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## FIN 413 – RISK MANAGEMENT

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**FIN 413 – RISK MANAGEMENT**Forward and Futures Prices**Topics to be covered**• Compounding frequency • Assumptions and notation • Forward prices • Futures prices • Cost of carry • Delivery options**Suggested questions from Hull**6th edition: #4.4, 4.10, 5.2, 5.5, 5.6, 5.14 5th edition: #4.4, 4.9, 5.2, 5.5, 5.6, 5.14**Compounding frequency**• Interest can be compounded with varying frequencies. • We will often assume that interest is compounded continuously. • Two rates of interest are said to be equivalent if for any amount of money invested for any length of time, the two rates lead to identical future values.**A(1+R)n**A 0 n Annual compounding • The interest earned on an investment in any one year is reinvested to earn additional interest in succeeding years. • R ≡ EAR, effective annual rate FV = A(1+R)n PV = A(1+R)-n A A(1+R)-n 0 n**Compounding m times per year**• The year is divided into m compounding periods. Interest earned in any compounding period is reinvested to earn additional interest in succeeding periods. • Rm ≡ the annual (or nominal) rate of interest compounded m times per year • Rm/m ≡ the effective rate of interest for each mth of a year**Compounding m times per year**FV = A(1+Rm/m)mn PV = A(1+Rm/m)-mn A(1+Rm /m)mn A 0 n A A(1+Rm /m)-mn 0 n**AeR∞n**A 0 n A Ae-R∞ n 0 n Continuous compounding R∞ ≡the annual rate of interest compounded continuously FV = lim A(1+Rm/m)mn m→∞ = AeR∞n PV = lim A(1+Rm/m)-mn m→∞ = Ae-R∞n**infinite decimal expansion**Euler’s number 2 < e < 3 e = 2.71828183…**Natural log function**Properties: -∞<ln(x)<∞, for 0<x<∞ ln(x)<0, for 0<x<1 ln(1) = 0 ln(x)>0, for x>1 ln(ax) = ln(a) + ln(x) ln(a/x) = ln(a) - ln(x) ln(ax) = xln(a) ln(ex) = xln(e) = x**Exponential function**Properties: ex>0, for -∞<x<∞ 0<ex<1, for x<0 e0 = 1 ex>1, for x>0 e-x = 1/ex exey= ex+y (ex)y = exy eln(x) = x**Short selling in the spot market**Involves selling securities that you do not own and buying them back later. When you initiate a short sale, your broker borrows the securities from another client and sells them on your behalf in the spot market. You receive the proceeds of the sale. Through your broker, you must pay the client any income received on the securities. At some later stage, you must buy the securities, close your short position, and return the securities to the client from whom you borrowed. Ignoring the income foregone, short selling yields a profit if the price of the security falls. Buy Sell**Example**Suppose you short sell IBM stock for 90 days. The cash flow are: Note: Short selling is the opposite of buying.**Analysis: forward prices**• Forward contracts are easier to analyze than futures contracts. • We begin our analysis with them. • We will consider forward contracts on the following underlying assets: • Assets that provide no income. • Assets that provide a known cash income. • Assets that provide a known yield. • Commodities • Later we will consider futures contracts.**Assumptions**There are some market participants (such as large financial institutions) that: - pay no transactions costs (brokerage fees, bid-ask spreads) when they trade. - are subject to the same tax rate on all profits. - can borrow or lend at the risk-free rate of interest. - exploit arbitrage opportunities as they arise. Note: The quality of any theory is a direct result of the quality of the underlying assumptions. The assumptions determine the degree to which the theory matches reality.**Notation**T : the time (in years) until the delivery date of a forward contract S (or S0): the current spot price of the asset underlying a forward contract K : the delivery price specified in a forward contract F (or F0): the current forward price f : the current value of a forward contract to the long -f : the current value of a forward contract to the short r : the risk-free interest rate (expressed as an annual, continuously compounded rate) for an investment maturing in T years Note: In practice, r is set equal to the LIBOR with a maturity of T years.**LIBOR**• LIBOR: London Interbank Offer rate • The rate at which large international banks are willing to lend to other large international banks for a specified period. • The rate at which large international banks fund most of their activities. • A variable interest rate. • A commercial lending rate, higher than corresponding Treasury rates.**Analysis**• Objective: to derive formulas for F and f. • We will use arbitrage pricing methods. • Note: The basis of any arbitrage is to sell what is relatively overvalued and to buy what is relatively undervalued.**Forward contract: UA provides no income**Examples: forward contracts on non-dividend-paying stocks and zero-coupon bonds. Proposition: F = SerT, in the absence of arbitrage opportunities Note: F = SerT > S**Forward contract: UA provides no income**Proposition: F = SerT, in the absence of arbitrage opportunities Proof: Suppose F > SerT. Arbitrage strategy (to be implemented today): • Buy one unit of the UA in the spot market by borrowing S dollars for T years at rate r. • Short a forward contract on one unit if the UA. At time T: • Sell the UA for F dollars under the terms of the forward contract. • Repay the bank SerT dollars. Arbitrage profit per unit of UA = [F – SerT ] > 0. S is bid up and F is bid down.**Forward contract: UA provides no income**Suppose F <SerT. Arbitrage strategy (to be implemented today): • Go long a forward contract on one unit if the UA. • Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars. Invest this for T years at rate r. At time T: • The proceeds from the sale/short sale have grown to SerT dollars. • Buy the UA for F dollars under the terms of the forward contract. • Return the UA to your portfolio or to the client from whom it was borrowed. Arbitrage profit per unit of UA = [SerT – F ] > 0. F is bid up and S is bid down. Thus: F =SerT**Alternative derivation of formula**• Spot transaction • Price agreed to. • Price paid/received. • Item exchanged. • Prepaid forward contract • Price agreed to. • Price paid/received. • Item exchanged in T years. • Forward contract • Price agreed to • Price paid/received in T years. • Item exchanged in T years.**Alternative derivation of formula**Underlying asset provides no income: FP = S Explanation: With a prepaid forward contract, as compared to a spot transaction, physical exchange of the asset is delayed T years. But since the asset, by assumption, pays no income to the holder, the holder neither receives nor foregoes income due to the delay. F = FP erT = SerT Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to S, by erT) compensates the short for the delay.**Forward contract: UA provides no income**Proposition: f = S – Ke-rT Proof: In general: f = (F – K )e-rT We derived: F = SerT Thus: f = (SerT – K )e-rT = S – Ke-rT Also: -f = -(F – K )e-rT= (K – F )e-rT = Ke-rT - S**The value today of the UA in the spot market.**The value today of the price that the long has agreed to pay for the asset in T years. Forward contract: UA provides no income We derived: f = S – Ke-rT Thus: f > 0 iff S > Ke-rT K 0 T**The value today of the price that the short has agreed to**receive in T years for the UA. The value today of the UA in the spot market. Forward contract: UA provides no income We derived: -f = Ke-rT – S Thus: -f > 0 iff Ke-rT > S K 0 T**Example: #5.9, page 121**T = 1 year S = $40 r = 10% (a) F = SerT = $40e(0.10×1) = $44.21 f = S – Ke-rT = $40 – $44.21e-(0.10×1) = 0 0 1**Example (continued)**(b) T = ½ year S = $45 r = 10% F = S erT = $45e(0.10×0.5) = $47.31 f = S – Ke-rT = $45 – $44.21e-(0.10×0.5) = $2.95 0 0.5 1**Creating a forward contract synthetically**A security is “created synthetically” by assembling a portfolio of traded assets that replicates the payoff to the security. A long position in aforward contract can be created synthetically by: • Buying the UA with borrowed funds. • Buying a call option and writing a put option.**Creating a forward contract synthetically**Method 1: Consider a forward contract on a stock with a delivery date in T years. The stock will pay no dividends during the next T years. The forward contract can be created synthetically by buying the stock with borrowed funds. r≡ the annual, continuously compounded rate at which funds can be borrowed. S0 ≡ the current price of the stock.**Value of stock, ST**ST -1 × what is owing to the bank = -1 × S0 erT Creating a forward contract synthetically Value at time T of a long position in a forward contract = fT= FT - K = ST – K= ST – S0erT Value at time T of replicating portfolio: fT ST**Forward contract: UA provides a known cash income**Examples: forward contracts on dividend-paying stocks and coupon bonds. I≡ the present value of the income to be received over the remaining life of the forward contract Proposition: F = (S – I )erT, in the absence of arbitrage opportunities**Forward contract: UA provides a known cash income**Note: F = (S – I )erT < SerT This price is lower than if the asset didn’t pay income.**Forward contract: UA provides a known cash income**Proposition: F = (S – I )erT, in the absence of arbitrage opportunities Proof: Suppose F > (S – I )erT. Arbitrage strategy (to be implemented today): • Buy one unit of the UA in the spot market by borrowing S dollars for T years at rate r. • Short a forward contract on one unit if the UA. Use the income from the asset to repay the loan. At time T: • Sell the UA for F dollars under the terms of the forward contract. • Repay the bank (S – I )erT dollars. Arbitrage profit per unit of UA = [F – (S – I )erT] > 0. S is bid up and F is bid down.**Forward contract: UA provides a known cash income**Suppose F < (S – I )erT. Arbitrage strategy (to be implemented today): • Go long a forward contract on one unit if the UA. • Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars. Invest this for T years at rate r. At time T: • The proceeds from the sale/short sale have grown to (S – I )erT dollars. • Buy the UA for F dollars under the terms of the forward contract. • Return the UA to your portfolio or to the client from whom it was borrowed. Arbitrage profit per unit of UA = [(S – I )erT – F] > 0. F is bid up and S is bid down. Thus: F = (S – I )erT**Alternative derivation of formula**• Spot transaction • Price agreed to. • Price paid/received. • Item exchanged. • Prepaid forward contract • Price agreed to. • Price paid/received. • Item exchanged in T years. • Forward contract • Price agreed to • Price paid/received in T years. • Item exchanged in T years.**Alternative derivation of formula**Underlying asset provides a known cash income: FP = S - I Explanation: With a prepaid forward contract, as compared to a spot transaction, physical exchange of the asset is delayed T years. As a result of the delay, the long foregoes income with present value I and the short receives this income. Thus, the price paid by the long and received by the short is reduced by amount I. F = FP erT = (S – I )erT Explanation: The forward contract allows the long to delay payment for T years and requires the short to delay receipt. The long can earn interest on the cash that would otherwise have been paid. The short foregoes this interest. The forward price (which is arrived at by multiplying the prepaid forward price, equal to S - I, by erT) compensates the short for the delay.**Forward contract: UA provides a known cash income**Proposition: f = S – I – Ke-rT Proof: In general: f = (F – K )e-rT We derived: F = (S – I )erT Thus: f = [(S – I )erT – K]e-rT = (S – I )– Ke-rT Also: -f = Ke-rT – (S – I )**The value today of the UA in the spot market.**The value today of the price that the long has agreed to pay for the asset in T years. The value today of the income the long foregoes as a result of delaying purchase of the asset for T years. Forward contract: UA provides a known cash income We derived: f = S – I – Ke-rT Thus: f > 0 iff S > Ke-rT + I K 0 T**The value today of the price at which the short has agreed**to sell the asset in T years. The value today of the income the short receives as a result of delaying sale of the asset for T years. The value today of the UA in the spot market. Forward contract: UA provides a known cash income We derived: -f = Ke-rT – (S – I) Thus: -f > 0 iff Ke-rT + I > S K 0 T**Example: #5.23, page 123**S = $50 r = 8% T = 6/12 (a) I = $1e-(0.08×2/12) + $1e-(0.08×5/12) = $1.9540 F = (S – I )erT = (50 – 1.9540)e(0.08×6/12) = $50.0068 -f = -(S – I – Ke-rT) = -(50 – 1.9540 – 50.0068e-(0.08×6/12)) = 0 $1 $1 6/12 0 2/12 5/12**Example (continued)**(b) S = $48 r = 8% T = 3/12 I = $1e-(0.08×2/12) = $0.9868 F = (S – I)erT = (48 – 0.9868)e(0.08×3/12) = $47.9629 -f = -(S – I – Ke-rT) = -(48 – 0.9868 – 50.0068e-(0.08×3/12)) = $2.00 $1 $1 6/12 0 2/12 3/12 5/12**Example (continued)**S = $50 T = 6/12 (a) I = $1e-(0.078×2/12) + $1e-(0.082×5/12) = $1.9535 $1 $1 6/12 0 2/12 5/12 Term structure of interest rates:**Forward contract: UA provides a known yield**Examples: forward contracts on stock portfolios and currencies. q≡ the average yield per annum expressed as a continuously compounded rate Proposition: F = Se(r-q)T, in the absence of arbitrage opportunities**Forward contract: UA provides a known yield**Note: F = Se(r-q)T < SerT This price is lower than if the asset didn’t pay income.**Forward contract: UA provides a known yield**Proposition: F = Se(r-q)T, in the absence of arbitrage opportunities Proof: Suppose F > Se(r-q)T. Arbitrage strategy (to be implemented today): • Buy one unit of the UA in the spot market by borrowing S dollars for T years at rate r. • Short a forward contract on one unit if the UA. Use the income from the asset to repay the loan. At time T: • Sell the UA for F dollars under the terms of the forward contract. • Repay the bank Se(r-q)T dollars. Arbitrage profit per unit of UA = [F – Se(r-q)T ] > 0. S is bid up and F is bid down.**Forward contract: UA provides a known yield**Suppose F <Se(r-q)T. Arbitrage strategy (to be implemented today): • Go long a forward contract on one unit if the UA. • Sell or short sell one unit of the UA. This leads to a cash inflow of S dollars. Invest this for T years at rate r. At time T: • The proceeds from the sale/short sale have grown to Se(r-q)T dollars. • Buy the UA for F dollars under the terms of the forward contract. • Return the UA to your portfolio or to the client from whom it was borrowed. Arbitrage profit per unit of UA = [Se(r-q)T – F ] > 0. F is bid up and S is bid down. Thus: F =Se(r-q)T