MAT 483
Mathematical Models in
Finance and Investments
Fall 2008
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
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?
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 value
Often a necessity to bring goods to a central place
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
Some general vocabulary
Futures / Forward Markets: Agree to terms at present time, but
currency exchange and delivery is many months or more into the
future
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
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
Futures Markets
Chicago Mercantile Exchange
Largest U.S. Futures Exchange
20 S. Wacker Drive, Chicago, IL
www.cme.com
Agricultural Products:
Beef, Dairy, Hogs, Lumber, Fertilizer
Financial Products:
Equity Index Futures (S&P 500, NASDAQ), Interest Rate Futures (TBill), Foreign Currency Futures (Euro)
Futures Markets
141 West Jackson Boulevard (Jackson and LaSalle)
www.cbot.com
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)
Futures Markets
See sample printed copies of corn and lean hog futures
contracts…
More vocabulary
Financial 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.
Interest rates: T-Bills
Start 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
Spot and Forward Rates
Spot 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?
Spot and Forward Rates
The 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%
Spot and Forward Rates
The 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
Spot and Forward Rates
Note 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= $104.58; cash flow = $7 in 1 year, $100 in 2
years
Price of Asset 3= $96.42; cash flow = $4 in 1 year, $4 in 2
years, $100 in 3 years
Spot and Forward Rates
Note 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 = 5.40%
2-Year spot rate = 5.64%
3-Year spot rate = 5.92%
Spot and Forward Rates
There 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
Other interest rate ideas
Yield 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
Other interest rate ideas
Equivalent 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)
Other interest rate ideas
Example: 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%
Other interest rate ideas
Example: 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