MAT 483 Mathematical Models in Finance and Investments Fall 2010.
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Transcript of MAT 483 Mathematical Models in Finance and Investments Fall 2010.
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MAT 483Mathematical Models in Finance and Investments
Fall 2010
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Financial Markets
Main ideas:
• Major types of financial instruments
• How are they…
• Initiated
• Traded
• Priced
• Quoted
• Some specific focus on the nature of interest rates
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Some general vocabulary
Market: a place where goods are bought and sold
This can be a very wide definition, covering many types of transactions
How do markets evolve?
What types of markets are there?
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Some general vocabulary
Barter Market: Most basic, exchange of goods and services
Most common accepted form of commerce prior to the invention of currency
1626: Peter Minuet trades beads, knives and kettles for...
Issues: Equality of valueOften a necessity to bring goods to a central place
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Some general vocabulary
Cash or Spot Market:
Purchase items with a currency, with delivery either immediate or delivered within a short amount of time
Most common type of market we know today
Goods and services can range from very simple to very complex financial instruments
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More vocabularyFinancial Market: A place where financial assets are bought and sold –
most common is a stock exchange like NYSE, but doesn’t need to be a physical location (National Association of Securities Dealers Automated Quotation System - NASDAQ)
Primary Market: Transaction between the creator of the security (issuer) and the first owner
Secondary Market: Owners sell the security to a new owner, typically in a financial market
Derivative Securities: Securities whose prices are dependent or “derived” from another security’s price – options, futures, etc.
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Some general vocabulary
Futures / Forward Markets: Agree to terms at present time, but currency exchange and delivery is in the future, often several months
The term “futures” is most commonly used when the asset being purchased is a standardized contract
The term “forward” is most commonly used when the asset being purchased is non-standardized; amount, quality shape and form need negotiated between the buyer and seller
The evolution of futures markets have given many industries an enhanced stability for risk management, budgeting, planning and confidence in production inputs and outputs
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Some general vocabulary
“Underlying” assets traded can be very diverse:
Tangible items: commodities, animals, production inputs, metals, stocks, bonds, etc ----- quoted in terms of the price of the item
Intangible items: Stock indexes, currency exchange rates, interest rates
The evolution of futures markets have given many industries an enhanced stability for risk management, budgeting, planning and confidence in production inputs and outputs
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Futures MarketsChicago Mercantile ExchangeCMEGroupLargest U.S. Futures Exchange20 S. Wacker Drive, Chicago, ILwww.cmegroup.com
Agricultural Products:Beef, Dairy, Hogs, Lumber, Fertilizer
Financial Products:Equity Index Futures (S&P 500, NASDAQ), Interest Rate Futures (T-
Bill), Foreign Currency Futures (Euro)
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Futures Markets
141 West Jackson Boulevard (Jackson and LaSalle)
Ceres, the Roman goddess of agriculture
Agricultural Products:
Corn, Soybeans, Wheat, Oats
Financial Products:
Equity Index Futures (Dow), Interest Rate Futures (Treasury Notes), Metal Futures (Gold, Silver)
Combined with CMEGroup in 2008
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Futures MarketsSee sample printed copies of lean hog and lumber futures
contracts…
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Interest rates: T-BillsStart with Treasury Bills or “T-Bills” – one of the most
common fixed income securities in U.S.
Primary market: issued by New York Fed every week for short term US financing – face amounts start at $10,000
Maturities up to 1 year are offered but not the same every week
Do NOT pay coupons – hence a “discount” or “zero-coupon” security – Repayment of face is only cash flow
Secondary Market extremely active – very liquid investments
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Spot and Forward RatesSpot rates: rates derived from the prices of interest rate
securities – usually zero-coupon securities like CD’s, money market securities, T-Bills, etc.
Forward rates: rates derived from spot rates that are implied for periods of time in the future
Example: If a one year CD yields 5.50%, a two-year CD yields 5.80% and a three-year CD yields 6.20%, then what does it imply about a one-year rate one year from today?
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Spot and Forward RatesThe facts imply there is some rate, f, that will be effective one
year from today for a one-year period that satisfies:
(1+.0550) * (1 + f) = (1 + .0580)2
f = .061 or 6.10%
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Spot and Forward RatesThe facts imply there is some rate, f, that will be effective
two years from today for a one-year period that satisfies:
(1+.0580)2 * (1 + f) = (1 + .0620)3
f = .070 or 7.00%
Note that these rates are “implied” – but may or may not come true – they are driven by the expectations of the market today in how they price the spot instruments
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Spot and Forward RatesNote that derivation of spot and forward rates is dependent
upon the set of assets in the marketplace.
Example:
Price of Asset 1= $89.60; cash flow = $100 in 2 years
Price of Asset 2= $96.25; cash flow = $7 in 1 year, $100 in 2 years
Price of Asset 3= $91.53; cash flow = $4 in 1 year, $4 in 2 years, $100 in 3 years
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Spot and Forward RatesNote that derivation of spot and forward rates is dependent upon the set of
assets in the marketplace.
Example:
Price of Asset 1= $89.60; cash flow = $100 in 2 years
Price of Asset 2= $96.25; cash flow = $7 in 1 year, $100 in 2 years
Price of Asset 3= $91.53; cash flow = $4 in 1 year, $4 in 2 years, $100 in 3 years
1-Year spot rate = 5.26%
2-Year spot rate = 5.64%
3-Year spot rate = 5.92%
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Spot and Forward RatesThere is a general relationship between the spot curve and the forward
curve dependent upon the characteristics of the spot curve….
If the spot curve is increasing, forward rates are greater than spot rates
If the spot curve is level, forward rates are equal to spot rates
If the spot curve is decreasing, forward rates are less than spot rates
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Other interest rate ideasYield to Maturity
In general the “effective yield”, or “yield to maturity” of a fixed income instrument is the interest rate that discounts the entire set of cash flows to the current time to get the current price
Since most bonds have coupons and then return the principal at maturity, there are many cash flows to consider
Generally an annuity discount factor used on the coupons
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Other interest rate ideasEquivalent Taxable Yield
Earnings on some investments are deemed to exempt from federal income tax, such as debt securities issued by states and local municipalities; these are often called municipal bonds or “muni” bonds
In order to compare alternative investment choices, investors must calculate the equivalent taxable yield on muni bonds
Equivalent Taxable Yield = Tax-Free Yield / (1 – Tax Rate)
Also, can calculate the tax rate where the investor becomes indifferent between taxable and tax-free yields
Tax Cutoff Bracket = 1 – (Tax-Free Yield / Taxable Yield)
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Other interest rate ideasExample: Investor has a tax rate of 32% and a Muni bond
yields 6%
Equivalent Taxable Yield = Tax-Free Yield / (1 – Tax Rate)
= .06 / (1 – .32)
= .06 / .68
= .0882 = 8.82%
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Other interest rate ideasExample: A muni bond has a yield of 5.25% versus a taxable investment
of 7.00%
Tax Cutoff Bracket = 1 – (.0525/ .0700)= 1 – .75= .25 = 25%
If the tax rate is less than 25%, say 10%, then the Equivalent Taxable Yield on the muni bond would be (.0525 / .90) = 5.83% and the taxable investment would be preferable
If the tax rate is greater than 25%, say 40%, then the Equivalent Taxable Yield on the muni bond would be (.0525 / .60) = 8.75% and the muni bond would be preferable