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- 1. Derivatives
- 2. Intro to my course summary <ul><li>If you are having to learn these concepts and techniques for the first time, and Peter Moles’ book is your guide, you have my sympathy. You are about to experience a great number of ‘WTF?’ moments. In addition to being poorly written and probably never edited, as a mathematics text, it suffers two fatal flaws: </li></ul><ul><ul><ul><li>It often does not fully explain what is being done or why . </li></ul></ul></ul><ul><ul><ul><li>It is full of mistakes : variables often don’t jibe with the discussion or the described methods don’t produce the results indicated. This is particularly maddening when you are checking the book to see of you are doing something correctly. The book is particularly sloppy with the precedence of mathematical operations and with mathematical precision. </li></ul></ul></ul>
- 3. 1. Fundamentals <ul><li>Financial risk management seeks to limit the effects of changes in financial variables (interest rates, currencies, commodity prices) on the ability of the firm to achieve its corporate objectives. </li></ul><ul><ul><li>There may be uncertainty about a particular outcome without there being any particular consequence. </li></ul></ul><ul><ul><li>The risk will be managed when a deviation from the expected outcome may be detrimental. </li></ul></ul><ul><li>Most individuals and many firms are risk averse . That is, they see the potential gains and losses, even if evenly distributed, as not equal. </li></ul>
- 4. Alternate Approaches <ul><li>Risk avoidance : risk is not assumed </li></ul><ul><li>Risk mitigation : loss prevention or control are used to reduce risk to acceptable levels </li></ul><ul><li>Risk retention : firm has the capacity to absorb the risk and the competency to manage its exposures </li></ul><ul><ul><li>Assumes that the firm is not retaining by default or out of ignorance (or both) </li></ul></ul>
- 5. Risk Management Process <ul><li>Identify the source of the risk exposure. </li></ul><ul><li>Quantify and / or assess the exposure. </li></ul><ul><ul><li>Top down </li></ul></ul><ul><ul><li>Building block: effects of risk factors are analyzed and aggregated </li></ul></ul><ul><li>Conduct a cost-benefit analysis of the impacts and possible adjustments. </li></ul><ul><li>Assess the firm’s capabilities to undertake its own insurance and hedging program. </li></ul><ul><li>Select the products and mix. </li></ul><ul><li>Keep the risk management process under review . </li></ul>
- 6. Portfolio of Projects <ul><li>One way to characterize the firm is as a portfolio of projects , funded by a mixture of debt and equity. </li></ul><ul><ul><li>Successful projects provide cash flows that allow the firm to grow and to reward the providers of the firm’s capital. </li></ul></ul><ul><ul><li>Losses hinder the execution of the firm’s business strategy. “A successful firm makes enough money to pay for its mistakes.” – Peter Drucker </li></ul></ul><ul><ul><li>Removing externalities allows managers to focus on producing shareholder value. </li></ul></ul>
- 7. Nature of Exposures <ul><li>Origin: internal, input or output </li></ul><ul><li>Transaction: the commitments chosen by the firm </li></ul><ul><li>Translation: assets or obligations in a foreign currency </li></ul><ul><li>Contingent: expected but non-contracted </li></ul><ul><li>Economic / Strategic: GDP growth, interest rates, foreign exchange, suppliers, customers, competitors </li></ul>
- 8. Risk Transfer and Reduction Accept Transfer Insure: indemnity, guarantee, options Hedge On Balance Sheet (Operational) Off Balance Sheet (Financial) Assess
- 9. 2. Building Blocks <ul><li>All of the terminal products (forwards, futures, swaps) have linear outcomes. The differences are in: </li></ul><ul><ul><li>How they handle performance risk </li></ul></ul><ul><ul><li>The degree of tailoring </li></ul></ul><ul><ul><li>The possibility of trading out of the position </li></ul></ul><ul><li>Options have non-linear outcome. </li></ul><ul><ul><li>Buyer has performance risk with the seller, but the seller has no risk with the buyer. </li></ul></ul><ul><ul><li>Allows choice of price level. </li></ul></ul>
- 10. Terminal Products <ul><li>In a forward , the buyer and seller agree to a price for an asset to be delivered on a set date in the future. </li></ul><ul><li>Futures were conceived as a means of eliminating credit risk from forwards. </li></ul><ul><ul><li>Daily mark-to-market out of margin funds (collateralization). Contract is effectively renegotiated each day at the new prevailing market rate. </li></ul></ul><ul><ul><li>Clearing house is the counterparty for all transactions, and guarantees performance. </li></ul></ul><ul><ul><li>Standardization increases liquidity at the expense of basis risks. </li></ul></ul><ul><li>A swap obliges two parties to exchange (or swap) a series of cash flows at specified intervals over a particular time period. </li></ul><ul><ul><li>This works like a bundle of forward contracts. </li></ul></ul>
- 11. Option Contracts <ul><li>Options confer on the owner the right, but not the obligation, to make a particular future transaction: a call gives the right to buy, and a put the right to sell. </li></ul><ul><li>The option price (without merchant markup) is one that which ensures, ex ante , the transaction is a fair one – that is, it has zero net present value. </li></ul><ul><li>An option is in fact a portfolio consisting of a forward contract on the underlying and a loan. The call option can be replicated dynamically by adjusting these two elements. </li></ul>
- 12. 3. Forward Contracts <ul><li>A forward contract is simply a bilateral commercial agreement negotiated today but with its execution or settlement deferred to some agreed date in the future. </li></ul><ul><ul><li>The economic rationale is that these add value by eliminating or reducing uncertainty. </li></ul></ul><ul><ul><li>The demand for forward contracts will be determined by the number of firms facing uncertainty about future prices. </li></ul></ul><ul><li>These can be tailored to match underlying risk. They are typically held to maturity (i.e., they are not unwound). </li></ul>
- 13. The Cost-of-Carry Model <ul><li>The model posits that forward prices will be determined by the costs of acquiring the asset in the cash market and holding it for delivery: </li></ul><ul><ul><li>FP t,T = CP t + [CP t x R t,T x (T-t)/365] + G t,T where FP is the forward price, CP is the cash price, R is the risk-free interest rate and G are storage and other related costs. </li></ul></ul><ul><ul><li>This provides an upper limit for the forward price, but is not a useful formulation for discovery markets , where prices are greatly influenced by availability. </li></ul></ul><ul><ul><li>Intermediate payments of interest or dividends (“ leakages ”) reduce the forward price. </li></ul></ul>
- 14. Volatility and Credit Exposure <ul><li>It is volatility in the potential outcome that creates performance risk (credit exposure). </li></ul><ul><ul><li>The greater the volatility, the greater the risk. </li></ul></ul><ul><ul><li>It is this same undesirable volatility in the price that makes the contract valuable to both parties. Each is willing to acquire the benefit of price certainty at the (perceived to be lower) cost of performance risk. </li></ul></ul>
- 15. Boundary condition (upper) <ul><li>There is an arbitrage opportunity when the costs of carry are less than the spread between cash and forward prices (i.e., forward is expensive ): </li></ul><ul><ul><li>CP t (1 + i B ) T-t + TC < FP t , where B is the borrowing rate and TC are transaction costs. </li></ul></ul><ul><ul><li>Buy the cash and sell the forward. </li></ul></ul>
- 16. Boundary condition (lower) <ul><li>There is an likewise an arbitrage opportunity when the replacement cost of the asset is less than the cash price minus the costs of carry (i.e., forward is cheap ): </li></ul><ul><ul><li>FP t < CP t (1 + i L ) T-t - TC, where L is the lending rate and TC are transaction costs. </li></ul></ul><ul><ul><li>Would-be transaction costs (e.g., storage) are subtracted because they are not incurred. </li></ul></ul><ul><ul><li>Sell the cash, invest proceeds and buy the forward. This is sometimes called a reverse cash and carry . </li></ul></ul>
- 17. Forward Rate Agreement <ul><li>An FRA is an interest rate transaction. It is called a forward rate agreement because, although it may be transacted today, it has a forward start. </li></ul><ul><li>It replicates a forward start deposit or loan, thus allowing the future borrower or lender to lock in an interest rate based on the rates quoted or implied today for the period between the settlement date and maturity date. </li></ul>transaction date settlement date maturity date loan or deposit period time
- 18. Forward Rate Agreement, 2 <ul><li>There is a settlement for two reasons: </li></ul><ul><ul><li>In common with a swap, the FRA has no delivery of the principal. It is still up to the borrower or lender to make his transaction on the settlement date. </li></ul></ul><ul><ul><li>Rates for the covered period may change between the transaction date and the settlement date. An amount is paid or received in order to restore the borrower or lender to the position he would have had with the locked-in rate. </li></ul></ul><ul><li>The borrower or lender is restored to his position as of the maturity date, but as the payment is actually made/ received on the settlement date, it is discounted according to the spot rate on the settlement date. </li></ul>
- 19. Forward Rate Agreement, 3 <ul><li>The settlement amount is the difference between the rates, times the number of years in the loan/deposit period, times the notional amount, all discounted back from the maturity date to the settlement date. </li></ul><ul><li>Annualized rates are quoted as whole numbers (e.g., 5 percent is 5 and not 0.05, thus requiring the division by 100. D is the number of days in the loan or deposit period. Basis is the number of days in the interest rate year, usually 360 or 365. </li></ul>[ R c – R s ] x D x A 100 basis Settlement amount = 1 + R s x D basis x 100
- 20. Forward Rate Agreement, 4 <ul><li>If you multiply the top and bottom terms by (basis x 100) then simplify: </li></ul><ul><li>This is the British Bankers Association formula. It is not as intuitive as the other, but is much less confusing to work with. </li></ul><ul><li>The future borrower will “buy” or “take” the FRA (i.e., pay the fixed rate); the future lender will “sell” or “place” the FRA (i.e., receive the fixed rate). </li></ul>( Settlement amount = (R s – R c ) x D x A (basis x 100) + (R s x D)
- 21. Forward Exchange Rates <ul><li>Forward rates for foreign exchange are projected using the theory of interest rate parity . </li></ul><ul><ul><li>The interest rate for the quoted currency is divided by the interest rate for the base currency. Multiplying by the spot exchange rate gives the forward outright . </li></ul></ul><ul><ul><li>If the spot rate is USD 1.5425/£, the one-year USD interest rate is 3.25% and the one-year GBP interest rate is 4.125%, the forward outright in one year is </li></ul></ul>$1.5425 x 1.0325 1.04125 = $1.5295
- 22. Forward Exchange Rates, 2 <ul><li>As the differentials between exchange rates are much more stable that the rates themselves, the custom in the foreign exchange markets is to quote the differential as a premium or discount to the spot rate. </li></ul><ul><ul><li>Nobody wants to risk a misplaced decimal, so the differential is multiplied by 10,000. </li></ul></ul><ul><ul><li>From the one-year projection on the previous slide: $1.5425 - $1.5295 = 0.0130 x 10,000 = 130 forward points or pips ) </li></ul></ul>
- 23. SAFE <ul><li>Although interest rates are required in the calculation, a Synthetic Agreement for Forward Exchange (SAFE) is a foreign exchange transaction. </li></ul><ul><li>One is notionally borrowing or lending an amount of the foreign currency in the future, for a known term, and wants to lock in the exchange rate or forward points differential at today’s rates. </li></ul><ul><li>The most common structures are: </li></ul><ul><ul><li>Exchange rate agreement (ERA) </li></ul></ul><ul><ul><li>Forward exchange agreement (FXA) </li></ul></ul>
- 24. SAFE, 2 <ul><li>As with the FRA, there is no delivery of principal, and the would-be borrower or lender will still have to make that transaction on the settlement date. </li></ul><ul><li>Without the transfer and re-transfer of principal amounts, this transaction has reduced credit exposure, and thus requires smaller reserve requirements. </li></ul>
- 25. SAFE, 3 <ul><li>Rates or forward points may change between the transaction date and the settlement date. An amount is paid or received in order to restore the borrower or lender to the position he would have had with a locked-in exchange rate or forward point spread. </li></ul><ul><li>The “buyer” of the SAFE sells the foreign currency at the settlement date and repurchases it at maturity (is the lender of the foreign currency). The “seller” of the SAFE purchases the foreign currency at the settlement date and resells at maturity (borrower of the foreign currency). </li></ul>
- 26. Exchange Rate Agreement <ul><li>The payoff of ERA = </li></ul><ul><li>That is, the difference in forward points between the contract date and the settlement date is discounted for the time between maturity and settlement. This is then multiplied by the notional principal. </li></ul><ul><li>The borrower or lender is restored to his position as of the maturity date, but as the payment is actually made/ received on the settlement date, it is discounted according to the spot rate on the settlement date. </li></ul>100 x basis i x (T m – T s ) 1 + (f c – f d ) notional principal x
- 27. Forward Exchange Agreement (FXA) <ul><li>An FXA covers both a change in the swap points and a change in the spot rate between the transaction date and the settlement date. </li></ul><ul><ul><li>Difference between contract rate + points and settlement rate + points is discounted for basis and for time between maturity and settlement, and multiplied by the notional maturity amount. </li></ul></ul><ul><ul><li>Subtract from this the difference between the contract and settlement rates, multiplied by the notional amount exchanged on the settlement date. (These notional amounts are different because we expect the original amount to change over time, e.g., we expect $1.00 delivered on settlement day to equal $1.02 when “re-exchanged”.) </li></ul></ul>[(s c + f c ) - (s d + f d )] 100 x basis i x (T m – T s ) Settlement amount = A m x – A s x (s c - s d ) 1 +
- 28. Quoting an FXA <ul><li>As with an FRA, a quote will be for 1 v. 4, 2 v. 6 etc. </li></ul><ul><li>A positive point spread ( f c – f d ) means that a payment is made to the buyer. This makes the bid – offer counter-intuitive. If the spread were 110 - 114: </li></ul><ul><ul><li>A larger f c makes it more likely that the term will be positive, therefore the buyer bids the higher number (i.e., 114). </li></ul></ul><ul><ul><li>A smaller f c makes it more likely that the term will be negative, therefore the seller offers the lower number (i.e., 110). </li></ul></ul>
- 29. 4. Futures <ul><li>The principal distinguishing features between the forward market and the exchange-traded futures market in the same asset relate not to fundamental differences in their economic effect, but to institutional arrangements dealing with counterparty risk and providing liquidity. </li></ul><ul><ul><li>The former is addressed by requiring all participants to post a performance bond and revaluing the position each day. </li></ul></ul><ul><ul><li>The latter is provided by restricting the number of maturity dates and standardizing the nature of the contracted instruments. </li></ul></ul>
- 30. Advantages / Disadvantages <ul><li>Transparency </li></ul><ul><li>Price discovery </li></ul><ul><li>Liquidity </li></ul><ul><li>Economic efficiency: bid – offer spread, transaction costs, leverage </li></ul><ul><li>Short position easily established </li></ul><ul><li>Positions are easily offset (fungible) </li></ul><ul><li>No credit exposure </li></ul><ul><li>Rounding error: actual exposure may not be a whole contract unit </li></ul><ul><li>Daily margining creates a mismatch of cash flows </li></ul><ul><li>Basis risks caused by standardization: commodity, timing and location </li></ul>
- 31. Minimizing Performance Risk <ul><li>The exchange’s clearing house acts as the counterparty for all contracts. </li></ul><ul><li>The exchange requires the posting of an initial margin amount, intended to approximate a maximum daily price move. </li></ul><ul><li>Contracts are marked to market at the end of each day, with the losses paid from the margin funds of the losers into the accounts of the winners. </li></ul><ul><li>If the margin funds on hand fall below the maintenance level, the exchange requires the posting of additional funds (“margin call”). </li></ul>
- 32. Convergence <ul><li>As the future moves towards expiry, the cost of carry will decline to the point where, at expiry, the two prices should be the same. </li></ul><ul><ul><li>This coming together of the cash and futures prices is known as convergence and it is the only time when the futures price and the cash price must necessarily be the same. </li></ul></ul><ul><ul><li>For markets in which they are no supply issues (convenience yields), the “fair” price of the future will be cash price plus the costs of carry. </li></ul></ul>
- 33. Basis Relationships <ul><li>Raw or simple basis: Cash – Future, i.e., S – F </li></ul><ul><li>Carry basis: Cash – theoretical Future, based on costs of carry, i.e., S – F* </li></ul><ul><ul><li>A comparison of the simple basis and carry basis will indicate whether the future is cheap or dear relative to cash. </li></ul></ul><ul><ul><li>This should also consider leakages such as interim payments of dividends or interest. </li></ul></ul><ul><li>Value basis: F – F* </li></ul><ul><li>As market prices change, the cash and futures prices will normally move in the same direction, but the basis will not be constant. This could be problematic for the hedger. </li></ul>
- 34. Effects of a Change in Basis <ul><li>As with the underlying itself, a market participant can be long or short the basis. This will be the same direction as the position in the underlying (cash). </li></ul><ul><li>Basis increase: </li></ul><ul><ul><li>Cash increases more quickly or decreases more slowly than futures </li></ul></ul><ul><ul><li>Favors the long basis holder and prejudices the short </li></ul></ul><ul><li>Basis decrease: </li></ul><ul><ul><li>Cash decreases more quickly or increases more slowly than futures </li></ul></ul><ul><ul><li>Favors the short basis holder and prejudices the long </li></ul></ul>If the basis on a commodity has gone from -10 to +10, has it widened or narrowed? This is why the text’s formulation is unhelpful. Also wrong.
- 35. Contango <ul><li>In the cost of carry model, the fair futures price is going to be driven by, inter alia , the financing cost. </li></ul><ul><ul><li>If a cash asset were valued at 100 and interest rates were 5%, with continuous compounding the futures contract three months out: </li></ul></ul><ul><ul><li>F t = S x e r x (T – t) = 100 x e 0.05 x 0.25 = 101.26 </li></ul></ul><ul><ul><li>Forward prices in these markets will be rising as we go further out in time, a condition called contango . </li></ul></ul>
- 36. Backwardation <ul><li>Seasonal influences or possible supply shortages, especially when there are no suitable substitutes, create a convenience yield. </li></ul><ul><ul><li>If this yield could be estimated with a parameter y , then the futures price could be projected: </li></ul></ul><ul><ul><li>F t = S x e (r – y)(T – t) </li></ul></ul><ul><ul><li>Forward prices in these markets will be falling as we go further out in time, a condition called backwardation . </li></ul></ul>
- 37. Tailing the Hedge <ul><li>The timing effects that arise from the margining system require the hedger to reduce the exposure on the futures contract by the expected reinvested income from the margin position – that is, tail the hedge . </li></ul><ul><ul><li>Tailed hedge = N x e -rt </li></ul></ul><ul><ul><li>If a “full” hedge would be for a position of 100 contracts, interest rates are 10% and the exposure period were 11 months, </li></ul></ul><ul><ul><li>Tailed hedge = 100 x e -0.1 x 0.9167 = 91.24 </li></ul></ul>
- 38. Interest Rate Futures <ul><li>The price quotation for Treasury bills is based on an index: 100 less the interest rate on the futures contract. </li></ul><ul><ul><li>If interest rates were 10 per cent on a three-month T-bill, the futures price would be 97.5, i.e., 100 – (10 x 0.25). </li></ul></ul><ul><li>Buy as protection from a lower rate. </li></ul><ul><ul><li>If interest rates fall, the price rises (a hedging gain). If interest rates rise, the price falls (a hedging loss). </li></ul></ul><ul><li>Sell as protection from a higher rate. </li></ul><ul><ul><li>If interest rates fall, the price rises (a hedging loss). If interest rates rise, the price falls (a hedging gain). </li></ul></ul>
- 39. 5. Swaps <ul><li>A swap is a bilateral agreement to exchange a series of cash flows. </li></ul><ul><ul><li>It has the appearance of a bundle of forwards, with the same linear (symmetric) payoff, but with the same fixed price term for each settlement. </li></ul></ul><ul><ul><li>It allows market participants to modify sets of connected cash flows by exchanging one form of exposure for another. </li></ul></ul><ul><ul><li>Payments are usually made on a net basis (that is, the differences). </li></ul></ul>
- 40. Swap Basics <ul><li>Fixed-Rate Payer </li></ul><ul><ul><li>Pays the fixed interest rate </li></ul></ul><ul><ul><li>Receives the floating interest rate </li></ul></ul><ul><ul><li>Has purchased (is long) the swap </li></ul></ul><ul><ul><li>Is short the bond market </li></ul></ul><ul><ul><li>Has a long-dated fixed-rate liability and a floating-rate asset </li></ul></ul><ul><li>Floating-Rate Payer </li></ul><ul><ul><li>Pays the floating interest rate </li></ul></ul><ul><ul><li>Receives the fixed interest rate </li></ul></ul><ul><ul><li>Has sold (is short) the swap </li></ul></ul><ul><ul><li>Is long the bond market </li></ul></ul><ul><ul><li>Has a long-dated fixed-rate asset and a floating-rate liability </li></ul></ul>
- 41. Interest Rate Swap <ul><li>The payoffs of an interest rate swap would imitate, for example, buying a fixed rate bond and financing the purchase with a floating rate note. </li></ul><ul><ul><li>The loan proceeds are used to purchase the bond. </li></ul></ul><ul><ul><li>The bond pays a fixed rate coupon. </li></ul></ul><ul><ul><li>Periodic interest payments are required on the note at the then-prevailing (floating) interest rate. </li></ul></ul><ul><ul><li>At bond maturity, the loan is retired. </li></ul></ul>
- 42. Cross-Currency Swap <ul><li>The parties to the swap exchange the underlying principal at the onset and return it ( not re-exchange it) at maturity. </li></ul><ul><ul><li>A obtains $15 million from counterparty B for £10 million. </li></ul></ul><ul><ul><li>Throughout the life of the swap, the parties service each other’s interest-rate payments. A floating-for-floating exchange is a cross-currency basis swap ; fixed-for-fixed is a cross-currency coupon swap . </li></ul></ul><ul><ul><li>At termination, A re-delivers the $15 million to B and receives £10 million. </li></ul></ul>
- 43. Asset – Liability Management <ul><li>An investment arbitrage exists when the synthetic alternative provides a positive net gain over the market equivalent. </li></ul><ul><ul><li>Synthetic floating-rate note : issue a bond, then swap to receive fixed and pay floating. </li></ul></ul><ul><ul><li>Synthetic bond : borrow at a floating rate, then swap to pay fixed and receive floating. </li></ul></ul><ul><ul><li>Synthetic floating-rate loan : buy a bond, then swap to pay fixed and receive floating. </li></ul></ul><ul><ul><li>Synthetic straight bond : buy a floating-rate note, then swap to receive fixed and pay floating. </li></ul></ul>
- 44. Swap Pricing <ul><li>In normal circumstances, it is possible for a counterparty to construct a riskless position on the other side of the swap. </li></ul><ul><li>The value to both sides of an at-market swap is such that neither is required to compensate the other when entering the transaction. </li></ul><ul><ul><li>The sum of the fixed cash flows, discounted at the appropriate zero-coupon rates, is equal to the sum of the expected floating rates similarly discounted. </li></ul></ul><ul><ul><li>That is, NPV = 0 </li></ul></ul>
- 45. Pricing a Swap <ul><li>At the outset, the net present value of the fixed-rate and floating rate payments are equal. </li></ul><ul><li>Any set of fixed cash flows can be seen as the sum of a series of zero-coupon bonds with matching cash flows. </li></ul><ul><li>The first step in valuation is to estimate the amounts of the floating rate payments for each future settlement date. </li></ul>
- 46. Pricing a Swap, 2 <ul><li>The forward floating rates could be derived from the zero coupon rates. If the zero coupon rates for 6 months and one year were 5.00 and 5.20 respectively, then </li></ul><ul><li>(1 + 0 z 0.5 ) 0.5 (1 + 0.5 f 1.0 ) 0.5 = (1 + 0 z 1.0 ) 1.0 </li></ul><ul><li>(1.05) 0.5 (1 + 0.5 f 1.0 ) 0.5 = (1.052) 1.0 </li></ul><ul><li>1.0247 x (1 + 0.5 f 1.0 ) 0.5 = 1.052 </li></ul><ul><li> (1 + 0.5 f 1.0 ) 0.5 = 1.0266 (= 1.0533, annualized) </li></ul><ul><li>Use the applicable zero-coupon rate to NPV each of the expected future payments, e.g., for the payment at the end of t = 5, NPV = payment / (1 + 0 z 5 ) 5 </li></ul>
- 47. Pricing a Swap, 3 <ul><li>Take the sum of the NPV’s of the floating payments. </li></ul><ul><ul><li>The fixed rate side has the same total NPV. </li></ul></ul><ul><li>The swap’s fixed rate is then related to the NPV by the annuity factor. </li></ul><ul><ul><li>NPV = r fixed x AF </li></ul></ul><ul><li>The annuity factor for the fixed payments is the sum of the zero-coupon factors used to discount the floating payments. </li></ul><ul><ul><li>Annuity factor = Σ 1 / (1 + 0 z t ) t </li></ul></ul><ul><li>Calculated rate may have to be annualized. </li></ul>
- 48. Annuity Factor <ul><li>Annuity factor = [1 – (1 / (1 + i ) t )] / i </li></ul><ul><li>If i = 4.5% and t = 10 then </li></ul><ul><ul><li>(1.045) 10 = 1.553 </li></ul></ul><ul><ul><li>1 / 1.553 = 0.6439 </li></ul></ul><ul><ul><li>1 – 0.6439 = 0.3561 </li></ul></ul><ul><ul><li>0.3561 / .045 = 7.9127 </li></ul></ul><ul><li>PV = coupon x AF </li></ul><ul><li>PV / AF = coupon </li></ul>
- 49. Bootstrapping <ul><li>Zero-coupon rates can be “bootstrapped” from the par yield curve with the formula s t = (Y t + 100) / (100 – (Y t x A t )) , where Y t is the coupon at time t and A t is the annuity rate applied to interim coupon payments. </li></ul><ul><li>s 2 = (8.98 + 100) / (100 – (8.98 x 0.913685)) = 1.187209 </li></ul><ul><li>The price relative s t is the equivalent zero-coupon payout. </li></ul><ul><li>The zero coupon yield is the n th root of s t . </li></ul><ul><li>The zero coupon discount factor = 1 / s t </li></ul><ul><li>The annuity factor is the sum of the zero-coupon discount factors for previous periods. </li></ul>Maturity Par yield (Y t ) A t s t ZC yield ZC discount factor 1 9.4469 - 1.094469 9.4469 0.913685 2 8.9800 0.913685 1.187209 8.9591 0.842312 3 8.6000 1.755997 1.279176 8.5534 0.781753
- 50. Cross-Subsidy Element <ul><li>For a swap to have a zero net present value, there will be periods when the value of the fixed is above that of the floating side payments. </li></ul><ul><ul><li>In general, an upward-sloping term structure means that the fixed price payer has a net payable position in the early periods versus a net receivable position in the later ones. </li></ul></ul><ul><ul><li>For a downward-sloping curve, the situation (for the fixed rate payer) is the opposite. </li></ul></ul>
- 51. Valuing a Seasoned Swap <ul><li>Once the swap rates are no longer at-market, the swap is said to be seasoned . </li></ul><ul><li>Fixed-price payments or receipts are discounted using the appropriate zero-coupon rate. </li></ul><ul><ul><li>This accomplishes the same valuation as replacement of the cash flows with new swaps, working from the longest dated, until there is only a present valuation. </li></ul></ul><ul><li>The value of a fixed-to-fixed cross-currency swap will be influenced by three factors: </li></ul><ul><ul><li>The first two are interest rates in the respective currencies. </li></ul></ul><ul><ul><li>The third is the change in the exchange rate. </li></ul></ul>
- 52. Amortizing Swap <ul><li>… has a structure in which the principal amount is reduced over time. </li></ul><ul><li>In most cases, the rates on the different notional elements comprising this swap will also be different, but a flat rate is desired. </li></ul><ul><li>For each element, calculate the present value of an interest rate annuity at that rate. Take the total of the PV’s. </li></ul><ul><ul><li>This is the present value of the interest that will be paid on the comprising swaps. </li></ul></ul>
- 53. Amortizing Swap, 2 <ul><li>Next, calculate the PV of 1% of the swap principal in each element, and total. </li></ul><ul><li>The total of the interest PV divided by the total of the principal PV gives the blended rate. </li></ul><ul><li>There is a residual risk for the market maker as there is a cash flow mismatch between the package and the swap elements used to create it. </li></ul>
- 54. Deferred Start Swap <ul><li>… is any swap with the start date of the contract deferred beyond the usual market terms for settlement (e.g., a four-year swap priced today but comes into effect one year from today). </li></ul><ul><ul><li>This is the equivalent of a five-year spot swap less a one-year spot swap: (swap rate for 5-year swap x annuity factor) – (swap rate for 1-year swap x annuity factor), all divided by the annuity factor for the 4-year swap (which is AF 5 – AF 1 ). </li></ul></ul><ul><ul><li>Pricing such a swap is equivalent to pricing the implied forward rate. </li></ul></ul>
- 55. Accreting & Rollercoaster Swaps <ul><li>The accreting swap is a package made up of an initial spot swap and a series of deferred-start swaps. </li></ul><ul><ul><li>Flat pricing is achieved in exactly the same manner as for an amortizing swap. </li></ul></ul><ul><li>A rollercoaster swap is a combination of an accreting and an amortizing swap. </li></ul>
- 56. Swap Credit Exposure <ul><li>The expected exposure in an interest rate swap peaks and then diminishes as: </li></ul><ul><ul><li>Interim payments are made </li></ul></ul><ul><ul><li>Interest rates revert to their mean </li></ul></ul><ul><li>The exposure on a cross-currency swap increases over time as: </li></ul><ul><ul><li>Rates trend away from the initial price (i.e., there is no reversion to a mean). This means that the interim interest payments or re-exchange of principal amounts is becoming less valuable to one of the parties, thus creating an exposure equal to the net change in value. </li></ul></ul>
- 57. Expected Loss <ul><li>In a lattice, we can calculate the expected loss associated with each node. In an interest rate swap, this would be the cost of replacing the swap at that time and interest rate. </li></ul><ul><ul><li>If we are paying the floating rate, there is no loss associated with rates that are equal to or higher than the swap’s fixed rate. </li></ul></ul><ul><ul><li>Otherwise, the loss = interest rate differential from the swap’s fixed rate x principal amount x annuity associated with the remaining term (given the number of payments that would be made and the interest rate associated with the node). </li></ul></ul><ul><ul><li>Multiply the value by the probability to get the expected loss associated with the node. </li></ul></ul>
- 58. Expected Loss, 2 <ul><li>As we do not know when a loss would occur, we assume it to be at the mid-point of each year. </li></ul><ul><ul><li>To calculate the loss at t = 1.5, we take the average of the probability-weighted losses for t = 1 and t = 2. </li></ul></ul><ul><ul><li>Discount by the swap’s fixed rate for 1.5 periods to obtain the present value. </li></ul></ul><ul><ul><li>The sum of these gives the exposure (i.e., maximum expected loss) at t = 0. </li></ul></ul><ul><li>Expected loss from default = φ of default x expected loss if default occurs </li></ul>
- 59. 6. Options Basics <ul><li>A purchased option provides a non-linear (asymmetric) payoff with some of the characteristics of insurance. </li></ul><ul><li>In exchange for the payment of a premium , the holder (buyer) of a call option has the right, but not the obligation, to buy a fixed quantity of the underlying asset at the strike price until expiry. </li></ul><ul><li>In exchange for the payment of a premium, the holder of a put option has the right, but not the obligation, to sell a fixed quantity of the underlying asset at the strike price until expiry. </li></ul>
- 60. More basics <ul><li>In each case, the writer (seller) of the option must take the opposite side of the transaction if the option is exercised. </li></ul><ul><li>An option may be capable of being exercised at any time until expiry ( American -style) or only at expiry ( European -style). </li></ul>
- 61. In, At or Out-of-the-Money <ul><li>In-the-money: the market price is above the strike price of a call or below the strike price of a put. </li></ul><ul><li>At-the-money: the market price is equal to the strike price. </li></ul><ul><li>Out-of-the-money: the market is price is below the strike price of a call or above the strike price of a put. </li></ul>
- 62. Intrinsic and Time Value <ul><li>The value of an option is comprised of intrinsic value and time value. </li></ul><ul><ul><li>The intrinsic value is the value of an option if exercised immediately, with a minimum of zero. </li></ul></ul><ul><ul><li>The remainder is the time value. This value declines as expiry approaches – a phenomenon known as time decay. </li></ul></ul>
- 63. Factors Affecting Value <ul><li>Time to expiry: more time provides more opportunity for conditions to favor the outcome. </li></ul><ul><li>Difference between the market price and the strike price: the amount by which an option is in or out-of-the-money. </li></ul><ul><li>Price volatility : the greater the degree of potential price movement and thus a favorable outcome. </li></ul><ul><li>Interest rates : the possibility of deferring a purchase allows interest to be earned in the interim. This works against the deferred sale (put). </li></ul><ul><li>Leakages : the underlying asset declines in value when interest or dividends are paid. </li></ul>
- 64. Put – Call Parity <ul><li>P + U = C + PV(K) </li></ul><ul><li>The put price plus the underlying asset price is equal to the call price (with the same strike) plus the present value of the strike. </li></ul>
- 65. Fundamental Strategies <ul><li>Purchased call </li></ul><ul><li>Purchased put </li></ul><ul><li>Written call </li></ul><ul><li>Written put </li></ul><ul><li>Written call with long position in underlying (synthetic short put) </li></ul><ul><li>Written put with short position in the underlying (synthetic short call) </li></ul>
- 66. Combination Strategies <ul><li>A vertical spread is a directional strategy with a purchased call and a written call (or a purchased put and a written put) with the same expiry but different strikes. A horizontal or time spread uses different expiries. </li></ul><ul><ul><li>A ratio spread uses different numbers of puts and calls (a strap if more calls, a strip if more puts). </li></ul></ul><ul><ul><li>A ladder purchases an additional deep out-of-the-money option. </li></ul></ul><ul><li>A straddle is the purchase or sale of a put and call with the same strike and expiry. </li></ul><ul><li>A strangle is the purchase or sale of a put and call, with the same expiry but different strikes. </li></ul>
- 67. Combination Strategies, 2 <ul><li>A butterfly spread can be comprised of all puts or all calls, positioned at three strikes. A long butterfly may buy calls at +1 and -1 (“the wings”) and sell two ATM (“the body”). If the market expires at the middle strike, the participant owns an in-the-money call, another that expires worthless and keeps the entire premium for writing two calls that expire worthless. </li></ul><ul><li>An iron butterfly uses the same three strikes, but is comprised of puts and calls. Strikes for a long strangle bracket the strike for the short straddle, e.g., shorts in the ATM put and call, long -1 put and long +1 call. </li></ul><ul><li>The condor/iron condor positions use four option strikes, with the middle options spread apart, creating a price range in which the trade can reach maximum profit potential. </li></ul>
- 68. 7. Option Pricing <ul><li>The value of an option is merely the present value of its expected payoffs. </li></ul><ul><ul><li>The pricing approach for options, like that of terminal instruments, is derived from the costs associated with replicating the payoffs. </li></ul></ul><ul><ul><li>The premium is the ex ante compensation from the buyer to the seller that ensures that the transaction has an initial zero net present value. </li></ul></ul>
- 69. The General Case <ul><li>The general case for pricing calls involves selling the call (C), holding delta (δ) units of the underlying asset (A), and borrowing an amount (B) for the period. </li></ul><ul><ul><li>Final asset value – options payout – borrowing cost = 0 </li></ul></ul><ul><ul><li>δuA – C u – Be r(T-t) = 0 (payout when the asset trades up) </li></ul></ul><ul><ul><li>δdA – C d – Be r(T-t) = 0 (payout when the asset trades down) </li></ul></ul><ul><li>This equality, looked at another way, requires that the premium received plus the amount borrowed be equal to the amount required to fund the holding of the asset. </li></ul>
- 70. Binomial Pricing of a Call <ul><li>C = δA + B </li></ul><ul><li>δ = (C u – C d ) /A(u – d) </li></ul><ul><ul><li>C u and C d are the values of the call where the asset price has risen (u) and fallen (d) </li></ul></ul><ul><ul><li>A(u – d) is the range in the values of the asset </li></ul></ul><ul><li>B = (dC u – uC d ) / e r(T-t) (u – d) </li></ul><ul><ul><li>Borrowed amount equals the product of the lower asset value and the call’s value when the price has risen less the product of the higher asset value and the call’s value when the price has fallen all divided by the range of asset prices scaled up for the time and interest rate. </li></ul></ul>
- 71. Binomial Pricing of a Call, 2 <ul><li>One-period C = [ρC u + (1 – ρ)C d ] / e r(T-t) </li></ul><ul><li>Two-period C = [ρ 2 C uu + 2ρ(1 – ρ)C ud + (1 – ρ) 2 C dd ] / e 2r(T-t) </li></ul><ul><li>Three period C = [ρ 3 C uuu + 3x2ρ(1 – ρ)C uud + 3ρx2(1 – ρ)C udd + (1 – ρ) 3 C ddd ] / e 3r(T-t) </li></ul>
- 72. Inputs for Binomial Pricing <ul><li>u = e σ x √(T – t) / N </li></ul><ul><ul><li>The size of upward steps (a scalar, so that in two periods, price = u 2 ; in three steps, price = u 3 ) </li></ul></ul><ul><li>d = e -σ x √(T – t) / N </li></ul><ul><ul><li>The size of downward steps </li></ul></ul><ul><ul><li>By definition, d = 1 / u </li></ul></ul><ul><li>a = e r x ((T – t) / N) </li></ul><ul><ul><li>Drift? </li></ul></ul><ul><li>ρ = (a – d) / (u – d) </li></ul><ul><ul><li>Probability of an up move is the ratio of the (drift – down move) / range </li></ul></ul>
- 73. Put – Call Parity Theorem <ul><li>The Law of One Price states that any two assets, or combination thereof, that have the same payoffs must trade at the same price in an efficient market. </li></ul><ul><ul><li>As a put exposure can be created synthetically from the underlying and calls, the Law will apply here. </li></ul></ul><ul><ul><li>Asset + Put = Call + PV(Exercise Price) </li></ul></ul><ul><ul><li>Put = Call + PV(Exercise Price) – Asset </li></ul></ul>
- 74. 8. The Black-Scholes Model <ul><li>The Black-Scholes model provides a closed-form analytic solution for options pricing. Its limitations are in its assumptions: </li></ul><ul><ul><li>The variance of returns (volatility) is constant over the life of the option. </li></ul></ul><ul><ul><li>Interest rates are constant. Funds can be borrowed or lent at the risk-free rate. </li></ul></ul><ul><ul><li>The asset price moves continuously (no jumps). </li></ul></ul><ul><ul><li>Asset returns follow a log-normal distribution. </li></ul></ul><ul><ul><li>No transaction costs or intermediate payments. </li></ul></ul><ul><ul><li>All assets are perfectly divisible. </li></ul></ul>
- 75. The Formula for a Call C = U 0 N(d 1 ) – Ke -r(T-t) N(d 2 ) where σ √ (T – t) σ √ (T – t) ln + [r + ] (T – t) ln + [r – ] (T – t) d 1 = U 0 K d 2 = U 0 K σ 2 2 σ 2 2 Note that the only difference between the d 1 and d 2 formulae is in this sign. Therefore, d 2 = d 1 – σ 2 (T – t) σ √ (T – t) = d 1 – σ √ (T – t)
- 76. Formula Components <ul><li>The first term U 0 x N(d 1 ) is the holding in the underlying asset. The second term Ke -r(T-t) x N(d 2 ) is the amount of borrowed funds. </li></ul><ul><li>N(d) are the cumulative normal probabilities (i.e., 0 =< ρ <= 1) based on a normal distribution with a mean of zero and a standard deviation of one. </li></ul><ul><ul><li>N(0) = 0.5 </li></ul></ul><ul><ul><li>N(d 1 ) + N(-d 1 ) = 1 </li></ul></ul><ul><ul><li>If d 1 = 1.5, N(d 1 ) = 0.93319 and N(-d 1 ) = 0.06681 </li></ul></ul><ul><ul><li>Therefore, U 0 N(d 1 ) – U 0 = U 0 N(-d 1 ) </li></ul></ul><ul><ul><li>Likewise, PV(K) – PV(K)N(d 2 ) = PV(K)N(-d 2 ) </li></ul></ul>
- 77. Deriving the Formula for the Put <ul><li>If P + U 0 = C + PV(K), and C = U 0 N(d 1 ) – PV(K)N(d 2 ), then: </li></ul><ul><ul><li>Isolating the put value, </li></ul></ul><ul><ul><li>P = U 0 N(d 1 ) – PV(K)N(d 2 ) + PV(K) - U 0 </li></ul></ul><ul><ul><li>Rearranging the terms, </li></ul></ul><ul><ul><li>P = PV(K) – PV(K)N(d 2 ) + U 0 N(d 1 ) - U 0 </li></ul></ul><ul><ul><li>Simplifying (see previous slide), </li></ul></ul><ul><ul><li>P = PV(K)N(-d 2 ) - U 0 N(-d 1 ) </li></ul></ul>
- 78. Time and Rate Inputs <ul><li>T – t: this is expressed as fractions of a year. </li></ul><ul><ul><li>Sometimes a distinction is made between trading days (~252/ year) and calendar days, based on evidence that prices are much less volatile when markets are closed. </li></ul></ul><ul><li>Risk-free rate: this is usually the US Treasury bill rate, which is always quoted in terms of its discount: </li></ul><ul><ul><li>T-bill = 100 – i d x [(T – t) / basis], where i d is the quoted discount rate and basis is usually 360 days. At 3% and 90 days to go, T-bill = 0.9925. Relative is the inverse = 1.00756, to the exponent 360/90 = 1.03058 </li></ul></ul><ul><ul><li>The model uses the continuously compounding rate, where r = ln(1 + r s ) = ln(1.03058) = 0.03013 </li></ul></ul>
- 79. Volatility <ul><li>Calculate periodic simple return: </li></ul><ul><li>r s = (P t – P t-1 )/ P t-1 </li></ul><ul><li>Convert to continuously compounding rates: </li></ul><ul><li>r c = ln(1 + r s ) </li></ul><ul><li>Calculate variance and standard deviation: </li></ul><ul><li>σ 2 = Σ(r t – r mean ) 2 / (n – 1) </li></ul><ul><li>σ = √σ 2 </li></ul><ul><li>Annualize by multiplying by the square root of the frequency, e.g., by √ 52 if the data was weekly. </li></ul><ul><li>The sample’s standard error diminishes as the sample size increases: SE = σ / √ 2n </li></ul>
- 80. Implied volatility <ul><li>This works the Black-Scholes model in reverse: taking the observable market price, strike price, interest rate and time to expiry to derive volatility. </li></ul><ul><li>An approximation: </li></ul><ul><li>σ ≈ C √ 2π / U √(T – t) </li></ul><ul><li>With a call priced at 6.5, an underlier at 100 and three months to go: </li></ul><ul><li>σ ≈ (6.5 x 2.5066) / (100 x 0.5) = 0.326 </li></ul>
- 81. 9. The Greeks <ul><li>Delta (δ) refers to the option price sensitivity to changes in the underlying asset. </li></ul><ul><ul><li>This describes the rate of change in the option value to the change in the asset value. </li></ul></ul><ul><ul><li>The rate of change is not linear. It is highest when the option is at-the-money, and lowest when it is deeply in- or out-of-the-money. This relationship is also not symmetrical: the time value for a deep in-the-money call is comprised mostly of the benefit of deferring purchase until expiry. </li></ul></ul><ul><ul><li>The delta of a portfolio is the weighted sum of the positive and negative deltas. </li></ul></ul><ul><ul><li>In the Black-Scholes formulation, this is represented by N(d 1 ) for the call and N(d 1 )-1 for the put. </li></ul></ul>
- 82. Gamma <ul><li>Gamma (γ) is the rate of change in delta. </li></ul><ul><ul><li>This is the second derivative of option value to price. </li></ul></ul><ul><ul><li>This measures the steepness of the delta curve. Since delta experiences its greatest change when at-the-money, so too does gamma. </li></ul></ul>N’(d 1 ) γ = (U 0 ) σ √(T – t) where N’(d 1 ) = √ 2π 1 -0.5(d 1 ) 2 e √ 2π 1 = 0.398942
- 83. Gamma, 2 <ul><li>If an option is being hedged with other options with different conditions, then merely matching deltas will not fully hedge the exposure. This is because the values for the written and purchased options will not change at the same rates. </li></ul><ul><li>The gamma of the portfolio can be zeroed by buying or selling options to offset, as the ratio of portfolio gamma/ option gamma requires. This, however, will also change the delta. Delta neutrality is then restored by buying or selling the underlying. </li></ul>
- 84. Theta, Lambda <ul><li>Theta (θ) is the sensitivity of the option to time. </li></ul><ul><ul><li>As the market price and futures price converge, time value decays – more quickly for at-the-money options, which have the highest time value. </li></ul></ul><ul><li>Lambda (λ) is the percentage change in the option price for a given percentage change in the asset price. </li></ul><ul><ul><li>This ‘gearing’ is an attractive feature for speculators. The highest lambda is for deep out-of-the-money options. </li></ul></ul><ul><ul><li>(U/K)N(d 1 ) for the call; (U/K)N(d 1 )-1 for the put. </li></ul></ul>θ call = θ put = [ N’(d 1 ) – r K e –r (T-t) N(d 2 ) ] [ N’(d 1 ) + r K e –r (T-t) N(-d 2 ) ] U 0 σ 2√(T – t) U 0 σ 2√(T – t)
- 85. Rho, Vega <ul><li>Rho (ρ) is the option sensitivity to interest rates. </li></ul><ul><li>Vega (not really a Greek letter) and its aliases refer to the effect of changes in volatility on value. </li></ul><ul><ul><li>This is the single most important determinant of option value. </li></ul></ul><ul><ul><li>The calculation, for puts and calls: </li></ul></ul>ρ call = (T – t) K e –r (T- t) N(d 2 ) ρ put = (T – t) K e –r (T- t) N(-d 2 ) v = U 0 √(T – t) N’(d 1 )
- 86. Sensitivity Summary Every option is a race between gamma and theta. Position Delta Gamma Theta Rho Vega Long call + + - + + Long put - + - - + Short call - - + - - Short put + - + + -
- 87. 10. Extensions to Option Pricing <ul><li>The Black – Scholes model can be modified to price options: </li></ul><ul><ul><li>When there is a value leakage (e.g., a dividend payment) </li></ul></ul><ul><ul><li>On an exchange rate between currencies </li></ul></ul><ul><ul><li>When the holder can ‘lock in’ an interest rate </li></ul></ul><ul><ul><li>When there is a possibility of early exercise </li></ul></ul>
- 88. Value Leakage <ul><li>The payment of a dividend on a stock reduces the stock’s value. If this occurs during the option period, it will increase the value of a put and decrease the value of a call. </li></ul><ul><ul><li>The model could employ the ex-dividend price: </li></ul></ul><ul><ul><li>S* = S – de -r(k – t) </li></ul></ul><ul><ul><li>subtracting from the price the present value of the dividend. </li></ul></ul>
- 89. Continuous Dividend Adjustment <ul><li>When dividends are being paid continuously (as on a stock index), Merton adjusts the price of the stock price, U, for the continuous dividend yield, q . </li></ul><ul><ul><li>Price U term becomes U x e –q(T – t) </li></ul></ul><ul><ul><li>In d 1 sub-equation, </li></ul></ul><ul><ul><li>r + σ 2 /2 term becomes r – q + σ 2 /2 </li></ul></ul><ul><ul><li>In d 2 sub-equation, </li></ul></ul><ul><ul><li>r - σ 2 /2 term becomes r – q – σ 2 /2 </li></ul></ul>
- 90. Currency Options <ul><li>The holder of a foreign currency is the recipient of payments equal to the risk-free interest rate in the foreign currency, r c . </li></ul><ul><ul><li>Replace the continuous dividend yield, q, in the previous example with r c . The domestic rate will be r. </li></ul></ul><ul><li>Note that calls and puts on currencies are the same: </li></ul><ul><ul><li>The call is right to receive the foreign currency in exchange for the base currency. </li></ul></ul><ul><ul><li>The put is the right to receive the base currency in exchange for the foreign currency. </li></ul></ul>
- 91. Currency Options, 2 <ul><li>If we were pricing the option against the forward rate rather than the spot rate, the continuous form of the cost of carry model would give the forward rate F at time T: </li></ul><ul><ul><li>F = FX e (r-rc)(T-t) </li></ul></ul><ul><ul><li>Substituting, </li></ul></ul><ul><ul><li>C = [F x N(d 1 ) – K x N(d 2 )] e -r(T-t) </li></ul></ul><ul><ul><li>P = [K x N(-d 2 ) – F x N(-d 1 )] e -r(T-t) </li></ul></ul><ul><ul><li>In the d 1 and d 2 sub-equations, the ln(U/K) term becomes ln(F/K) and the r term is removed. </li></ul></ul>
- 92. Options on Futures, Commodities <ul><li>For futures, the foreign currency risk-free yield is replaced with the continuously-compounding return a , representing the net cost of carry. </li></ul><ul><li>For commodities, a = r + w – d where </li></ul><ul><ul><li>r is the risk-free rate </li></ul></ul><ul><ul><li>w is storage costs, insurance and deterioration expressed as a yield </li></ul></ul><ul><ul><li>d is the convenience yield </li></ul></ul><ul><li>If the relationship between the underlying and its corresponding future price incorporate these returns: </li></ul><ul><ul><li>that is, if [ln(F/U)/(T – t )] = r + w – d </li></ul></ul><ul><ul><li>then d = r + w – [ln(F/U)/(T – t)] </li></ul></ul><ul><ul><li>and we replace the U term in the model and sub-equations with Ue (w - d)t </li></ul></ul>
- 93. Early Exercise <ul><li>The fundamental question is whether the capture of the dividend through early exercise justifies the loss of the remaining time value. </li></ul><ul><ul><li>Optimal when d > K (1 – e –r(T – t) ) </li></ul></ul><ul><ul><li>Not optimal when d <= K (1 – e –r(T – t) ) </li></ul></ul><ul><ul><li>The attraction increases as the option approaches expiry. </li></ul></ul><ul><li>The pseudo-American adjustment for dividend sees the call option is valued twice: </li></ul><ul><ul><li>Once for a European-style option expiring at time T, adjusted for the leakage </li></ul></ul><ul><ul><li>Once for the same option expiring at the time of the last dividend </li></ul></ul><ul><ul><li>The higher of the two prices is deemed to be the American option. </li></ul></ul>
- 94. Early Exercise, 2 <ul><li>Early exercise may be attractive for puts since it liberates value that can then be reinvested. </li></ul><ul><ul><li>This condition arises when the put is deep in-the-money, when interest rates rise and when volatility falls. </li></ul></ul><ul><ul><li>If the asset falls below some critical value relative to the strike price, the put value is simply the option’s intrinsic value. Above that critical value, there is a value adjustment to the European exercise put price. </li></ul></ul>
- 95. American Value Adjustment <ul><li>The AVA for a put is based on its strike relative to that critical value where the put is worth only its intrinsic value. </li></ul><ul><li>For any put for which we wish to calculate the AVA, we first calculate three constants: </li></ul><ul><ul><li>M = 2r / σ 2 </li></ul></ul><ul><ul><li>W = 1 – e -r(T-t) </li></ul></ul><ul><ul><li>q 1 = ½ [ -(M – 1) - √ (M – 1)2 + 4 ( ) ] </li></ul></ul>M W
- 96. American Value Adjustment, 2 <ul><li>In the next step, in-the-money strike values U A are tested iteratively until K – U A = Put euro + a 1 (that is, until the exercised price is equal to the theoretical price for the European option plus the a 1 value). </li></ul><ul><li>a 1 = - ( ) N(d 1 A ) </li></ul><ul><li>Then* AVA = (U / U A ) </li></ul>q 1 U A q 1 * In neither the example in the text nor the Case Study can you use these methods to come to the proffered result.
- 97. Interest Rate Options (IROs) <ul><li>It is surely illogical to use the Black-Scholes option-pricing model, with its assumption of a constant risk-free interest rate, to value an option that is based on changes in interest rates! </li></ul><ul><li>Interest rates do not follow a lognormal distribution – there is a strong reversion to the mean. </li></ul><ul><li>Price volatility falls as the instrument moves toward maturity – the pull to par. This again runs counter the assumptions of the Black-Scholes model. </li></ul>
- 98. Interest Rate Forwards & Futures <ul><li>The methodology using a forward rate is the same as on the slide “Currency Options, 2”, above. </li></ul><ul><ul><li>The forward rates must first be converted to prices. </li></ul></ul><ul><li>Likewise, futures are quoting prices and not rates. </li></ul><ul><ul><li>To have a call on the interest rate requires that one own the put on the futures price . </li></ul></ul><ul><ul><li>As the writer often does receive the premium upfront, the formulae simplify to be based on stochastic rates, without an imbedded interest rate assumption. </li></ul></ul><ul><ul><li>C = FP ∙ N(d 1 ) - K ∙ N(d 2 ) </li></ul></ul><ul><ul><li>P = K ∙ N(-d 2 ) - FP ∙ N(-d 1 ) </li></ul></ul>
- 99. Pricing a Fraption (option on a Forward Rate Agreement) <ul><li>There are two time periods to consider: the life of the option and the interest rate protection period. </li></ul><ul><li>The rates are converted to values: </li></ul><ul><ul><li>The notional principal is multiplied by the protection period to get the cap notional principal </li></ul></ul><ul><ul><li>the underlying (U) is the interest payable on the cap notional principal at the market rate </li></ul></ul><ul><ul><li>the strike (K) is the interest payable on the cap notional principal at the FRA forward rate. </li></ul></ul><ul><li>These values are then inserted into the Black-Scholes model as modified to price an option on a forward. </li></ul>
- 100. Adjusting Bond Volatility <ul><li>We expect price volatility on a bond to decrease as it approaches maturity. </li></ul><ul><ul><li>Schaefer and Schwartz show that the relationship of bond duration to volatility is constant. </li></ul></ul><ul><ul><li>Therefore, if observed volatility is adjusted by duration, one can arrive at the correct forward volatility. </li></ul></ul><ul><ul><li>κ (kappa) = ~σ /(U α-1 )∙D , where ~σ is the observed volatility , U is the underlying, D is duration and α is estimated at 0.5 </li></ul></ul><ul><ul><li>The corrected forward volatility is then σ = (κ ∙ U α-1 ) D </li></ul></ul>
- 101. Complex Options <ul><li>Payoff : digital (also called Boolean, asset-or-nothing, and cash-or-nothing) </li></ul><ul><li>Singularity : premium is paid only if the option expires in-the-money </li></ul><ul><li>Path dependency : barrier (knock outs, knock ins); lookback (buy or sell at best price in period); Asian (average in period); ratchet (strike is periodically set at prevailing price, thus capturing intrinsic value); shout (holder announces to the writer what the strike price will be) </li></ul><ul><li>Compound : include cacall – a call on a call, a caput – a call on a put; and chooser – whether the option in a future period will be a call or a put. </li></ul>
- 102. Complex Options, 2 <ul><li>Multivariate : basket (portfolio v. strike); rainbow (best of or worst of assets listed, or a spread between assets); quantity-adjusted or ‘quanto’ (e.g., struck in one currency but paid in another) </li></ul><ul><li>Timing : forward start (strike to be set versus asset value at a future date); Bermuda (series of option exercise dates) </li></ul><ul><li>Embedded or ‘embeddoes’: as part of another financial instrument, e.g., a range floating-rate note comprised of a written cap and purchased floor </li></ul>
- 103. 11. Hedging and Insurance <ul><li>The basic principle of hedging is to match two opposing sensitivities in such a way that their value changes cancel out. </li></ul><ul><ul><li>Terminal instruments are typically used to hedge price risk or to make tactical allocations among assets. </li></ul></ul><ul><ul><li>Options, with their asymmetric payoff, have an element of insurance. They can be used to hedge a contingent income stream or liability, or to take a view, especially on future volatility. </li></ul></ul>
- 104. Hedging Costs <ul><li>For hedging to be cost-effective: </li></ul><ul><ul><li>it has to eliminate a large part of the change in value of the underlying position, and </li></ul></ul><ul><ul><li>it must do so at a lower cost than alternative approaches. </li></ul></ul><ul><li>Hedging is costly in that the user usually pays the bid – ask spread on transactions. In order to minimize these costs, the user should take advantage of natural offsets and portfolio effects, and hedge only the net position. </li></ul>
- 105. Hedge Ratio <ul><li>A forward contract may provide a “perfect hedge”, albeit with a loss of flexibility and by taking on credit exposure. </li></ul><ul><li>In using an imperfect hedge, the objective is to find the minimum-risk hedge ratio (h). </li></ul><ul><ul><li>ΔP cash + hΔP hedge = 0 </li></ul></ul><ul><ul><li>As a rule, the naïve one-to-one approach (i.e., h = 1) works best when the cash position is nearly equivalent to the characteristics of the futures. The wider the discrepancy, the greater is the hedging error that will result. </li></ul></ul>
- 106. Hedge Ratio, 2 <ul><li>If the two sides are less than perfectly correlated, we can find the minimum-variance hedge ratio of the portfolio by: </li></ul><ul><ul><li>h = ρ cash,hedge x (σ cash / σ hedge ) </li></ul></ul><ul><ul><li>where ρ cash,hedge is the correlation coefficient between the cash instrument and the hedging instrument and σ cash and σ hedge are the standard deviations of the cash and hedge instruments. This defines the regression line . </li></ul></ul>
- 107. Strip and Stack Hedging <ul><li>If an exposure covers more than one futures contract period, the ideal solution is to match the hedging instrument to the exposure period, a process known as a strip hedge . </li></ul><ul><li>The alternative stack hedge approach is used when there is no liquidity in the contracts with longer time to expiry. The procedure is to ‘stack up’ the hedge using the nearby contracts and roll forward the position, reducing the hedge as required over the exposure period. </li></ul><ul><ul><li>This makes an implicit assumption that all changes in the yield curve are parallel shifts. </li></ul></ul>
- 108. Convergence in Interest Rates <ul><li>An interest rate future is of obligations to buy or sell the three-month rate prevailing at contract expiry. </li></ul><ul><ul><li>The obligation is priced at 100 – i d x [(T – t) / basis], where i d is the quoted discount rate and basis is usually 360 days. If the applicable was 6.0%, a the contract would be priced at 98.50. </li></ul></ul><ul><ul><li>The difference between the contract rate and the spot rate is the current basis. As the contract rate must converge with the spot rate at expiry, this basis must shrink to zero. </li></ul></ul><ul><ul><li>If the forward curve were upward sloping, then even in the absence of a change in rates, we expect the futures price to increase with the passage of time. </li></ul></ul>
- 109. Parallel Shifts and Twists <ul><li>If a futures contract had been used to hedge an interest rate exposure, a parallel shift in the forward curve would not affect the outcome. </li></ul><ul><li>A twist changes the shape of the yield curve, and thus the basis relationship. The effect of convergence may create a basis gain or loss. The solution lies in setting up a spread position to mitigate the risk. </li></ul>
- 110. Interest Rate Spreads <ul><li>Buy the basis: </li></ul><ul><ul><li>Go “long” the spread: buy nearby contract and sell the deferred </li></ul></ul><ul><ul><li>Gain if the nearby increases more quickly / decreases more slowly than the deferred (curve steepens) </li></ul></ul><ul><li>Sell the basis: </li></ul><ul><ul><li>Go “short” the spread: sell the nearby contract and buy the deferred. </li></ul></ul><ul><ul><li>Gain if the deferred increases more quickly / decreases more slowly than the nearby (curve flattens) </li></ul></ul>
- 111. Interest Rate Spreads, 2 <ul><li>The portion of the original position to spread is: (time to expiry of deferred contract – time to maturity of hedge) / time between contracts, e.g. (4 – 2) / 3. </li></ul><ul><li>If the original hedge was a short, then the spread will short the nearby and vice versa. </li></ul><ul><ul><li>For a 40 contract short position in the deferred, the spread would be 2/3 x 40 = ~27, which is -27 in the nearby and +27 in the deferred (net -27 nearby and -13 deferred) </li></ul></ul>
- 112. Dynamic Hedging <ul><li>A share portfolio’s market risk is modified using stock index futures as a hedge. </li></ul><ul><ul><li>A portfolio’s sensitivity to market effects is known as its beta (β). </li></ul></ul><ul><ul><li>The beta is found by regressing the portfolio return against the index return. </li></ul></ul><ul><ul><li>N stock index futures = β </li></ul></ul>Value of portfolio Value of futures contract
- 113. Modifying Interest-Rate Sensitivity <ul><li>Calculate the price value of a basis point (PVBP) for the asset or portfolio: </li></ul><ul><li>(D x MVP x .0001) / (1 + y/f), where: </li></ul><ul><ul><li>D is Macaulay duration </li></ul></ul><ul><ul><li>MVP is the market value of the portfolio </li></ul></ul><ul><ul><li>y is the annualized yield to maturity </li></ul></ul><ul><ul><li>f is the frequency of payments per year </li></ul></ul><ul><ul><li>This effectively measures cost using the modified duration (sensitivity measure) for a 1 bp move. </li></ul></ul>
- 114. Modifying Interest-Rate Sensitivity, 2 <ul><li>Divide the difference between the starting and target positions by the PVBP for the futures contract, which should include an adjustment for the cheapest-to-deliver (CTD) eligible bond versus the notional underlying bond. </li></ul><ul><li>The result indicates the number of futures contracts required to effect the modification. </li></ul><ul><ul><li>If increased exposure was desired, the sign is positive; if a hedge, then negative. </li></ul></ul><ul><ul><li>A futures contract protects the nominal value for a fixed period of three months. Since time is a factor in the value, if the underlying is more or less than that three months, the position must be scaled accordingly. </li></ul></ul>
- 115. Modifying Interest-Rate Sensitivity, 2 <ul><li>Divide the difference between the starting and target positions by the PVBP for the futures contract. </li></ul><ul><li>The result indicates the number of futures contracts required to effect the modification. </li></ul><ul><ul><li>If increased exposure was desired, the sign is positive; if a hedge, then negative. </li></ul></ul><ul><ul><li>A futures contract protects the nominal value for a fixed period of three months. As time is a factor in the cost/ revenue of an interest rate, an underlying of more or less than that three months will require that the position be scaled accordingly. </li></ul></ul>
- 116. Portfolio Insurance <ul><li>With portfolio insurance, a portion of the potential return from an uninsured portfolio is surrendered in order to guarantee a minimum portfolio value. </li></ul><ul><li>Rather than purchase puts, a floor can be replicated with a synthetic put strategy. This involves modifying the proportion of the portfolio held between a safe asset (with zero sensitivity to the market) and risky assets (those exposed to market fluctuations). </li></ul><ul><li>No cash outlays are required, although there are transaction costs. </li></ul>
- 117. Portfolio Insurance, 2 <ul><li>As prices increase, more assets are transferred from the risk-free asset to the risky asset. As prices decline, more funds are placed in the risk-free asset. </li></ul><ul><li>In constant proportions portfolio insurance (CPPI), </li></ul><ul><li>Value in the risky asset = κ (Value of portfolio – Value of floor) </li></ul><ul><li>where κ is a multiplier that determines how quickly assets are transferred (higher being faster) </li></ul><ul><li>The strategy requires the resetting of the proportion at a fixed percentage change in the portfolio or in the cushion (difference between portfolio and floor), e.g.., at intervals of 5%. </li></ul>
- 118. Removing Market Risk <ul><li>A stock may be the target of a takeover bid. This development would increase its market price without affecting the broader market. </li></ul><ul><li>While holding the shares, there is a generalized risk of loss. </li></ul><ul><li>A possible market loss could be hedged for the beta times the value of the holding in the target. If one million shares were trading at 95 p, and the beta was 1.5, an offsetting market exposure = £ 950,000 x 1.5 = £1,425,000. </li></ul><ul><li>With the FTSE index at 3891, and each point for a futures contract worth £ 10, this is a value of £ 38,910 / contract. Dividing into the exposure gives the contract size of an offsetting position: £1.425m / 38,910 = 36.6 contracts (~37). </li></ul><ul><li>Buy 37 out-of-the-money puts. </li></ul>
- 119. Removing Market Risk, 2 <ul><li>A delta hedge takes the strategy further by selling calls or buying puts with an equivalent underlying value. </li></ul><ul><li>The product of the beta and portfolio value is divided by the notional value of the futures contract, as before. This number is then divided by the delta corresponding to the option’s strike. </li></ul><ul><ul><li>£10 million portfolio with a beta of 1.2; FTSE index at 3891.1 (each point worth £10) and an October 3725 put with a delta of -0.30 and priced at 15.5 = 12 000 000 / (38 911 x 0.30) = 1028 puts x £10 x 15.5 = £ 159 430 outlay </li></ul></ul>
- 120. 12. Using the Product Set <ul><li>Terminal Instruments </li></ul><ul><li>(Forwards, Futures, Swaps) </li></ul><ul><li>Benefits </li></ul><ul><li>No upfront cost. </li></ul><ul><li>Forward will provide tailored end date and amounts. </li></ul><ul><li>Costs </li></ul><ul><li>Lost opportunity to participate in favorable price move. </li></ul><ul><li>Transaction can be at only one price. </li></ul><ul><li>Futures may not provide exact offset and have basis risk. </li></ul><ul><li>Credit risk on forwards and swaps. </li></ul>
- 121. The Product Set, 2 <ul><li>Options </li></ul><ul><li>Benefits </li></ul><ul><li>Provide one-way protection against adverse changes. </li></ul><ul><li>Allow participation in favorable price move. </li></ul><ul><li>Can be priced at one or more strike prices. </li></ul><ul><li>Costs </li></ul><ul><li>Upfront premium payment (although compound options or a vertical spread may reduce cost). In currency markets, this spread is called a ‘cylinder’. </li></ul><ul><li>Credit risk upon exercise. </li></ul>
- 122. Do-It-Yourself Forward <ul><li>The key is to the transaction is to have the price determined today , even if components have not run their course until the future period. </li></ul><ul><li>Company has separate obligation to repay the loan; both loan and deposit are subject to credit risk </li></ul><ul><li>These transactions can affect the firm’s balance sheet and accounting ratios, and thus their credit rating </li></ul>
- 123. Do-It-Yourself Forward, 2 <ul><li>e.g., British company is owed USD 5.0 million in six months, and wishes to hedge. </li></ul><ul><ul><li>Discount USD receivable to PV using six-month USD offer rate. (Don’t forget that the rate quoted is annualized, so use r x ½ ) </li></ul></ul><ul><ul><li>Take USD loan for PV amount at USD offer rate. This rate has to be “corrected” from a 360 to a 365 day year – which factor operates only on the rate r and not the “1 +” part. </li></ul></ul><ul><ul><li>Convert to £ using bid side of spot rate </li></ul></ul><ul><ul><li>Deposit £ at £ bid rate </li></ul></ul>
- 124. 13. Asset-Liability Management <ul><li>Against a one-year loan that is repriced quarterly, a bank could borrow funds: </li></ul><ul><ul><li>on a one-year term that is repriced quarterly: match funded </li></ul></ul><ul><ul><li>on a three-month term, and then seek new funding at the end of that term: funding gap </li></ul></ul><ul><ul><li>on a one-year term at a fixed rate: long funded </li></ul></ul><ul><ul><li>on a one-year term that is repriced monthly: short funded </li></ul></ul>
- 125. Rate Sensitivity Gap <ul><li>The exposure to interest rates – the rate sensitivity gap – is the difference in the repricing characteristics of the assets to the liabilities. </li></ul><ul><ul><li>Positive gap : assets are repricing before liabilities </li></ul></ul><ul><ul><li>Negative gap : liabilities are repricing before assets </li></ul></ul><ul><li>While the bank may seek to increase / decrease its rate-sensitive assets (RSA) or liabilities (RSL), their customers, based on their own expectations, are trying to shift their interest rate risk to the bank. </li></ul>
- 126. Maturity-Gap Approach <ul><li>The bank’s assets and liabilities are bucketed by time period. Within each bucket: </li></ul><ul><ul><li>Balance sheet assets </li></ul></ul><ul><ul><li>Balance sheet liabilities </li></ul></ul><ul><ul><li>Off balance-sheet instruments </li></ul></ul><ul><ul><li>Net </li></ul></ul><ul><li>This would show the size and periods in which the bank is exposed to interest rate changes. </li></ul><ul><ul><li>While it shows the volume of assets exposed, it does not show the volume of that exposure. </li></ul></ul>
- 127. Cash-Gaps Method <ul><li>This is a refinement of the maturity gap approach. </li></ul><ul><ul><li>The net cash flows (positive or negative) are accumulated in each bucket. </li></ul></ul><ul><ul><li>These are present-valued using the zero-coupon rate. </li></ul></ul><ul><ul><li>The present value is recalculated for a level (usually 1 bp) shift or rotational shift. </li></ul></ul><ul><li>This requires sophisticated and accurate capture of the cash flows from the underlying assets and liabilities, but has superior accuracy because it is modeling these directly. </li></ul>
- 128. The Funding-Gap Method <ul><li>Within each time bucket, we calculate the net of assets less liabilities. </li></ul><ul><ul><li>The PV of the net is calculated assuming a zero-coupon interest rate. </li></ul></ul><ul><ul><li>The PV is recalculated for parallel shift (1 bp higher) and rotational shift (1 bp higher at, say, 5 years). </li></ul></ul><ul><ul><li>The sum across buckets indicates the level risk of a 1 bp change. </li></ul></ul>
- 129. Level and Rotational Shift <ul><li>If we have a level shift risk of £120,000 and a rotational shift risk of £145,000: </li></ul><ul><ul><li>If the risks are perfectly correlated, the total risk is £120,000 + £145,000 = £265,000 </li></ul></ul><ul><ul><li>If the risks are perfectly uncorrelated, the total risk is </li></ul></ul><ul><ul><li>√ (120,000) 2 + (145,000) 2 = £188,215 </li></ul></ul>
- 130. ALM for Real Assets <ul><li>Foreign currency cash flows are projected </li></ul><ul><li>The sensitivity of cash flows and factor costs are estimated versus the base currency </li></ul><ul><ul><li>If a commodity was repriced immediately in the foreign currency due to changes in the base currency, it was given a zero rating (e.g., oil or refined products “shadow priced” in USD) </li></ul></ul><ul><ul><li>If a commodity was insensitive to such changes, it was given a 100 rating (e.g., local labor costs) </li></ul></ul><ul><ul><li>This analysis would also attempt to estimate indirect exposures due to responses from competitors, suppliers and customers </li></ul></ul>
- 131. ALM for Real Assets, 2 <ul><li>Future cash flows in the foreign currency were adjusted for their currency sensitivity by multiplying them by their sensitivity factor. </li></ul><ul><ul><li>This created a division of exposures to the local currency and the base currency (unexposed) </li></ul></ul><ul><ul><li>A foreign currency exposure of $150 million and a sensitivity of 60 produces a foreign currency exposure of $90 million and a base currency exposure of $60 million. </li></ul></ul><ul><li>A hedging decision is made on the net foreign exposure only. </li></ul>
- 132. 14. Duration Measures <ul><li>Macaulay’s duration is the weighted average maturity (the book calls it the discounted mean term ) of the cash flows in an asset or liability, with the weights given by the present value of the cash flows. </li></ul><ul><ul><li>A zero coupon bond would have a duration equal to its term, since all of the value in the bond is paid at maturity. </li></ul></ul><ul><ul><li>A bond with intermediate cash flows will have a shorter duration. How much shorter depends on the rates used to discount the flows: higher interest rates imply shorter duration and lower rates imply higher duration. </li></ul></ul><ul><ul><li>Likewise, the larger the periodic flows relative to the total, the shorter the duration and vice versa: the coupon effect . </li></ul></ul>
- 133. Loan Types <ul><li>Bullet : repays all the principal at maturity </li></ul><ul><li>Amortizing ( annuity ): level payments are made in each period consisting of both interest and principal </li></ul><ul><li>Balloon : initial periods are interest only and later periods include both interest and principal </li></ul><ul><li>Zero-coupon (“zero”): principal and all of the interest is repaid at maturity </li></ul>
- 134. Calculating Duration <ul><li>A three year 10% coupon bond has cash flows of 10, 10, and 110. If the discount rate is 8%, the PV of these cash flows = (10/1.08) + (10/1.08 2 ) + (110/1.08 3 ) = 9.2593 + 8.5734 + 87.3215 = 105.1542 </li></ul><ul><li>Duration = [(9.2593 x 1) + (8.5734 x 2) + (87.3215 x 3)] / 105.1542 = 2.7424 years </li></ul><ul><li>If the cash flows were uneven, each can be discounted as if it were a zero of that maturity. </li></ul>
- 135. Calculating Duration, 2 (1 + r t ) n+1 – (1 + r t ) – (r t x n) 100 x n r t 2 (1 + r t ) n (1 + r t ) n Duration = PV ] + C [ coupons principal PV i = C [ ] + 100 (1 + r i ) n r i 1 - (1 + r i ) n 1
- 136. Calculating Duration, 3 <ul><li>If the frequency of the (same) payments was doubled, the duration would be halved. A quarterly payment would have 0.25 x duration of an instrument with the same payments annually. </li></ul>
- 137. Modified Duration <ul><li>Macaulay’s duration assumes continuous compounding. Modified duration is an adjustment to reflect discrete compounding intervals. </li></ul><ul><li>Modified duration = </li></ul><ul><li>Macaulay’s duration /(1 + (r i / f )) </li></ul><ul><ul><li>If the Macaulay’s duration is 4.46 for an instrument with annual payments when discounted at 6 percent, the modified duration is: 4.21 = 4.46 / (1.06) </li></ul></ul><ul><ul><li>As the frequency of the payments gets larger, the smaller the adjustment due to discrete intervals. If payments were monthly, modified duration is: 4.438 = 4.46 / (1.005) </li></ul></ul>
- 138. Modified Duration, 2 <ul><li>Modified duration is a measure of the value sensitivity of the present value of cash flows to a change in interest rates. The higher the modified duration, the greater the volatility of the PV. </li></ul><ul><ul><li>Percentage change in price = - Modified duration x Change in yield </li></ul></ul><ul><ul><li>One percent change in PV = 1 / Modified duration </li></ul></ul><ul><ul><li>If modified duration is 4.21, a 10 bp change in rates impacts prices: 0.421% = -4.21 x 0.10 </li></ul></ul><ul><ul><li>If the PV is 1000 for an instrument with a modified duration of 4.21, then a 10 bp change would increase / decrease the value by: $1,000 x 0.421% = +/- $4.21 </li></ul></ul>
- 139. Modified Duration, 3 CF 1 n(100 - ) r i 2 (1 + r i ) n (1 + r i ) n+1 [ 1 - ] + CF r i m.d. = PV
- 140. Error in Modified Duration Estimate <ul><li>Modified duration underestimates price increases and overestimates price decreases. When the asset is owned, this is when the rates go down and up, respectively. </li></ul><ul><ul><li>Although the value impact up or down is predicted to be the same size, a move of that size represents a larger proportional decrease than it does a proportional increase. </li></ul></ul><ul><ul><li>This arises because of an attempt to approximate a curved price / yield relationship with a tangent line. </li></ul></ul>
- 141. Convexity <ul><li>Duration measures the rate of change in value for a rate in change in interest rates. </li></ul><ul><ul><li>It is the first derivative of the curve, taken at the present value of the instrument. </li></ul></ul><ul><ul><li>Convexity is the second derivative, and therefore measures the rate of change in duration for a change in the underlying rate (like gamma). </li></ul></ul><ul><ul><li>A value change is going to be more precisely estimated when we consider that, not only has the underlying changed, but so has the rate of change in the duration. </li></ul></ul>
- 142. Convexity, 2 2 x CF 1 2 x CF x n n(n + 1)(100 – [ ]) (r i ) 3 (1 + r i ) n r i 2 (1 + r i ) n+1 (1 + r i ) n+2 [ 1 - ] + Convexity = CF r i 2 x PV This term = 0 when trading at par
- 143. Convexity, 3 <ul><li>Convexity percentage value change = Convexity x (change in r) 2 x 100% </li></ul><ul><ul><li>If modified duration predicts a change in value of =/- 4.21% for a 100 bp change, and convexity is 11.4594, to each we add 0.114594 % </li></ul></ul><ul><ul><li>Increase in rates: -4.21 + 0.114594 = -4.095406 </li></ul></ul><ul><ul><li>Decrease in rates: 4.21 + 0.114594 = 4.324594 </li></ul></ul><ul><li>As the discount rate increases and duration decreases, so too does the correction provided by convexity decrease. As with duration, convexity is greatest for a zero-coupon asset or liability. </li></ul>
- 144. Uses for Duration <ul><li>An anticipated interest rate increase would decrease the PV of assets, therefore an investor would attempt to reduce duration . </li></ul><ul><li>An anticipated interest rate decrease would increase the PV of assets, therefore an investor would attempt to increase duration . </li></ul><ul><li>The opposite cases apply for liabilities. </li></ul>
- 145. Hedging Interest-Rate Risk <ul><li>The asset manager will have a target for the mean term of liabilities that have to be matched in terms of the fund’s assets, process known as immunization . </li></ul><ul><ul><li>If interest rates are expected to rise, making asset duration < liability duration results in asset out-performance. </li></ul></ul><ul><ul><li>If interest rates are expected to fall, making asset duration > liability duration results in asset out-performance. </li></ul></ul>
- 146. Hedging Interest-Rate Risk, 2 <ul><li>An attempt to hedge a loan (an asset) would require a short position (a liability) whose interest rate sensitivity (modified duration) would be equal in size, and opposite in sign. </li></ul><ul><ul><li>-MD L / MD H = -ΔL / ΔH = -1 </li></ul></ul><ul><ul><li>This term is known as the duration hedge ratio . The ratio of the modified durations indicates the ratio of the present value of the loan to the corresponding hedge. </li></ul></ul>
- 147. Duration-Gap Analysis <ul><li>Duration-gap analysis is used to measure risk in terms of a financial institution’s equity value. </li></ul><ul><ul><li>This is the difference between the weighted average duration of the assets and that of the fixed liabilities, divided by the equity. </li></ul></ul><ul><ul><li>With assets of 2200 at a duration of 2.72, and liabilities of 2000 at a duration of 1.14, [(2200 x 2.72) – (2000 x 1.14)] / 200 = 18.52 </li></ul></ul><ul><ul><li>The sign indicates whether shareholders gain or lose from an increase in interest rates. </li></ul></ul>
- 148. Limitations of Methodology <ul><li>Assumes that there is a constant discount rate across all maturities. </li></ul><ul><li>Assumes that changes to the term structure of interest rates are parallel (no twisting). </li></ul><ul><li>Methodology assumes that there are no options in the portfolio, or that they are so far out of the money that they can be ignored. </li></ul>
- 149. Approximation Methods <ul><li>Duration: Subtract the PV associated with a small increase in rates from the PV associated with a similar decrease in rates, all divided by the product of two times the original price times the rate increment. </li></ul><ul><ul><li>(P - - P + )/ (2 x P 0 x δ i ) </li></ul></ul><ul><ul><li>For a 6 percent par loan, with a ten basis point change: (100.4224 – 99.5799) / (2 x 100 x 0.001) = 4.21245 </li></ul></ul>
- 150. Approximation Methods, 2 <ul><li>Convexity: Subtract two times the original price from the sum of the PVs associated with a small decrease / increase in rates, all divided by the product of two times the original price times the rate increment squared. </li></ul><ul><ul><li>(P - + P + - 2 P 0 ) / (2 x P 0 x δ i 2 ) </li></ul></ul><ul><ul><li>For a 6 percent par loan, with a ten basis point change: (100.4224 + 99.5799 - 200) / (2 x 100 x 0.001 2 ) = 11.45944 </li></ul></ul>
- 151. 15. Immunization / Liability Funding <ul><li>In asset-liability management (ALM), the central risk is that there may be insufficient assets available in the future to meet maturing liabilities. </li></ul><ul><ul><li>Insolvency (bankruptcy): the PV of assets is less than the PV of liabilities. </li></ul></ul><ul><ul><li>Illiquidity : The PV of assets is equal to or greater than liabilities, but these cannot readily be realized (although, presumably, one could borrow against this asset value). </li></ul></ul>
- 152. Immunization <ul><li>Immunization seeks to match known cash flows from an investment portfolio to expected liabilities. </li></ul><ul><ul><li>Adjustments are made to make the portfolio behave like a zero coupon bond with a maturity equal to the decision maker’s investment horizon. This is achieved when the Macaulay’s duration is equal to the investment horizon. </li></ul></ul><ul><li>The Macaulay duration of a portfolio changes: </li></ul><ul><ul><li>With the passage of time </li></ul></ul><ul><ul><li>Due to the fluctuation of interest rates </li></ul></ul>
- 153. Rebalancing the Immunized Portfolio <ul><li>Balance requires that two related but opposing interest rate risks should be fully offsetting: </li></ul><ul><ul><li>Price (market) risk arises when a bond is sold before maturity. </li></ul></ul><ul><ul><li>Reinvestment risk arises upon the receipt of the coupons or income stream. </li></ul></ul><ul><ul><li>An increase in rates would aid reinvestment at the cost of market risk. A decrease in rates aids market risk at the cost of reinvestment risk. </li></ul></ul>
- 154. Rebalancing, 2 <ul><li>Immunization risk : Unless the portfolio is actually comprised of zero-coupon bonds of the same maturity as the investment horizon, any twisting in the curve may upset the market / reinvestment balance. </li></ul><ul><ul><li>Like Macaulay duration itself, there is an assumption that all changes to the interest rates are parallel shifts. </li></ul></ul><ul><li>The rebalancing decision considers: </li></ul><ul><ul><li>The degree of divergence </li></ul></ul><ul><ul><li>Transaction costs </li></ul></ul>
- 155. Rebalancing, 3 <ul><li>A portfolio with interim cash flows requires a smaller initial outlay, traded off against the burden of managing additional cash flows. </li></ul><ul><ul><li>The greater dispersion of cash flows increases the immunization risk. </li></ul></ul><ul><ul><li>A non-parallel shift (a twist) that lowered short-term rates and raised longer term rates would produce less income from interim cash flows and a capital loss on the security. Convexity measures the portfolio risk to non-parallel shifts. See next slide for formula for M 2 . </li></ul></ul><ul><ul><li>As we narrow the dispersion around the target (a barbell), the immunization risk declines. </li></ul></ul>
- 156. Immunization Risk (Fong and Vasicek) 1 CF 1 (1 – H) 2 CF 2 (2 – H) 2 CF n (n – H) 2 I 0 (1 + y) (1 + y) 2 (1 + y) n M 2 = ( + + … + ) I 0 = initial investment CF t = cash flow at time t H = investment horizon y = yield on portfolio n = time to last cash flow <ul><li>Where there is only one cash flow at time H, 1 – H = 0 therefore M 2 = 0 </li></ul><ul><li>M 2 ≈ Convexity – Duration, therefore a larger convexity has a greater exposure to twists in rates </li></ul>
- 157. Credit Risk and Embedded Options <ul><li>Default risk is equivalent to underperforming the asset value at the investment horizon. Different responses: </li></ul><ul><ul><li>Invest only in assets with the highest credit quality. </li></ul></ul><ul><ul><li>Select only instruments with an acceptable trade-off between risk and return. </li></ul></ul><ul><ul><li>Over-collateralize the portfolio to take account of potential losses. </li></ul></ul><ul><li>There are also trade-offs in owning a callable security, with a higher return compensating for the potential re-investment risk. </li></ul><ul><ul><li>Response is to monitor assets and replace those subject to significant call risk. </li></ul></ul>
- 158. Liability Funding <ul><li>Liability funding uses the matching approach to equate a known liability stream with a set of assets in such a way that, at each maturing liability, there is a corresponding cash flow from the assets (multi-period immunization). </li></ul><ul><ul><li>The matching of asset duration and liability duration is necessary but not sufficient – assets must be decomposed into segments so that each liability part is immunized. </li></ul></ul><ul><ul><li>PV of assets must be => PV of liability. </li></ul></ul><ul><ul><li>The asset durations must dominate (i.e., must have a greater dispersion) those of the liabilities or the assets will be subject to price risk – as when an asset must be realized at market values without a corresponding gain in reinvested income. </li></ul></ul>
- 159. Cash Flow Matching <ul><li>… (also known as dedication ) is a bootstrapping approach to hedging a set of liabilities by finding assets that match exactly the future cash outflows. </li></ul><ul><ul><li>This process is like the replacement of cash flows in a swap, starting with the longest-dated cash flow, and working to t = 0. </li></ul></ul><ul><li>Symmetric cash matching allows short-term borrowing to satisfy a liability. </li></ul><ul><li>A hybrid approach, variously known as combination matching or horizon matching, uses cash flow matching for earlier liabilities and immunization for the more distant. </li></ul>

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