    21/97

    View full-size slide

  22. Upward vs Downward Sloping Rate Curve
    ▪ For an upward sloping yield curve:
    >
    ▪ For a downward sloping yield curve
    <
    22/97

    View full-size slide

  23. The instantaneous forward rate for a maturity is the forward rate that applies for a very short time period starting at .
    If is the -year rate and + is the ( + )-year rate (continuous compounding) we have
    = lim
    →0
    +
    ( + ) −

    + −
    = lim
    →0
    +


    +
    +


    =
    +


    =



    +


    =




    = න
    0


    or equivalently

    =
    1


    0


    Instantaneous Forward Rate
    23/97

    View full-size slide

  24. = − = − ׬
    0


    or equivalently
    − = = න
    0


    where
    ▪ is the discount factor
    ▪ is the zero rate (continuous compounding)
    ▪ is the instantaneous forward rate
    ▪ is misured according to Act/365 convention (or another strictly monotone day count…why?)
    We have 0 =1
    ▪ < 1 for any > 0 (assuming positive interest rates)
    ▪ 1
    > 2 if 1
    < 2 (assuming positive interest rates)
    ▪ > 0 for any
    Discounts and rates
    24/97

    View full-size slide

  25. Treasury and Libor
    ▪ Treasury rates: rates on instruments issued by a government in its own currency.
    ▪ LIBOR rate: the rate of interest at which a bank is prepared to deposit money with another bank (the second bank must typically
    have a AA rating).
    ▪ LIBID rate: the rate which a AA bank is prepared to pay on deposits from another bank.
    ▪ Eurodollar futures and swaps are used to extend the LIBOR yield curve beyond one year.
    25/97

    View full-size slide

  26. Overnight and repo
    ▪ Overnight rate: the rate of interest at which a bank is prepared to lend to another bank for just one day.
    ▪ Repo rates: Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today
    for X and buy them back in the future for a slightly higher price Y
    ◦ The financial institution obtains a loan
    ◦ The rate of interest is calculated from the difference between X and Y and is known as the repo rate
    26/97

    View full-size slide

  27. The Risk-Free Rate
    ▪ The short-term risk-free rate traditionally used by derivatives practitioners was LIBOR
    ▪ Now the overnight indexed swap rate is increasingly being used instead of LIBOR as the risk-free rate
    27/97

    View full-size slide

  28. Interest Rate Basics
    2. Deposits

    View full-size slide

  29. Deposits
    Interest Rate Deposits are OTC zero coupon contracts that start at reference date 0 (today or spot), span the length corresponding
    to their maturity and pay the (annual, simply compounded) interest accrued over the period with a given rate fixed at 0
    ▪ Maturities
    o O/N (overnight), T/N (tomorrow-next), S/N (spot-next)
    o 1W or SW (spot-week)
    o 1M, 2M, 3M, 6M, 9M, 12M
    ▪ Deposit Rates are
    o illiquid
    o not collateralized (to be explained later)
    o not representative of Libor/Euribor fixings (which are the underlying of collateralized interest rate derivatives)
    29/97

    View full-size slide

  30. Linear Interpolation (1/2)
    Used to calculate stub period Libor fixings and Deposit rates
    30/97

    View full-size slide

  31. Linear Interpolation (2/2)
    ▪ Euribor 2M (62 days) → 3% = 300 bps
    ▪ Euribor 3M (92 days) → 3.60% = 360 bps
    60 bps increase in 30 days
    60/30=2 bps per day
    ▪ Euribor 80D (80=62+18 days)
    300 bps + (2 bps ∗ 18) = 336 bps = 3.36%
    31/97

    View full-size slide

  32. Interest Rate Basics
    3. Forward Rate Agreements

    View full-size slide

  33. Forward Rate Agreement: Definition
    ▪ A Forward Rate Agreement (FRA) is an OTC agreement that a certain rate will apply to a certain principal during a certain future
    time period
    ▪ FRAs pay the difference between a given strike and the underlying Euribor fixing
    ▪ 4x7 stands for 3M Euribor fixing in 4 months time
    ▪ The market quotes FRA strips with different fixing dates and Libor/Euribor tenors
    33/97

    View full-size slide

  34. Forward Rate Agreement: Key Results
    A FRA is equivalent to an agreement where interest at a predetermined rate is exchanged for interest at the market rate
    A FRA can be valued by assuming that the forward LIBOR interest rate is certain to be realized
    This means that the value of a FRA is the present value of the difference between the interest that would be paid at rate and the
    interest that would be paid at rate
    34/97

    View full-size slide

  35. Valuation Formulas (1/2)
    If the period to which a FRA applies lasts from 1 to 2, we assume that and are expressed with a compounding frequency
    corresponding to the length of the period between 1
    and 2
    With an interest rate of , the interest cash flow at time 2 is

    (1
    , 2
    )
    With an interest rate of , the interest cash flow at time 2 is

    (1
    , 2
    )
    35/97

    View full-size slide

  36. Valuation Formulas (2/2)
    ▪ When the rate will be received on a principal of the value of the FRA is the present value of
    (

    )(1
    , 2
    )
    received at time 2
    ▪ When the rate will be paid on a principal of the value of the FRA is the present value of
    (

    )(1
    , 2
    )
    received at time 2
    36/97

    View full-size slide

  37. Example (1/2)
    ▪ A FRA entered into some time ago ensures that a company will receive 4% (s.a.) on $100 million for six months starting in 1 year
    ▪ Forward LIBOR current estimation for the period is 5% (s.a.)
    ▪ The 1.5 year rate is 4.5% with continuous compounding
    The value of the FRA (in $ millions) is
    100 × 0.04 − 0.05 × 0.5 × −0.045×1.5 = −0.467
    37/97

    View full-size slide

  38. Example (2/2)
    If the six-month interest rate in one year turns out to be 5.5% (s.a.) there will be a payoff (in $ millions) of
    100 × 0.04 − 0.055 × 0.5 = −0.75
    in 1.5 years.
    Considering that
    1 + 0.055 × 0.5 = 1.0275
    the transaction might be settled at the one-year point for an equivalent discounted payoff of
    −0.75
    1.0275
    = −0.73
    38/97

    View full-size slide

  39. Forward Rate Agreement: real payoff
    ( − )(1
    , 2
    )
    1
    (1 + × (1
    , 2
    ) )
    ▪ Notional
    ▪ Libor Fixing
    ▪ FRA strike
    39/97

    View full-size slide

  40. Interest Rate Basics
    4. Interest Rate Futures

    View full-size slide

  41. Interest Rate Futures (1/2)
    ▪ Futures are exchange-traded contracts similar to OTC FRAs. Any profit and loss is regulated through daily marking to market
    (margining process). Such standard characteristics reduce credit risk and transaction costs, thus enhancing a very high liquidity.
    ▪ The most common contracts insist on Euribor3M and expire every March, June, September and December (IMM dates). The first
    front contract is the most liquid interest rate instrument, with longer expiry contracts having decent liquidity up to about the 8th
    contract.
    41/97

    View full-size slide

  42. Interest Rate Futures (2/2)
    ▪ There are also the so called serial futures, expiring in the upcoming months not covered by the quarterly IMM futures. The first
    serial contract is quite liquid, especially when it expires before the front contract.
    ▪ Futures are quoted in terms of prices instead of rates. The relation is
    = 100 −
    To mimic the bond behavior of increasing prices for decreasing rates (see later)
    42/97

    View full-size slide

  43. Eurodollar Futures
    ▪ A Eurodollar is a dollar deposited in a bank outside the United States.
    ▪ Eurodollar Futures are futures on the 3-month Eurodollar deposit rate (same as 3-month LIBOR rate)
    ▪ It is margined every day
    ▪ It is settled in cash
    ▪ When it expires (on the third Wednesday of the delivery month) the final settlement price is 100 minus the actual three month
    Eurodollar deposit rate
    43/97

    View full-size slide

  44. Eurodollar Futures: Formula for Contract Value
    ▪ One contract is on the rate earned on $1 million
    ▪ A change of one basis point or 0.01% in a Eurodollar futures quote corresponds to a contract price change of $/€ 25
    1,000,000 for 3 months (t = 0.25)
    1,000,000 × × 0.25 = 250,000 ×
    1 basis point = 0.01% = 0.0001
    250,000 × 1 = 25
    44/97

    View full-size slide

  45. Example (1/2)
    ▪ Suppose you buy (take a long position in) a contract on Nov 1
    ▪ The contract expires on Dec 21
    ▪ The prices are as shown in table
    ▪ How much do you gain or lose
    o on the first day
    o on the second day
    o over the whole time until expiration
    Date Quote
    Nov 1 97.12
    Nov 2 97.23
    Nov 3 96.98
    ……. ……
    Dec 21 97.42
    45/97

    View full-size slide

  46. Example (2/2)
    If on Nov 1 you know that you will have $1 million to invest for three months on Dec 21, the contract locks in a rate of
    100 − 97.12 = 2.88%
    In the example you earn
    100 − 97.42 = 2.58%
    on $1 million for three months (= $6,450) and make a gain day by day on the futures contract of
    30 × $25 = $750
    46/97

    View full-size slide

  47. Convexity adjustment (1/3)
    Once upon a time…
    Because of daily marking to market an investor short on a futures contract
    ▪ when the futures price increases (rate decreases) has a loss that can be funded at lower rate
    ▪ when the futures price decreases (rate increases) has a profit that can be invested at higher rate
    Short futures, long FRA generates free money: there is a convexity to be compensated with lower futures price (higher rate)

    =
    +
    47/97

    View full-size slide

  48. Convexity adjustment (2/3)
    Forward rate volatility and its correlation to the spot rate have to be accounted for. Easiest evaluation using Ho-Lee:

    =
    − 0.5 × 2 × 1
    × 2
    where
    ▪ 1 is the start of period covered by the forward/futures rate
    ▪ 2 is the end of period covered by the forward/futures rate (90 days later that 1)
    ▪ is the standard deviation of the change in the short rate per year
    Market practice: Hull-White (Bloomberg: fixed mean reversion, rough volatility evaluation)
    48/97

    View full-size slide

  49. Convexity Adjustment (3/3)
    = 0.012
    49/97
    Maturity of Futures (yrs) Convexity Adjustment (bps)
    2 3.2
    4 12.2
    6 27.0
    8 47.5
    10 73.8

    View full-size slide

  50. Extending the LIBOR Zero Curve
    ▪ LIBOR deposit rates define the LIBOR zero curve out to one year
    ▪ Eurodollar futures can be used to determine forward rates and the forward rates can then be used to bootstrap the zero curve
    50/97

    View full-size slide

  51. Interest Rate Basics
    5. Bonds

    View full-size slide

  52. Bond Price Quotes
    Cash price = Quoted price + Accrued Interest
    ▪ The quoted price is called clean price because it is free from the deterministic price change effect of accruing interest
    ▪ The cash price is also called dirty price
    52/97

    View full-size slide

  53. Bond Yield
    The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
    Suppose that the market price of the bond in our example equals its theoretical price of 98.39. The bond yield (continuously
    compounded) is given by solving
    3−×0.5 + 3−×1.0 + 3−×1.5 + 103−×2.0 = 98.39
    to get
    = 0.0676 = 6.76%
    53/97

    View full-size slide

  54. Par Yield
    The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value.
    In our example we solve

    2
    −0.05×0.5 +

    2
    −0.058×1.0 +

    2
    −0.064×1.5 + 100 +

    2
    −0.068×2.0 = 100
    to get
    = 6.87
    (with semiannual compounding)
    54/97

    View full-size slide

  55. Duration
    Duration of a bond that provides cash flow at time is
    = σ=1



    where
    ▪ is the bond price
    ▪ is the bond yield (continuously compounded)
    55/97

    View full-size slide

  56. Key Duration Relationship (1/2)
    Duration is important because it leads to the following key relationship between the change in the yield on the bond and the change in
    its price


    = −∆
    56/97

    View full-size slide

  57. Key Duration Relationship (2/2)
    When the yield is expressed with compounding times per year the expression
    ∆ = −

    1 +


    is referred to as the “modified duration”
    M =

    1 +


    57/97

    View full-size slide

  58. Convexity
    The convexity of a bond is defined as
    =
    1

    2
    2
    =
    σ
    =1


    2


    This leads to a more accurate relationship


    = −∆ +
    1
    2
    (∆)2
    When used for bond portfolios it allows larger shifts in the yield curve to be considered, but the shifts still have to be parallel
    58/97

    View full-size slide

  59. Bond Portfolios
    ▪ The duration for a bond portfolio is the weighted average duration of the bonds in the portfolio with weights proportional to prices
    ▪ The key duration relationship for a bond portfolio describes the effect of small parallel shifts in the yield curve
    ▪ What exposures remain if duration of a portfolio of assets equals the duration of a portfolio of liabilities?
    59/97

    View full-size slide

  60. Bond Pricing
    Assuming a given zero rate curve
    To calculate the cash price of a bond we discount each cash flow with the appropriate zero rate
    In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is
    3−0.05×0.5 + 3−0.058×1.0 + 3−0.064×1.5 + 103−0.068×2.0 = 98.93
    Maturity (years) Zero rate (cont. comp.)
    0.5 5.0
    1.0 5.8
    1.5 6.4
    2.0 6.8
    60/97

    View full-size slide

  61. Bond Data Example
    Bond prices are usually fixed by the market (marked-to-market)
    * Half the stated coupon is paid every six months
    61/97
    Bond Principal Time to Maturity
    (yrs)
    Coupon per year
    ($)*
    Bond price
    ($)
    100 0.25 0 97.5
    100 0.50 0 94.9
    100 1.00 0 90.0
    100 1.50 8 96.0
    100 2.00 12 101.6

    View full-size slide

  62. Zero Curve Bootstrap Using Bond Data (1/2)
    An amount 2.5 can be earned on 97.5 during 3 months.
    Because
    100−0.10127×0.25 = 97.5
    the 3-month rate is 10.127% with continuous compounding
    Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding
    62/97

    View full-size slide

  63. Zero Curve Bootstrap Using Bond Data (2/2)
    To calculate the 1.5 year rate we solve
    4−0.10469×0.5 + 4−0.10536×1.0 + 104−×1.5 = 96
    to get
    = 0.10681 = 10.681%
    Similarly the two-year rate is 10.808%
    63/97

    View full-size slide

  64. Zero Curve Calculated Trom The Bond Data
    9
    10
    11
    12
    0 0.5 1 1.5 2 2.5
    Zero Rate (%)
    Maturity (yrs)
    10.127
    10.469 10.536
    10.681 10.808
    64/97

    View full-size slide

  65. Interest Rate Basics
    6. Interest Rate Swaps

    View full-size slide

  66. Interest Rate Swaps
    ▪ A swap is an agreement to exchange cash flows at specified future times according to certain specified rules
    ▪ Vanilla swaps are OTC contracts in which two counterparties agree to exchange fixed against floating rate cash flows
    ▪ The EUR market quotes standard plain vanilla swaps starting at spot date with annual fixed leg versus floating leg indexed to 6M
    (or 3M) Euribor rate
    ▪ Swaps can be regarded as weighted portfolios of 6M (or 3M) FRA
    66/97

    View full-size slide

  67. An Example of a “Plain Vanilla” Interest Rate Swap
    An agreement by Microsoft to
    ▪ receive 6-month LIBOR
    ▪ pay a fixed rate of 5% per annum every 6 months
    for 3 years on a notional principal of $100 million
    Next slide illustrates cash flows that could occur (day count conventions are not considered)
    67/97

    View full-size slide

  68. One Possible Outcome for Cash Flows to Microsoft
    Date LIBOR Floating
    Cash Flow
    Fixed Cash
    Flow
    Net Cash
    Flow
    Mar 5, 2012 4.20%
    Sep 5, 2012 4.80% +2.10 −2.50 −0.40
    Mar 5, 2013 5.30% +2.40 −2.50 −0.10
    Sep 5, 2013 5.50% +2.65 −2.50 + 0.15
    Mar 5, 2014 5.60% +2.75 −2.50 +0.25
    Sep 5, 2014 5.90% +2.80 −2.50 +0.30
    Mar 5, 2015 +2.95 −2.50 +0.45
    68/97

    View full-size slide

  69. Typical Uses of an Interest Rate Swap
    Converting a liability from
    ▪ fixed rate to floating rate
    ▪ floating rate to fixed rate
    Converting an investment from
    ▪ fixed rate to floating rate
    ▪ floating rate to fixed rate
    69/97

    View full-size slide

  70. Intel and Microsoft (MS) Transform a Liability
    Intel MS
    LIBOR
    5%
    LIBOR+0.1%
    5.2%
    70/97

    View full-size slide

  71. Financial Institution is Involved
    ▪ Financial Institution has two offsetting swaps
    F.I.
    LIBOR LIBOR
    LIBOR+0.1%
    4.985% 5.015%
    5.2%
    Intel MS
    71/97

    View full-size slide

  72. Intel and Microsoft (MS) Transform an Asset
    Intel MS
    LIBOR
    5%
    LIBOR-0.2%
    4.7%
    72/97

    View full-size slide

  73. Financial Institution is Involved
    Intel F.I. MS
    LIBOR LIBOR
    4.7%
    5.015%
    4.985%
    LIBOR-0.2%
    73/97

    View full-size slide

  74. Quotes by a Swap Market Maker
    Maturity Bid (%) Offer (%) Swap Rate (%)
    2 years 6.03 6.06 6.045
    3 years 6.21 6.24 6.225
    4 years 6.35 6.39 6.370
    5 years 6.47 6.51 6.490
    7 years 6.65 6.68 6.665
    10 years 6.83 6.87 6.850
    74/97

    View full-size slide

  75. Confirmations
    ▪ Confirmations specify the terms of a transaction
    ▪ The International Swaps and Derivatives Association (ISDA) has developed Master Agreements that can be used to cover all
    agreements between two counterparties
    ▪ Governments now require central clearing to be used for most standardized derivatives
    75/97

    View full-size slide

  76. The Comparative Advantage Argument
    ▪ AAACorp wants to borrow floating
    ▪ BBBCorp wants to borrow fixed
    Fixed Floating
    AAACorp 4.0% 6 month LIBOR − 0.1%
    BBBCorp 5.2% 6 month LIBOR + 0.6%
    AAACorp BBBCorp
    LIBOR
    LIBOR+0.6%
    4.35%
    4%
    76/97

    View full-size slide

  77. The Swap when a Financial Institution is Involved
    AAACorp F.I
    .
    BBBCorp
    4%
    LIBOR LIBOR
    LIBOR+0.6%
    4.33% 4.37%
    77/97

    View full-size slide

  78. Telescopic property of the Libor Leg (1/2)
    Once upon a time…
    The forward rate , between and is given purely by the discount factors at and :
    ()
    1
    1 + , −
    = ()
    , =
    1

    ()

    − 1
    Hence the cash flow at time is set at time and its PV is given by
    , − = − ()
    78/97

    View full-size slide

  79. Telescopic property of the Libor Leg (2/2)
    Once upon a time…
    If you sum up all the cash flows then the intermediate discount factors all drop out and the PV of the floating leg is
    = ෍
    =0



    , +1
    +1

    +1
    =
    = ෍
    =0


    − +1
    = 0

    79/97

    View full-size slide

  80. Using Swap Rates to Bootstrap the LIBOR/Swap Zero Curve
    ▪ Consider a new swap where the fixed rate is the swap rate
    ▪ When principals are added to both sides on the final payment date the swap is the exchange of a fixed rate bond for a floating rate
    bond
    ▪ The floating rate bond is worth par:
    ∗ + ∗
    =
    = ∗ 0

    + ∗
    =
    = ∗ 0
    =
    ▪ The swap is worth zero, so the fixed-rate bond must therefore also be worth par: This shows that swap rates define par yield
    bonds that can be used to bootstrap the LIBOR (or LIBOR/swap) zero curve
    ∗ + ∗
    =
    ֜ =
    80/97

    View full-size slide

  81. Example of Bootstrapping the LIBOR/Swap Curve
    6-month, 12-month and 18-month zero rates are 4%, 4.5% and 4.8% with continuous compounding.
    Two-year swap rate is 5% (2.5% in 6 months)
    2.5−0.04×0.5 + 2.5−0.045×1.0 + 2.5−0.048×1.5 + 102.5−×2.0 = 100
    The 2-year zero rate is
    = 4.953%
    81/97

    View full-size slide

  82. Valuation of an Interest Rate Swap
    ▪ Initially Interest Rate Swaps are worth close to zero
    ▪ At later times they can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond
    ▪ Alternatively, they can be valued as a portfolio of Forward Rate Agreements (FRAs)
    82/97

    View full-size slide

  83. Overnight Indexed Swaps (1/2)
    ▪ Fixed interest rate is exchanged for the overnight rate
    ▪ The overnight rate is compounded and paid at maturity
    ▪ On both legs there is
    o a single payment for maturity up to 1Y
    o yearly payments with short stub for longer maturities
    83/97

    View full-size slide

  84. Overnight Indexed Swaps (2/2)
    ▪ Fixed rate for a period is exchanged for the geometric average of the overnight rates
    ▪ Should OIS rate equal the LIBOR rate? A bank can
    o Borrow $100 million in the overnight market, rolling forward for 3 months
    o Enter into an OIS swap to convert this to the 3-month OIS rate
    o Lend the funds to another bank at LIBOR for 3 months
    ...but it bears the credit risk of another bank in this arrangement
    ▪ The OIS rate is now regarded as a better proxy for the short-term risk-free rate than LIBOR
    ▪ The excess of LIBOR over the OIS rate is the LIBOR-OIS spread
    84/97

    View full-size slide

  85. Swaps & Forwards
    ▪ A swap can be regarded as a convenient way of packaging forward contracts
    ▪ Although the swap contract is usually worth close to zero at the outset, each of the underlying forward contracts are not worth zero
    85/97

    View full-size slide

  86. Credit Risk
    ▪ A swap is worth zero to a company initially. At a future time its value is liable to be either positive or negative
    ▪ The company has credit risk exposure only when its value is positive
    ▪ Some swaps are more likely to lead to credit risk exposure than others
    o What is the situation if early forward rates have a positive value?
    o What is the situation when early forward rates have a negative value?
    86/97

    View full-size slide

  87. Other Types of Swaps
    ▪ Floating-for-Floating Interest Rate Swaps
    ▪ Amortizing Swaps
    ▪ Step Up Swaps
    ▪ Forward Swaps
    ▪ Constant Maturity Swaps
    ▪ Compounding Swaps
    ▪ LIBOR-in-Arrears Swaps
    ▪ Accrual Swaps
    ▪ Diff Swaps
    ▪ Cross Currency Interest Rate Swaps
    87/97

    View full-size slide

  88. Basis Swaps
    ▪ Interest Rate (Single Currency) Basis Swaps are usually floating vs floating swaps with different tenors on the two legs
    ▪ The EUR market quotes standard plain vanilla basis swaps as portfolios of two regular fixed-floating swaps with the floating legs
    paying different Euribor indexes. The quotation convention is to provide the difference (in basis points) between the fixed rates of
    the two regular swaps.
    ▪ Basis is positive and decreasing with maturity, reflecting the preference of market players for receiving payments with higher
    frequency (e.g. 3M instead of 6M, 6M instead of 12M, etc.) and shorter maturities
    ▪ Basis swaps allow to imply levels for non-quoted swaps on Euribor 1M and 12M from the quoted swap rates on Euribor 6M
    88/97

    View full-size slide

  89. Interest Rate Basics
    7. Indexes

    View full-size slide

  90. Interest Rate Indexes
    ▪ An Index summarizes the value of a basket of financial instruments and its variations over time: its value is the weighted average
    (according to a certain method) of the prices of instruments included in the portfolio
    ▪ An Interest Rate Index is based on the interest rate of a financial instrument or basket of financial instruments an it serves as a
    benchmark used to calculate the interest rate charged on financial products
    90/97

    View full-size slide

  91. Ibor Indexes (1/2)
    ▪ Ibor indexes are indexes related to interbank lending between one day and one year
    ▪ They are usually computed as the trimmed average between rates contributed by participating banks
    ▪ The rates are banks' estimates but usually do not refer to actual transactions
    ▪ The most common usage of those indexes in interest rate derivatives is in swaps and caps/floors
    91/97

    View full-size slide

  92. Ibor Indexes (2/2)
    Examples
    ▪ Euribor: rate at which Euro interbank term deposits are offered by one prime bank to another prime bank within the EMU zone.
    ▪ ICE LIBOR (London Inter Bank Offered Rate): benchmark rate produced for five currencies (CHF, EUR, GBP, JPY, USD) with
    seven maturities quoted for each (ranging from overnight to 12 months) producing 35 rates each business day. It provides an
    indication of the average rate at which a LIBOR contributor bank can obtain unsecured funding in the London interbank market for a
    given period, in a given currency.
    92/97

    View full-size slide

  93. Swap Indexes
    ▪ Swap indexes are benchmarks for interest rate swap rates
    ▪ They are usually computed as trimmed average between mid-market rates contributed by participating banks
    ▪ The most common usage of these indexes is in Constant Maturity Swaps (CMS)
    Example
    ▪ ISDAFIX: a benchmark for annual swap rates for interest rate swap transactions; it represents average mid-market rates for vanilla fixed-
    for-floating interest rate swaps in three major currencies (USD, EUR, GBP) at selected maturities on a daily basis.
    93/97

    View full-size slide

  94. Overnight Indexes (1/2)
    ▪ Overnight indexes are indexes related to interbank lending on a one day horizon
    ▪ Most indexes are for overnight loans and some for tomorrow/next loans
    ▪ The rates are computed as a weighted average of actual transactions
    ▪ The most common usage of those indexes in interest rate derivatives is in overnight indexed swaps
    94/97

    View full-size slide

  95. Overnight Indexes (2/2)
    Examples
    ▪ Eonia (Euro Over Night Index Average): the effective overnight reference rate for the Euro, computed as a weighted average of all
    overnight unsecured lending transactions in the interbank market, undertaken in the European Union and European Free Trade
    Association (EFTA) countries.
    ▪ Sonia (Sterling Over Night Index Average): the weighted average rate to four decimal places of all unsecured sterling overnight
    cash transactions brokered in London by contributing WMBA member firms between 00:00 hrs and 16:15 hrs UK time with all
    counterparties in a minimum deal size of £25 million.
    95/97

    View full-size slide

  96. Interest Rate Basics
    8. Bibliography

    View full-size slide

  97. Bibliography
    ▪ J. Hull - Option, Futures, and Other Derivatives (8th edition) - chapters 4, 6, 7
    ▪ M. Marchioro - Pricing Simple Interest Rate Derivatives - www.marchioro.org
    ▪ www.quantlib.org
    97/97

    View full-size slide