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§ Finance: The Economics of allocating resources across time or reallocation of recourses in § Finance: The Economics of allocating resources across time or reallocation of recourses in time , Such as borrowing and Lending money raising capital. § Financial Investment: Borrowing / Lending / Buying shares ……etc ( Needs provision of funds) *Real Assets Investment → Building factories, buying equipment to be used in production. (needs information) that financial markets provide to interested investors. *Financial Markets allow participants to reallocate resources across time

The Present Value (Discount value) : amount of money you must invest or lend The Present Value (Discount value) : amount of money you must invest or lend at the present time to end up with a particular amount of money in the future. M. B. finding a present value of future cash flow is called discounting cash flow. -Present value is an accurate representation of what the financial market does when it sets a price on a financial asset. *Opportunity costs : alternative to buying certain security or doing certain investment.

(The Cost OF Doing This Instead) Of Something Else (Opportunity Foregone) *Present Wealth: - (The Cost OF Doing This Instead) Of Something Else (Opportunity Foregone) *Present Wealth: - Present value of all present and future resources (cash Flow).

Investing in real assets: productive machinery, product line. . . Etc to create new Investing in real assets: productive machinery, product line. . . Etc to create new future cash flows that did not previously exist. -We must give up some resources in order to undertake investment. -If the present value of the amounts we give up is greater than the present value of what we gain from investment (i. e Bad investment ) the investment will decrease our present wealth. Investment decision depends on the effect on present value of wealth

Net Present Value. Present value of difference between an investments cash inflows and outflows. Net Present Value. Present value of difference between an investments cash inflows and outflows. a) Net present value £ 770/1. 10 -£ 550 = 700 – 550 = £ 150 which is = change in present wealth b) NPV is a reflection of how much investment differs from its opportunity cost or NPV is the present value of the future amount by which the returns from the investment exceed opportunity cost of the investor.

In Our Example: The investment cost ( @10%) = £ 550 X 1. 1 In Our Example: The investment cost ( @10%) = £ 550 X 1. 1 = £ 605 While investment returned £ 770 i. e difference is £ 770 - £ 605 = £ 165. 00 PV of difference = £ 165/1. 1 = £ 150. 00 Which is once again the NPV

∴ NPV of Investment 1 - Present value of all investment present and future ∴ NPV of Investment 1 - Present value of all investment present and future cash flow discounted at the opportunity cost of cash flow 2 - Is the change in present wealth of investor who choose a positive NPV investment. 3 - Is the discounted value of the amount by which the investment cash flow differs from those of its opportunity cost.

Internal Rate of Return (IRR) 1 - IRR is the average per-period rate of Internal Rate of Return (IRR) 1 - IRR is the average per-period rate of return on money invested. 2 -IRR is calculated by finding the discount rate that would cause NPV of investment to be Zero. 3 -To use IRR, we compare it with the return available on an equal Risk investment of comparable cash flow timing.

In Our Example: NPV = O = -£ 550 + £ 770 / (1 In Our Example: NPV = O = -£ 550 + £ 770 / (1 + IRR) ∴ ( 1 + IRR) = £ 770 / £ 550 ( 1 + IRR) = 1. 4 IRR = 0. 4 i. e. 40% ∴ IRR is larger than opportunity cost which is 10% from comparable Risk and timing investment.

1 + IRR = £ 594/ £ 550 = 1. 08 IRR= 0. 08 1 + IRR = £ 594/ £ 550 = 1. 08 IRR= 0. 08 or 8% (i. e. Less the opportunity cost of 10%) ∴ Investment should be rejected

Multiple – Period Finance t 0 Period 1 t 1 period 2 t 2 Multiple – Period Finance t 0 Period 1 t 1 period 2 t 2 Ii Cf 2 = CF 0(1+i)2 CF 2= 100 (1. 21) = £ 121. 00 =10% m =2 CF 0 = £ 100 PV = CF 2 / (1+i)2 = £ 121/(1. 10)2 = £ 100 ∴ FV = PV (1 + i)n PV = FV/ (1 +i)n

- Compound Interest Exchange rate between two time points where interest is not only - Compound Interest Exchange rate between two time points where interest is not only earned on original investment (principle) but also on interest earned. Example : CF 2= CF 0 +CF 0 (i 1) + CF 0 (i 2) + CF 0(i 1)(I 2) i. e. £ 121 = £ 100 + £ 100 (10%)(10%) Or CF 2 =CF 0(1+i)2 £ 121 = £ 100 (1 +10%)2 -The general arithmetic of interest compounding (for number of times period) =CF 0 (1+ ⅰ/m)mt → Where m number of times period And – t- is–number of periods

Multiple Period Cash Flow The Present Value of Stream of future cash flow is Multiple Period Cash Flow The Present Value of Stream of future cash flow is the sum of present values of each of the future cash flow. PV = CF 1 + CF 2 + CF 3 (1+i 1)1 (1+i 2)2 (1+i 3)3 Multiple period Investment Decision NPV = (CF 0) + CF 1 + CF 2 + CF 3 (1 + i 1)1 (1+i 2)2 (1+i 3)2 NPV must include all present and future cash flow associated with investment

As IRR is the discount rate that causes NPV to equal Zero. Solving for As IRR is the discount rate that causes NPV to equal Zero. Solving for IRR is the technique called Trial & Error NPV = 0 = (CF 0) + CF 1 + CF 2 + CF 3 (1+IRR)1 (1+IRR)2 (1+IRR)3

Using Financial Table PV PVA FV PVA Due = = FV (factor) PMT (factor) Using Financial Table PV PVA FV PVA Due = = FV (factor) PMT (factor) PV (factor) PMT (factor ) X (1+i) Present value of perpetuity *Perpetuity: PV = FV → Constant period cash flow i → Constant interest (discount) rate. *Limitation: Value Error. if cash flow is to cease by 40 years £ 100/0. 10/(1. 10)40=£ 22. 10 A value of error of t 22. 10 out of £ 1000 = or 2. 21 percent

∴Perpetuity valuation can be convenient in longlived assets Growth Perpetuity present-value calculation. PV = ∴Perpetuity valuation can be convenient in longlived assets Growth Perpetuity present-value calculation. PV = CF / (i- G) Example : £ 100/0. 10 -0. 05 = £ 2000. 00 Limitation: (Equation does not work if growth rate is equal or larger than the discount rate)

*The yield to Maturity It is the IRR of the Bonds promised cash flow *The yield to Maturity It is the IRR of the Bonds promised cash flow used to discount promised cash flow to equal market price of the Bond. -The coupon effect on the YTM Government Bonds Coupon rate 8% 8% 8% 4% 12% Maturity Price t 1 t 2 t 3 t 3 Yield £ 1029 £ 1039 £ 1029 £ 983 £ 1136 The yield curve → yield at various maturities 5% 5. 96% 6. 90% 6. 9% 6. 85% A B C D E

Interest Rate Risk & Duration § Effect of Interest Rate Risk; variability of values Interest Rate Risk & Duration § Effect of Interest Rate Risk; variability of values and therefore in wealth due to changes in interest rates. Reasons of interest rate changes *Change in inflation *changes in creditworthiness of bond issuer *changes in return available on real asset investment. § Duration: extent bond is subject to interest rate risk. i. e. how much bond value will go up and down as interest change. Or “exposure” of the value of a bond to changes in interest rates.

Duration number of periods into future where bonds value , on average is generated. Duration number of periods into future where bonds value , on average is generated. The greater duration of bond the more it react to changes in interest rates. Calculation by weighting the time points from which cash flow is generated by proportion of total value generated at each time.

Example: Cash Flow Bond c Bond D Interest price t 0 £ 1029 £ Example: Cash Flow Bond c Bond D Interest price t 0 £ 1029 £ 923 t 1 £ 80 £ 40 5% Duration Bond c =1 [80/(1. 05)/1029)+2[80/(1. 06)2/1029]+ 3 [1080/(1. 07)3/1029]=2. 78 Duration Bond D =1 [40/(1. 05)/923)+2[40/(1. 06)2/923]+ 3 [1040/(1. 07)3/923]=2. 88 t 2 £ 80 £ 40 6% t 3 £ 1080 £ 1040 7%

§ Relationship between risk and return Return SML Return i Risk § The higher § Relationship between risk and return Return SML Return i Risk § The higher the Risk , The higher the required return. § SML can be used to generate risk-adjusted discount rates to be used in Financial decisions.

Risk and Rate of Return ØPortfolio Rate of Return : Average weighted return on Risk and Rate of Return ØPortfolio Rate of Return : Average weighted return on the portfolio as a whole. ØStandard deviation : reflection of risk inherent in the portfolio. It measures the extent of possible outcomes are likely to be different from the mean outcome

Risk , Return and Diversification Return Assets B 10% 45% 20% Assets A Probability Risk , Return and Diversification Return Assets B 10% 45% 20% Assets A Probability 55% 7% 65% 12% 35%

§ Asset A average return 0. 10 X 0. 45 +0. 20 X 0. § Asset A average return 0. 10 X 0. 45 +0. 20 X 0. 55 = 15. 5% Standard deviation of Return (0. 10 -0. 155 )2 X 0. 45+(0. 20 -0. 155)2 X 0. 55 Variance = 0. 00247 ∴standard deviation = σ√ 0. 00247 =4. 97% Asset B average Return Standard Deviation 8. 75 2. 39%

The correlation coefficient , the covariance Direct way to find portfolio risk by dealing The correlation coefficient , the covariance Direct way to find portfolio risk by dealing with the interrelatedness of Assets returns. It is a number that can take values from -1 (perfect negative relatedness) to +1 ( perfect positive relatedness). *The more positively related securities in portfolio, the less gain from diversification.

Summery Portfolio Risk 1 - Portfolio risk is not likely the average risk of Summery Portfolio Risk 1 - Portfolio risk is not likely the average risk of underlying Assets. 2 -Measurment of relatedness of individual Assets returns is the correlation coefficient of paired return of assets within the portfolio.

Risk of Average Portfolio Diversifiable & Un-diversifiable Risk (M) Number of Securities in Portfolio Risk of Average Portfolio Diversifiable & Un-diversifiable Risk (M) Number of Securities in Portfolio

Beta Coefficient ( Regression coefficient) B coefficient express relationship between the return expected from Beta Coefficient ( Regression coefficient) B coefficient express relationship between the return expected from security and that expected from the market as whole ßj= Standard deviation of return j X Correlation of j with the Market Standard deviation of Market Return

i. e = σj Pjm σm Same as σj σm Pjm σ 2 m i. e = σj Pjm σm Same as σj σm Pjm σ 2 m ∴ß j = σj m σ 2 m Covariance j with Market variance of Market

§ Example: If Market return move from 12% to 14% a security with return § Example: If Market return move from 12% to 14% a security with return of 15% and ß of 1. 3 will move to 15% +1. 3 (14%-12%) = 17. 6%

Security Market line (SML) the Market Model Objective : having quantitative mechanism for setting Security Market line (SML) the Market Model Objective : having quantitative mechanism for setting risk adjusted returns necessary for co. investment decision SML ERj M Er (M) 1. 0 E ( rj) = rf +{ E(rm) –r. F } X ßj -SML allows estimating discount rates and opportunity cost for co. investment

Learning Points 1 - Total risk can be separated to diversifiable and systematic (Market) Learning Points 1 - Total risk can be separated to diversifiable and systematic (Market) Risk. 2 - Un-diversifiable Risk is related to underlying market factors that is common to all assets and securities. 3 - Systematic Risk is measured by Beta coefficient (standard deviation X Correlation with Market) 4 -SML dictate set of risk adjusted returns available in the market 5 - SML from basis for evaluating internal co. investment where it should offer returns in excess of capital supplier opportunity.

Market Terminology -Spot Rate –Interest Rate –Cross Rate. - FX Forward - Options –Swap Market Terminology -Spot Rate –Interest Rate –Cross Rate. - FX Forward - Options –Swap –Hedging

Exchange Rate & Low OF One Price (Purchasing Power Parity) Same thing can not Exchange Rate & Low OF One Price (Purchasing Power Parity) Same thing can not sell for different prices at the same time. Impact of Demand & Supply Factors on currencies for different prices quoted across countries for same thing Exchange rates : Portray relationship in wealth exchanges across national borders (Now) Interest Rates: Portray wealth exchanges across time (in Future)

* Spot & Forward Exchange Rates Example: FX Quotation on July 22, 1997 £/$ * Spot & Forward Exchange Rates Example: FX Quotation on July 22, 1997 £/$ $/£ 0, 6266_____ 1. 5960 0, 6337 _____ 1. 5780 Spot Forward 180 days Sterling is at forward discount or dollars is at forward premium *Forward Exchange Rates & Interest Rates Interest rate parity interact with forward exchange market in order for no arbitrage opportunities to exist. Example: $i = 6. 5% £i = 8. 94% spot $/£ 1. 5960 £/$ 0, 6266

1 - amount borrowed ( six month $ 100. 000 @ 6. 5%) = 1 - amount borrowed ( six month $ 100. 000 @ 6. 5%) = 100 000 = $ 96900, 00 (PV) (1. 065)½ 2 - £ purchase spot = $ 96 900 X 0, 6266 = £ 60717. 00 3 -£ invested 6 month @ 8. 94% =60717 X(1. 894)½=£ 63373. 00 Review $ 100. 000 (Borrowed) should equal £ 63373 in 6 month period i. e Forward exchange rate should be £/$ 0, 6337

FX Forward § Example : Spot €$ FWD points ∴FWD Buying 1. 2750 50 FX Forward § Example : Spot €$ FWD points ∴FWD Buying 1. 2750 50 _______ 1. 2800 1. 27 50/55 50/60 FWD Selling 1. 2755 60 ____ 1. 2815

Currency Swap § Simultaneous spot buying/selling against opposite forward selling/Buying of a foreign Exchange. Currency Swap § Simultaneous spot buying/selling against opposite forward selling/Buying of a foreign Exchange. Example: * €/$ spot 1. 2850/55 *Swap cost 30/35 -at 55&30 we buy and sell € against $ i. e on same date we buy € spot at ← 1. 2855 and sell € FWD at 30 ______________ 1. 2885

Interest Rate Swap (IRS) § An interest rate swap is an exchange of interest Interest Rate Swap (IRS) § An interest rate swap is an exchange of interest flows in the same currency for maturities of over one year. These swaps never involve an exchange of principal amounts. § The common type of interest rate swap is one in which fixed Rate payments are exchange for floating rate payments (based on reference rate such as Libor ) § Swaps can also be arranged. In which one floating rate is swapped for an other (e. g 3 month Libor against 6 month Libor) the notional principal amount should be at least US $ 5 Million.

Interest rate swaps are available for a Varity of periods from one year up Interest rate swaps are available for a Varity of periods from one year up to a maximum of ten years. All interest rate swap transactions between the customers and the bank is governed by single standard agreement( the ISDA)

§ Example: A company has taken out a bank loan for Swiss Franc 10 § Example: A company has taken out a bank loan for Swiss Franc 10 Million at a fixed rate of 7 percent for 5 years. This money is used to buy a floating rate note at ( Libor +3/8 percent) with a 6 month rollover. ( if the floating rate investment is left unhedged, the company stand to make a loss as soon as Libor fall below 6 5/8 percent. ) Making an interest rate swap in which the company pays a floating rate and receives a fixed rate can cover this risk. Let us assume that the bank quotes (7. 5 -7. 6%) against 6 month Libor for the 5 years swap in our example. This means that banks is prepared to pay 7. 5% in return for receiving flat Libor If the bank were the floating rate payer , it would demand a rate of 7. 6%.

IRS § Example I Libor Swap Bank CO. 7. 5% CHM 10 mm @ IRS § Example I Libor Swap Bank CO. 7. 5% CHM 10 mm @ 7% Libor +3/8 Loan 5 year FRN Bank

§ Example II A company that had borrowed $ 20 million on a floating § Example II A company that had borrowed $ 20 million on a floating rate basis with interest at ½ percent per annum over 6 month Libor. The company is concerned the interest rates will rise and wants to fix the rate on the loan for the balance of the term. The company can enter into an interest rate swap and agree to pay a fixed rate for remaining term of the loan and in return receive 6 month Libor on a floating basis. If the fixed rate was 7 ¾ percent , then it has effectively created a fixed rate loan with a cost of 8 1/4 percent.

IRS § Example 2 Fixed 7. 75 % CO. Swap Bank LIBOR $ 20 IRS § Example 2 Fixed 7. 75 % CO. Swap Bank LIBOR $ 20 mm Cost id fixed 7. 75+0. 50=8. 25% 6 month Libor +0. 50

Options: having the right ( not the obligation) to buy or to sell at Options: having the right ( not the obligation) to buy or to sell at certain agreed terms for a fixed period of time Options terminology : *Contingent claim *Underlying Asset *Striking/ exercise price *Expiration date *Profitable option In the Money Ex: - option to buy underlying asset @ £ 1. 25 while market price is £ 1. 50

Out the Money : Example Call option by @ £ 1. 50 while Market Out the Money : Example Call option by @ £ 1. 50 while Market price is £ 1. 25 *Option Holder (Beneficiary ) *Option Writer (Issuer) *Covered option (Covered call) option writer owns the underlying Asset. * Naked option (naked call) Writer does not own the underlying asset.

Put option allows holder to sell at fixed price for a fixed period of Put option allows holder to sell at fixed price for a fixed period of time Put has a positive (+) exercise value if underlying asset value is less than striking (exercise) price While call option has a positive exercise value if underlying Asset value is higher than the striking price.

Value Of a Simple Option Example; (A) *Current price of underlying security So=£ 1. Value Of a Simple Option Example; (A) *Current price of underlying security So=£ 1. 50 *Probability of price increase q= 0. 6 *Upward Multiplier of underlying security U=2 *Downward Multiplier of underling security d=0. 5 ﺍ ﺣﺘﻤﺎ ﻻﺕ ﺳﻌﺮ ﺍ ﺗﻐﻴ ﻷ ﻴﺮ ﺼﻞ . 6 q =0 i. e £ 1. 50 1 -q=0. 4 Uso= 2 X£ 1. 50 = £ 3. 00 dso=0. 5 X£ 1. 50 =£ 0. 75

(B) ∴ Value of Call Option : Co= Current Market Value of the option (B) ∴ Value of Call Option : Co= Current Market Value of the option Cu= option payoff if underlying security price is up Cd= option payoff if underlying security price is down X = Striking (exercise Price . 6 q =0 i. e Co= 1 -q = 0. 4 Cu=max =( 0. Uso-X) =max= (0. 3 -1. 25) =1. 75 Cd= max (0. Dso-X) = max (0. 75 -1. 25) =0

Result It the underlying Security X increase in price to Uso £ 3. 00 Result It the underlying Security X increase in price to Uso £ 3. 00 the £ 1. 25 call option will have an exercise value (co) of £ 1. 75 and no payoff if option is out of the money and the option has 60% chance of paying £ 1. 75 and 40% chance of paying nothing

shares Y Purchased sold y or Z Borrowed mone lent Money (C) Option Valuation shares Y Purchased sold y or Z Borrowed mone lent Money (C) Option Valuation ( Binomial model Case) Technique of calculating market value of an option 1 - Get Y = Cu –Cd So (U-d) = £ 1. 75 -£ 0. 0 = 0. 7778 £ 1. 50 (2 -0. 5) 2 - Get Z = Ucd–dcu = 2(£ 0. 0)– 0. 5(£ 1. 75) = £ 0. 5303 (u-d)(1+rf) (2 -0. 5)(1+0. 10) 3 -Co =Y so + Z = 0. 7778 X£ 1. 50 - £ 0. 5303 =£ 0. 6364 Assuming European option.

Forward Interest Rates: Set of interest rates that begins at some future point other Forward Interest Rates: Set of interest rates that begins at some future point other than now (t 0). PV = CF 1 + CF 2 (1+i 1) (1+i 2)2 = £ 80 + £ 1080 (1. 05) (1. 06)2 = £ 76 + £ 961 = £ 1037

i. e. of the £ 1037 invested at t 0 £ 76. 00 Produces i. e. of the £ 1037 invested at t 0 £ 76. 00 Produces t 80 at t 1 for 5% £ 961 Produces t 1080 at t 2 for 6%

Forward interest Rates Example: Bond t 0 1037 t 1 £ 80 t 2 Forward interest Rates Example: Bond t 0 1037 t 1 £ 80 t 2 £ 1080 £ 1037 invested at t 0 @ 5 % ∴ The amount invested at t 1 ( before interest payment ) £ 1037 X 1. 05 = £ 1089 after £ 80. 00 payment at t 1 the amount invested is thus £ 1089 - £ 80 = £ 1009 amount invested in bond B at t 1 ∴ Value at t 1* ( 1+ 1∫ 2) = CF 2 £ 1009 ( 1 + 1∫ 2) = £ 1080 ∴ 1∫ 2 = 7%

Relationship between spot interest rates and forward interest rates (1+ i 2)2 =(1+ ∫ Relationship between spot interest rates and forward interest rates (1+ i 2)2 =(1+ ∫ 1 ) 0 (1+ ∫ ) 1 2 i. e. (1+ spot Rate) = geometric means of (1+ Forward Rates) Example : (1+ i 3)3 = (1+ 0∫ 1 ) (1+ 1∫ 2 )(1+ 2∫ 3 ) (1. 07)3= (1. 05)(1. 07)(1+ 2∫ 3 ) (1+ 2∫ 3) = (1. 07)3 /[(1. 05)(1. 07)] = 1. 09 2∫ 3 = 9%

Conclusion of Relationships PV = CF 1/(1+i) + CF 2/(1+i 2)2 + CF 3/(1+i Conclusion of Relationships PV = CF 1/(1+i) + CF 2/(1+i 2)2 + CF 3/(1+i 3)3 = CF 1/(1+ 0∫ 1) + CF 2/(1+ 0∫ 1)(1+ 1∫ 2) + CF 3/(1+ 0∫ 1 )(1+ 1∫ 2)(1+ 2∫ 3) =CF 1/1+YTM + CF 2/(1+YTM)2 + CF 3/(1+YTM)3

£ 923 = £ 40/(1. 05) + £ 40/(1. 06)2 + £ 1040/(1. 07)3 £ 923 = £ 40/(1. 05) + £ 40/(1. 06)2 + £ 1040/(1. 07)3 £ 923 = £ 40/(1. 05) + £ 40/(1. 05)(1. 07) + £ 1040/(1. 05)(1. 07)(1. 09) £ 923 = £ 40/(1. 069) + £ 40/(1. 069)2 + £ 1040/(1. 069)3

Interest Rate Futures Undertaking a real asset investment because of positive NPV could be Interest Rate Futures Undertaking a real asset investment because of positive NPV could be badly affected if actual future interest rate increase. Example: t 0 - £ 1700 t 1 £ 1000 t 2 £ 1000 i 1=10% And i 2 =11% 1)NPV = (£ 1700)+ £ 1000/(1. 1) + £ 1000/(1. 11)2 = + £ 20. 71

And 2) ∴ (1+i 2)2 =(1+i 1)(1+ ∫ ) =(1+i 2)2/(1=i 1) (1+ ∫ And 2) ∴ (1+i 2)2 =(1+i 1)(1+ ∫ ) =(1+i 2)2/(1=i 1) (1+ ∫ ) =(1. 11)2/(1. 10) ∫ =12. 009% 1 2 1 2

Now, If Risk that 1∫ 2 increase to 15% i. e. ∴ 1 - Now, If Risk that 1∫ 2 increase to 15% i. e. ∴ 1 - (1+i 2)2 = (1+i 1)(1+1∫ 2) (1+i 2)2 =(1. 10)(1. 15) (1+i 2)2 = (1. 265) i 2 = 12. 4722% 2 -NPV = (£ 1700) + £ 1000/(1. 124722)2 NPV = -£ 0. 40 i. e. The positive NPV of investment has become negative due to the increased interest rate applicable to t 2 cash flow N. B. : LOSS WILL BE 20. 71+0. 40 =21. 1

Hedge against interest Rate Risk by selling interest Rate futures contract. Commit to sell Hedge against interest Rate Risk by selling interest Rate futures contract. Commit to sell at a fixed price a security at t 1. So IF 1∫ 2 increases, the price of such security will decline, but since we have a contract to sell that security at a fixed higher price -the value of the contract will increase to offset the decrease of the NPV of investment. (i. e. avoiding risk of interest Rate changes)

Interest Rate Futures. Example: i 1 =10% i 2=11% t 2 CF £ 1000 Interest Rate Futures. Example: i 1 =10% i 2=11% t 2 CF £ 1000 ∴(1+i 2)2 = (1+i 1)(1+ ∫ ) 1+ ∫ = (1+i 2)2/(1+i 1) 1+ ∫ = (1. 11)2 /(1. 10) ∫ = 12. 009% t 1 future price =£ 1000/(1+ ∫ ) =£ 1000/1. 2000 =£ 892. 79 T 1 CF £ 1000 1 2 1 2 1 2

To hedge against interest rate risk you commit to sell £ 1000 t 2 To hedge against interest rate risk you commit to sell £ 1000 t 2 cash flow at t 1 for £ 892. 79. N. B. no cash actually changed at t 0 where investment undertaken futures contract sold –meaning you only made a commitment to sell the security at t 1.

If 1∫ 2 increases to 15% beside decline of NPV from £ 20. 71 If 1∫ 2 increases to 15% beside decline of NPV from £ 20. 71 to (£ 0. 40) i. e. a wealth loss of £ 21. 11 However a) The new t 1 value of t 2 £ 1000 is £ 1000/1. 15 = £ 869. 57 b) Contract value increase at t 1 £ 892. 79 - £ 869. 57 = £ 23. 22 c) Discounting that value increase to t 0 @ 10% ∴ increase in contract value at t 0 £ 23. 22/1. 10 = 21. 11 Conclusion: increase in value of interest rate futures contract at t 0 equals the decrease in value of investment NPV

N. B : If 1∫ 2 re to decrease the NPV would increase while N. B : If 1∫ 2 re to decrease the NPV would increase while futures contract value would decrease to offset NPV increase i. e. : Hedging means losing both bad and good chances arising out of interest.