Interest Rate

					Futures market

• Forward contract - an agreement
  between two parties involving the future
  delivery of a particular quantity of an
  asset at a price agreed upon today.
• Buyers and sellers are obliged to deliver
  or take delivery.
• No money is exchanged until
• This may introduce default risk for some
  forward contracts.                          2
• Example: You buy a forward contract to
  receive delivery of Euros at an exchange rate
  of 1 $/€ in three months.
• If in three months the spot rate is 1.2 $/€, you
  gain, since you get 1 Euro for each dollar
  through your forward contract, but the Euros
  are currently worth more: 1.2 dollars for each
• To think about it another way, you get one
  Euro for each dollar, whereas if you had
  waited, your dollar would have bought only
  0.83 Euros.
• Futures contract - similar to a forward
  contract but it is entered into on an
  organized exchange and has
  standardized features (contract size,
  delivery date, acceptable grade of the
  commodity, etc.)

• They have very low transactions costs.
  Commissions can run as low as .05% of
  the value of the contract.                4
• Rice contracts - 17th Century Japan
• 1848 CBOT established
• Financial Futures
  – 1972 foreign currencies at CME
  – 1975 interest rate futures at CBOT
  – 1982 stock index futures at KBT, CME, NYFE

   Foreign Exchange Futures
• Futures markets
  – Chicago Mercantile (International Monetary
  – London International Financial Futures
  – MidAmerica Commodity Exchange
• Active forward market

             Types of Contracts
• Agricultural commodities
• Metals and minerals (including energy
• Foreign currencies
• Financial futures
  – Interest rate futures
  – Stock index futures
  futures contract specifications can be found on:

• Frozen Pork Bellies Futures Ticker Symbol: PB
• Trading Unit': 40,000 lbs. USDA-inspected 12-14, 14-16
  pound or 16-18 pound (at a 21/2c discount) Pork Bellies
• Price Quote: $ per hundred pounds (or cents/pound)
• Min Price Fluct: $.025 $10.00/tick
• Daily Price Limit: $2.00 $800.00/contract
• Contract Months: Feb, Mar, May, Jul, Aug
• Trading Hours: 9:10 am-1:00 pm (Chicago Time)
• Last day: 9:10 am-12:00 pm
• Last Day of Trading: The business day immediately
  preceding the last 5 business days of the contract month.
• Delivery Days: Any business day of the contract month.
• Delivery Points: The CME Clearing House or a current list
  of approved warehouses.

       Key Terms for Futures
• Futures price - agreed-upon price at maturity
  – Note that this is different than the price of a security,
    which is the price paid today for the security. A futures
    contract involves no exchange of money at the outset.
• Long position - agree to purchase
• Short position - agree to sell
• Profits on positions at maturity
   Long = future spot price minus futures price
   Short = futures price minus future spot price

       Key Difference in Futures

• Secondary trading - liquidity.
• Standardized contract units.
• Clearinghouse warrants performance.
  – Unlike forwards, there is no default risk with
  – Sellers do not have to be concerned about
    evaluating the credit risk of every different buyer,
    since the clearing house guarantees all
           Trading Mechanics
 • Clearinghouse - acts as the
   counterparty to all buyers and sellers.
     – Obligated to deliver or supply delivery
     – Stands in the middle of the transaction
       between the long and short position
              $                           $
 Long                                               Short
Position                                           Position
           commodity                   commodity
              ?                           ?
           Margin Accounts
• When buying a contract, the buyer must post
  a performance bond; that is, deposit money in
  a margin account, usually 20 percent of the
  value of the contract.
• Note that this margin account is NOT the same
  as a stock margin account in which a buyer is
  making a down payment and borrowing funds
  from the broker to complete the sale.
• Since the account earns interest, there is no
  cost to you, the buyer, of posting the
  performance bond.
   Margin and Trading Arrangements
• Initial Margin - funds deposited to provide
  capital to absorb losses (more than one-day
  price moves)
• Marking to Market - each day the profits or
  losses from the new futures price are reflected in
  the account (“daily settlement.”). This is
  calculated by taking the difference between the
  closing futures price at the end of the day minus
  the previous closing price.
• Maintenance or variation margin - an
  established value below which a trader‟s margin
  may not fall.
• Bears any residual credit risk, that exists
1) futures prices move so dramatically that
  the amount required to mark to market is
  larger than the balance of an individual's
  margin account, and
2) the individual defaults on payment of the
 Daily Settlement - an example
• Suppose that the current futures price of
  gold for delivery 4 days from now is
  $293.50 per ounce.
• Over the next 3 days, the price evolves as
  follows, and the daily settlements are
  calculated accordingly for a long position:

 Day        Futures Price Profit (loss)/ounce
  1               295.20          +1.70
  2               294.60          -0.60
  3               293.00         -1.60
 Net Profit = -0.50
• Suppose an Australian futures speculator buys one
  SPI (share price index) futures contract on the
  Sydney Futures Exchange (SFE) at 11:00am on
  June 6. At that time, the futures price is 2300.
• At the close of trading on June 6, the futures price
  has fallen to 2290 (what causes futures prices to
  move is discussed below).
• Underlying one futures contract is $25 x Index, so
  the buyer's position has changed by $25(2290-
• Since the buyer has bought the futures contract and
  the price has gone down, he has lost money on the
  day and his broker will immediately take $250 out of
  his account. This immediate reflection of the gain or
  loss is known as marking to market.                16
• Where does the $250 go?
• On the opposite side of the buyer's buy order,
  there was a seller, who has made a gain of $250
  (note that futures trading is a zero-sum game -
  whatever one party loses, the counterparty
• The $250 is credited to the seller's account.
• Suppose that at the close of trading the following
  day, the futures price is 2310. Since the buyer
  has bought the futures and the price has gone
  up, he makes money.
• In particular, $25(2310-2290)=+$500 is credited
  to his account. This money, of course, comes
  from the seller's account.
        Forwards and Futures
         Profits and Losses
• A futures contract can be considered a series of
  one-day forward contracts.
• It‟s as if you close out your position each day
  and then open up a new position.
• The price of a futures contract will approach
  the spot price as the futures contract nears
  its maturity date.
• The capital gains and losses on the futures
  contract are realized over the life of the contract
  rather than at the end, which is the case with a
  forward.                                           18
         Closing out Positions
• Option One: You can reverse the trade
  – Go long (enter into a contract to buy) if closing out a
    short position.
  – Go short (enter into a contract to sell) if closing out a
    long position.
  – The difference in the prices in the two contracts will
    be your profit.
  – Example: you have been long corn futures for 5 days.
    You want out. You enter into a short contract for the
    same amount. The long and short positions cancel
    at the clearinghouse.
        Closing out Positions

• Option Two: You can take or make
  – Financial futures though are “cash settled.”

• Most trades are reversed and do not
  involve actual delivery.

• Deliverable quality: Greasy wool futures.
• Delivery must be made at approved warehouses in the
  major wool selling centres throughout Australia.
• For wool to be deliverable, it must possess the
  relevant measurement certificates issued by the
  Australian Wool Testing Authority (AWTA) and
  appraisal certificates issued by the Australian Wool
  Exchange Limited (AWEX).
• In particular, it must be good topmaking merino fleece
  with average fibre diameter of 21.0 microns, with
  measured mean staple strength of 35 n/ktx, mean
  staple length of 90mm, of good colour with less than
  1.0% vegetable matter.

• Because any particular bale of wool is unlikely to exactly
  match these specifications, wool within some
  prespecified tolerance is deliverable.
• In particular, 2,400 to 2,600 clean weight kilograms of
  merino fleece wool, of good topmaking style or better,
  good colour, with average micron between 19.6 and 22.5
  micron, measured staple length between 80mm and
  100mm, measured staple strength greater than 30 n/ktx,
  less than 2.0% vegetable matter is deliverable.
• Premiums and discounts for delivery that does not match
  the exact specifications of the underlying contract are
  fixed on the Friday prior to the last day of trading for all
  deliverable wools above and below the standard, quoted
  in cents per kilogram clean.

          Trading Strategies
• Speculation -
  – You go short if you believe price will fall.
  – You go long if you believe price will rise.
• Hedging -
  – long hedge - protecting against a rise in
     • e.g. for an input to production
  – short hedge - protecting against a fall in
     • e.g. for an output commodity
           Corn Example
• You are a farmer who wants to sell your
  corn harvest in September. You would like
  to lock into a price now.
• The current futures price is $2.26 ¼ per
  bushel, and you think that this is an
  acceptable lock-in price.
• Contract size is 5000 bushels.
• You go short 10 contracts and deposit
  (.2)(10×5000)(2.26 1/4) = $22,625 in a
  margin account.
• This means you agree to sell 50,000
  bushels of corn to the CBOT on              24
  September for 2.26 ¼ per bushel.
               Corn Example
• September comes around and the current corn price is
• You sell your corn to a buyer for $1.
• But you make money on your futures contract.
• Since the futures price equals the spot price when the
  contract comes due, the futures price is also $1.
   – At this point your margin account will have gained
     value by the amount (2.26 1/4 - 1)(50,000).
• So at this point you enter into a long contract. The
  two contracts (one long and one short) cancel out,
  you don’t have to deliver any corn to the CBOT, and
  you pocket the gain to your margin account.
  Forward and Futures Pricing
• There are two ways to acquire an asset for
  some date in the future
  – Purchase it now and store it
  – Take a long position in futures
• These two strategies must have the same
  market determined costs.
• Otherwise we could do arbitrage.

   Determining Forward Prices
• Consider a perfect hedge:
  – Hold an underlying asset and short a forward
    contract on that asset.
• This combined position has no risk!
  – You knew the terms of the contract from the start.
  – At the maturity date, you deliver the underlying asset in
    exchange for the forward price.
• Therefore, a perfect hedge should return the
  riskless rate of return.
• This idea can be used to develop a relationship
  between forward (or futures) prices and spot prices.
   Basis & Convergence Property
• On the delivery date, the futures price =
  spot price or FT = PT or FT - PT = 0
• gain or loss (marking to market) on a
  long position = FT - FT-1
• the sum of all daily settlements = FT -
• given convergence, this means sum of
  all daily settlements = PT - F0
Futures prices versus expected
       future spot price
 Futures prices


      Expectations Hypothesis

               Normal Backwardation

                                Delivery date
     Futures Prices vs. Expected
•   Expectations Hypothesis
                                 Suppliers are natural hedgers
•   Normal Backwardation F < E[P ]
•   Contango Purchasers are the natural hedgers => F > E[PT]
•   Modern Portfolio Theory

         P0 = E[PT]/(1+k)T
         P0 = F/(1+Rf)T                    If beta > 0,
         E[PT]/(1+k)T = F/(1+Rf)T          then k > Rf
         E[PT]* (1+Rf)T /(1+k)T = F        F < E[PT]

 Are forward and future prices the
• Marking to market complicates valuation
  for futures
• However if interest rats do not change
  then forwards and futures should have the
  same price
• We are going to assume that interest rates
  do not change.

           Hedge Example
• An investor owns an S&P 500 fund that
  has a current value equal to the index
  level of 1000.
• Assume dividends of $26 will be paid
  on the index at the end of the year.
• Assume that the futures price for a
  contract that calls for delivery in one
  year is $1050.
• The investor hedges by shorting one
  futures contract.
   Rate of Return for the Hedge
• If we denote the one-year futures price today
  in the market by F0, the dividend by D, and the
  spot price by S0, we can write the return on
  the strategy as:

        ( F0  D)  S 0
• The way to understand this is to think about $ in and $ out.
• He gets F0 when he delivers his S&P fund. He gets D in
  dividends, and he pays S0 for the fund.
• Plugging in the numbers from the example we get.
                    (1050  26)  1000
                                        7.6%
                          1000                                   33
  General Spot-Futures Parity
• Since this strategy is
  risk-free, its return
  must be equal to the       ( F0  D)  S 0
  risk free rate.                             rf

• Rearranging terms,
  we get                   F0  S0 (1  rf )  D  S0 (1  rf  d )
• Continues
  compounding should
  be used…how              dD
  would the formula
  look then?

    Intuition behind the Pricing
• One can interpret the formula
F0=S0(1+rf -d) as follows.
• If the futures contract is priced correctly, you
  should be indifferent between:
   – arranging today to buy the underlying asset at
      a cost of F0 one year from now; and,
   – paying S0 to buy the asset today,
      foregoing interest of rfS0 on your money
      (over the next year), but receiving dS0 (at the
      end of the year) as a benefit to holding on to
      the asset for the year.
       Arbitrage Opportunities

• If spot-futures parity is not observed, then
  arbitrage is possible.
• If the futures price is too high:
  1. Short the futures
  2. Borrow the cost of the stocks at the risk free
  3. Buy the index

  For example, if F0=1055, this strategy would
    yield 1055+26-1000(1.076) = 5.
      Arbitrage Opportunities
• If the futures price is too low, the reverse
  is true:
  1. Go long futures
  2. Short the index
  3. Invest the proceeds at the risk free rate.

  For example, if F0=1045, this strategy would
     yield 1000(1.076)-1045-26 = 5.

       Stock Index Contracts
• Available on both domestic and
  international indexes.
• Advantages over direct stock purchase:
  – lower transaction costs
  – better for timing or allocation strategies
  – takes less time to acquire the portfolio

   Using Stock Index Contracts to
     Create Synthetic Positions
• We have seen how to create a synthetic lending
  strategy by holding the index and taking a short
  position in index futures.
• By playing with this relationship, we can see
  how to create a synthetic index purchase:
   – Go long the index future and lend.
• Also, we can see that being long a forward or
  futures contract is equivalent to buying the
  underlying stocks in the index and borrowing to
  finance this purchase.
  Using Stock Index Futures as a
          Partial Hedge
• We saw how to create a completely riskless
  portfolio by combining a long position in stocks
  with a short position in index futures.
• A portfolio manager may also want to take a
  short position in an index futures contract to
  reduce temporarily the portfolio's exposure to
  the market (but not eliminate all exposure).
• This allows the manager to take advantage of
  her superior stock-picking ability, and avoids
  the triggering of high transaction costs (from
  selling off stock and buying it again later) and of
  capital gains taxes.                              40
 Using Stock Index Futures as a
         Partial Hedge
• The number of futures contracts to go long
  or short is determined by how much you
  want to change the portfolio‟s level of risk.
• You can measure the change in risk by the
  amount you want to change the portfolio‟s
  beta or by the amount you want to change
  the portfolio‟s level of total investment.
• These two approaches are equivalent.

Using Stock Index Futures as a
        Partial Hedge
• Changing beta
                             portfolio value
Number of contracts 
                      index valu e  contract multiple

• Changing value
                                portfolio value
 Number of contracts 
                         index valu e  contract multiple

• A portfolio manager whose $450 million
  portfolio currently has a beta of 1.2
  believes that the market may fall in
  the next couple of months and wants
  to reduce the portfolio beta to 0.8.
• How many contracts should she short?
• Assume that the S&P 500 is now at
• If the market falls by 1%, the S&P index
  will fall by 10 points.                    43
• Since S&P futures contracts come in multiples of
  $500 (per point), the drop translates into a
  change of $5,000 per contract.
• The manager's portfolio would fall by
   – $5.4 million (=1.2*450*1%) if the beta stayed
     at 1.2,
   – $3.6 million (=0.8*450*1%) if beta falls to 0.8.
• Thus, to reduce the loss by $1.8 million (= 5.4-
  3.6), the manager should short 360 contracts
Hedging Foreign Exchange Risk
  A US firm wants to protect against a
  decline in profit that would result from a
  decline in the pound:
• Estimated profit loss of $200,000 if the
  pound declines by $.10.
• Short or sell pounds for future delivery to
  avoid the exposure.

Pricing on Foreign Exchange
Interest rate parity theorem
      Developed using the US Dollar and
      British Pound
              1  rUS   
     F0  E0 
             1 r       
                   UK   
          F0 is the forward price
          E0 is the current exchange rate
       Text Pricing Example

rus = 5%    ruk = 6%     E0 = $1.60 per pound
T = 1 yr
             1.05 
  F0  $1.60        $1.585
             1.06 

If the futures price varies from $1.58 per pound
arbitrage opportunities will be present.

         Hedge Ratio for Foreign
           Exchange Example
Hedge Ratio in pounds
=ch. in value of unprotected position / profit on 1 futures position
$200,000 per $.10 change in the pound/dollar exchange rate
$.10 profit per pound delivered per $.10 in exchange rate
= 2,000,000 pounds to be delivered
 Hedge Ratio in contacts
 Each contract is for 62,500 pounds or $6,250 per a $.10 change
         $200,000 / $6,250 = 32 contracts
         Interest Rate Futures
• Idea: to separate security-specific decisions from
  bets on movements in the entire structure of
  interest rates
• Domestic interest rate contracts
  – T-bills, notes and bonds
  – municipal bonds
• International contracts
  – Eurodollar
• Hedging
  – Underwriters
  – Firms issuing debt                            49
  Uses of Interest Rate Hedges
• Owners of fixed-income portfolios
  protecting against a rise in rates.
• Corporations planning to issue debt
  securities protecting against a rise in rates.
• Investor hedging against a decline in rates
  for a planned future investment.
• Exposure for a fixed-income portfolio is
  proportional to modified duration.

  Hedging Interest Rate Risk: Text

Portfolio value     = $10 million
Modified duration    = 9 years
If rates rise by 10 basis points (.1%)
Change in value = ( 9 ) ( .1%) = .9% or $90,000
Present value of a basis point (PVBP) = $90,000 / 10 = $9,000
per basis point

Hedge Ratio: Text Example

       PVBP for the portfolio
     PVBP for the hedge vehicle (contract
     size is 1000)
 =    $9,000
                  = 100 contracts
      Commodity Futures Pricing

General principles that apply to stock apply to commodities.
       Carrying costs are more for commodities.
       Spoilage is a concern.

   F0  P0 (1  rf )  C
    Where; F0 = futures price P0 = cash price of the asset
           C = Carrying cost    c = C/P0
    F0  P0 (1  rf  c)
• Interest rate swap
  – Variable for fixed
• Foreign exchange swap
  – One currency for another
• Credit risk on swaps
  – Exists but not very problematic
  – Why not?
• Swap Variations
  –   Interest rate cap
  –   Interest rate floor
  –   Collars – combines caps and floors
  –   Swaptions- an option on a swap       54
  Pricing on Swap Contracts
•Swaps are essentially a series of forward contracts.
•One difference is that the swap is usually structured
with the same payment each period while the forward
rate would be different each period.
•Using a foreign exchange swap as an example, the
swap pricing would be described by the following
       F1         F2           F*         F*
                                   
    (1  y1 ) (1  y2 ) 2
                            (1  y1 ) (1  y2 ) 2

Portfolio performance

• Complicated subject
• Theoretically correct measures are difficult
  to construct
• Different statistics or measures are
  appropriate for different types of
  investment decisions or portfolios
• The nature of active management leads
  to measurement problems

  Dollar- and Time-Weighted
Dollar-weighted returns
• Internal rate of return considering the cash
  flow from or to investment
• Returns are weighted by the amount
  invested in each stock
Time-weighted returns
• Not weighted by investment amount
• Equal weighting

     Text Example of Multiperiod
Period          Action
 0         Purchase 1 share at $50
 1         Purchase 1 share at $53
      Stock pays a dividend of $2 per share
 2    Stock pays a dividend of $2 per share
           Stock is sold at $108 per share

         Dollar-Weighted Return
Period          Cash Flow
 0              -50 share purchase
 1              +2 dividend -53 share purchase
 2              +4 dividend + 108 shares sold

     Internal Rate of Return:

                   51      112
          50            
                (1  r ) (1  r ) 2

         r  7.117%
Time-Weighted Return

     53  50  2
r1               10%
     54  53  2
r2               5.66%

Simple Average Return:
      (10% + 5.66%) / 2 = 7.83%

          Averaging Returns

Arithmetic Mean:             Text Example Average:
      r                    (.10 + .0566) / 2 = 7.83%
        t 1 n

 Geometric Mean:             Text Example Average:
                    1/ n
     r   (1  rt )  1    [ (1.1) (1.0566) ]1/2 - 1
          t 1                     = 7.81%
 Comparison of Geometric and
     Arithmetic Means
• Past Performance - generally the
  geometric mean is preferable to
• Predicting Future Returns- generally
  the arithmetic average is preferable to
  – Geometric has downward bias

                  1 2
        r g  ra  
                  2                         63
        Abnormal Performance
What is abnormal?
Abnormal performance is measured:
• Benchmark portfolio
• Market adjusted
• Market model / index model adjusted
• Reward to risk measures such as the Sharpe
  E (rp-rf) / p

Factors That Lead to Abnormal

 • Market timing
 • Superior selection
   – Sectors or industries
   – Individual companies

  Risk Adjusted Performance:
1) Sharpe Index
         rp - rf

rp = Average return on the portfolio
rf = Average risk free rate
p = Standard deviation of portfolio
              M2 Measure
• Developed by Modigliani and Modigliani
• Equates the volatility of the managed
  portfolio with the market by creating a
  hypothetical portfolio made up of T-bills
  and the managed portfolio

           M2 Measure: Example
Managed Portfolio: return = 35%       standard deviation = 42%
Market Portfolio: return = 28%        standard deviation = 30%
      T-bill return = 6%
Hypothetical Portfolio:
30/42 = .714 in P (1-.714) or .286 in T-bills
(.714) (.35) + (.286) (.06) = 26.7%
How do we get the weights?
Since this return is less than the market, the managed
portfolio underperformed

   Risk Adjusted Performance:
2) Treynor Measure      rp - rf
rp = Average return on the portfolio
rf = Average risk free rate
ßp = Weighted average for portfolio
 Risk Adjusted Performance:
3) Jensen’s Measure
        p= rp - [ rf + ßp ( rm - rf) ]
 p = Alpha for the portfolio
rp = Average return on the portfolio
ßp = Weighted average Beta
rf = Average risk free rate
rm = Avg. return on market index port.
            Appraisal Ratio

Appraisal Ratio = p / (ep)

 •Appraisal Ratio divides the alpha of the
 portfolio by the nonsystematic risk
 •Nonsystematic risk could, in theory, be
 eliminated by diversification
Which Measure is Appropriate?
It depends on investment assumptions
1) If the portfolio represents the entire investment
   for an individual, Sharpe Index compared to the
   Sharpe Index for the market.
2) If many alternatives are possible, use the
   Jensen or the Treynor measure

The Treynor measure is more complete because it
  adjusts for risk


• Assumptions underlying measures limit
  their usefulness
• When the portfolio is being actively
  managed, basic stability requirements are
  not met
• Practitioners often use benchmark
  portfolio comparisons to measure
        Market Timing

Adjusting portfolio for up and down
  movements in the market
• Low Market Return - low ßeta
• High Market Return - high ßeta

 Example of Market Timing
     rp - rf
               * *
              * *
             ** *
          * **
         * *
     * * *
** *
 * * * *                       rm - rf
Steadily Increasing the Beta
      Performance Attribution
• Decomposing overall performance into
• Components are related to specific
  elements of performance
• Example components
  – Broad Allocation
  – Industry
  – Security Choice
  – Up and Down Markets

       Process of Attributing
          to Components
Set up a „Benchmark‟ or „Bogey‟ portfolio
• Use indexes for each component (equity,
  fixed income, etc…)
• Use target weight structure

        Process of Attributing
           to Components
• Calculate the return on the „Bogey‟ and on
  the managed portfolio
• Explain the difference in return based on
  component weights or selection
• Summarize the performance differences
  into appropriate categories

             Formula for Attribution
     rB   wBi rBi
             i 1
     rp   w pi rpi
             i 1
                       n             n
     rp  rB   w pi rpi   wBi rBi 
                      i 1          i 1

      (w
      i 1
              pi pi r  wBi rBi )

Where B is the bogey portfolio and p is the managed portfolio
    Contributions for Performance

    Contribution for asset allocation     (wpi - wBi) rBi
+   Contribution for security selection   wpi (rpi - rBi)
=   Total Contribution from asset class wpirpi -wBirBi

 Complications to Measuring
• Two major problems
  – Need many observations even when portfolio
    mean and variance are constant
  – Active management leads to shifts in parameters
    making measurement more difficult
• To measure well
  – You need a lot of short intervals
  – For each period you need to specify the
    makeup of the portfolio


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