futures market l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Futures market PowerPoint Presentation
Download Presentation
Futures market

Loading in 2 Seconds...

play fullscreen
1 / 81

Futures market - PowerPoint PPT Presentation


  • 442 Views
  • Uploaded on

Futures market Forwards 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 settlement.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Futures market' - Rita


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
forwards
Forwards
  • 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 settlement.
  • This may introduce default risk for some forward contracts.
example
Example
  • 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 Euro.
  • 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
Futures
  • 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.
history
History
  • 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
Foreign Exchange Futures
  • Futures markets
    • Chicago Mercantile (International Monetary Market)
    • London International Financial Futures Exchange
    • MidAmerica Commodity Exchange
  • Active forward market
types of contracts
Types of Contracts
  • Agricultural commodities
  • Metals and minerals (including energy contracts)
  • Foreign currencies
  • Financial futures
    • Interest rate futures
    • Stock index futures

futures contract specifications can be found on:

http://www.duke.edu/~charvey/options/futures/f_idx.htm

slide8
Frozen Pork Bellies FuturesTicker 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 contracts
Key Terms for Futures Contracts
  • 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
Key Difference in Futures
  • Secondary trading - liquidity.
  • Standardized contract units.
  • Clearinghouse warrants performance.
    • Unlike forwards, there is no default risk with futures.
    • Sellers do not have to be concerned about evaluating the credit risk of every different buyer, since the clearing house guarantees all transactions.
trading mechanics
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

Position

Short

Position

Clearinghouse

commodity

?

commodity

?

margin accounts
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
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.
clearinghouse
Clearinghouse
  • Bears any residual credit risk, that exists because:

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 balance.

daily settlement an example
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 -1.60Net Profit = -0.50
slide16
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(2290-2300)=-$250.

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.

slide17
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 gains).
  • 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
Forwards and FuturesProfits 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.
closing out positions
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 positions20
Closing out Positions
  • Option Two: You can take or make delivery.
    • Financial futures though are “cash settled.”
  • Most trades are reversed and do not involve actual delivery.
slide21
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.

slide22
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
Trading Strategies
  • Speculation -
    • You go short if you believe price will fall.
    • You golong if you believe price will rise.
  • Hedging -
    • long hedge - protecting against a rise in price
      • e.g. for an input to production
    • short hedge - protecting against a fall in price
      • e.g. for an output commodity
corn example
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 September for 2.26 ¼ per bushel.
corn example25
Corn Example
  • September comes around and the current corn price is $1.
  • 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
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
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
Basis & Convergence Property
  • On the delivery date, the futures price = spot price or FT = PTor FT - PT = 0
  • gain or loss (marking to market) on a long position = FT - FT-1
  • the sum of all daily settlements = FT - F0
  • given convergence, this means sum of all daily settlements = PT - F0
futures prices versus expected future spot price

Futures prices

Contango

Expectations Hypothesis

Normal Backwardation

Time

Delivery date

Futures prices versus expected future spot price
futures prices vs expected spot
Futures Prices vs. Expected Spot
  • Expectations Hypothesis
  • Normal Backwardation
  • Contango
  • Modern Portfolio Theory

Suppliers are natural hedgers

F < E[PT]

Purchasers are the natural hedgers => F > E[PT]

P0 = E[PT]/(1+k)T

P0 = F/(1+Rf)T

E[PT]/(1+k)T = F/(1+Rf)T

E[PT]* (1+Rf)T /(1+k)T = F

If beta > 0,

then k > Rf

F < E[PT]

are forward and future prices the same
Are forward and future prices the same?
  • 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
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
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:
  • 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.
general spot futures parity
General Spot-Futures Parity
  • Since this strategy is risk-free, its return must be equal to the risk free rate.
  • Rearranging terms, we get
  • Continues compounding should be used…how would the formula look then?
intuition behind the pricing formula
Intuition behind the Pricing Formula
  • 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
Arbitrage Opportunities
  • If spot-futures parity is not observed, then arbitrage is possible.
  • If the futures price is too high:
    • Short the futures
    • Borrow the cost of the stocks at the risk free rate
    • Buy the index

For example, if F0=1055, this strategy would yield 1055+26-1000(1.076) = 5.

arbitrage opportunities37
Arbitrage Opportunities
  • If the futures price is too low, the reverse is true:
    • Go long futures
    • Short the index
    • 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
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
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
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.
using stock index futures as a partial hedge41
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.
example43
Example
  • A portfolio manager whose $450 million portfolio currently has a beta of 1.2believes that the market may fall in the next couple of months andwants to reduce the portfolio beta to 0.8.
  • How many contracts should she short?
  • Assume that the S&P 500 is now at 1000.
  • If the market falls by 1%, the S&P index will fall by 10 points.
example44
Example
  • 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 (1,800,000/5,000).
hedging foreign exchange risk
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 futures
Pricing on Foreign Exchange Futures

Interest rate parity theorem

Developed using the US Dollar and British Pound

where

F0 is the forward price

E0 is the current exchange rate

text pricing example
Text Pricing Example

rus = 5% ruk = 6% E0 = $1.60 per pound T = 1 yr

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

hedge ratio for foreign exchange example
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
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
uses of interest rate hedges
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 example
Hedging Interest Rate Risk: Text Example

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
Hedge Ratio: Text Example

PVBP for the portfolio

PVBP for the hedge vehicle (contract size is 1000)

$9,000

$90

H =

=

= 100 contracts

commodity futures pricing
Commodity Futures Pricing

General principles that apply to stock apply to commodities.

Carrying costs are more for commodities.

Spoilage is a concern.

Where; F0 = futures price P0 = cash price of the asset

C = Carrying cost c = C/P0

swaps
Swaps
  • 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
pricing on swap contracts
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 formula.
introduction
Introduction
  • 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 returns
Dollar- and Time-Weighted Returns

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 returns
Text Example of Multiperiod Returns

PeriodAction

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
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:

time weighted return
Time-Weighted Return

Simple Average Return:

(10% + 5.66%) / 2 = 7.83%

averaging returns
Averaging Returns

Arithmetic Mean:

Text Example Average:

(.10 + .0566) / 2 = 7.83%

Geometric Mean:

Text Example Average:

[ (1.1) (1.0566) ]1/2 - 1

= 7.81%

comparison of geometric and arithmetic means
Comparison of Geometric and Arithmetic Means
  • Past Performance - generally the geometric mean is preferable to arithmetic
  • Predicting Future Returns- generally the arithmetic average is preferable to geometric
    • Geometric has downward bias
abnormal performance
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 Measure:

E (rp-rf) / p

factors that lead to abnormal performance
Factors That Lead to Abnormal Performance
  • Market timing
  • Superior selection
    • Sectors or industries
    • Individual companies
risk adjusted performance sharpe

rp = Average return on the portfolio

  • rf = Average risk free rate

= Standard deviation of portfolio

return

p

Risk Adjusted Performance: Sharpe

1) Sharpe Index

rp - rf

p

m 2 measure
M2 Measure
  • Developed by Modigliani and Modigliani
  • Equates the volatility of the managed portfolio with the marketby creating a hypothetical portfolio made up of T-bills and the managed portfolio
m 2 measure example
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 treynor

rp = Average return on the portfolio

  • rf = Average risk free rate
  • ßp = Weighted average for portfolio
Risk Adjusted Performance: Treynor

2) Treynor Measure

rp - rf

ßp

risk adjusted performance jensen
Risk Adjusted Performance: Jensen

3) Jensen’s Measure

= rp - [ rf + ßp ( rm - rf) ]

p

= Alpha for the portfolio

p

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

Appraisal Ratio = ap / s(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
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

limitations
Limitations
  • 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 performance
market timing
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
Example of Market Timing

rp - rf

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

rm - rf

*

*

*

Steadily Increasing the Beta

performance attribution
Performance Attribution
  • Decomposing overall performance into components
  • Components are related to specific elements of performance
  • Example components
    • Broad Allocation
    • Industry
    • Security Choice
    • Up and Down Markets
process of attributing performance to components
Process of Attributing Performance 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 performance to components78
Process of Attributing Performance 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
Formula for Attribution

Where B is the bogey portfolio and p is the managed portfolio

contributions for performance
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 performance
Complications to Measuring Performance
  • 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