In this chapter
- the time value of money
- an introduction to equities, commodities, currencies and indices
- fixed and floating interest rates
- futures and forwards
- no-arbitrage
The time value of money
$1 today is worth more than $1 in a year’s time
simple vs. compound interest
Simple interest is when the interest you receive is based only on the amount you invest initially, whereas compound interest is when you also get interest on your interest.
Discrete vs. continuous compounded
\[ (1 + \frac{r}{m})^m = e^{m \log{(1 + \frac{r}{m})}} \sim e^r \]
\[ M(t + dt) - M(t) \approx \frac{dM}{dt}dt + \dots \]
\[ \frac{dM}{dt}dt = rM(t)dt \]
\[ M(t) = M(0)e^{rt} \]
I can relate cashflows in the future to their present value by multiplying by this factor.
\[ e^{-r (T - t)} \]
Equities
This is the ownership of a small piece of a company.
You could raise capital to realize this idea by selling off future profits in the form of a stake in your new company. The investor in the company gives you some cash, and in return you give him a contract stating how much of the company he owns.
In this book I am going to take the point of view that prices have a large element of randomness. This does not mean that we cannot model stock prices, but it does mean that the modeling must be done in a probabilistic sense.
Dividends
To the average investor, the value in holding the stock comes from the dividends and any growth in the stock’s value. Dividends are lump sum payments, paid out every quarter or every six months, to the holder of the stock.
Stock splits
Three-for-one stock split simply means that instead of holding one stock valued at $900, I hold three valued at $300 each.
Commodities
Commodities are usually raw products such as precious metals, oil, good products etc.
Currencies
Exchange rate, the rate at which one currency can be exchanged for another. This is the world of foreign exchange, or Forex, or FX for short.
Although the fluctuation in exchange rates is unpredictable, there is a link between exchange rates and the interest rates in the two countries. If the interest rate on dollars is raised while the interest rate on pounds sterling stays fixed we would expect to see sterling depreciating against the dollar for a while. Central banks can use interest rates as a tool for manipulating exchange rates, but only to a degree.
Indices
For measuring how the stock market/economy is doing as a whole, there have been developed the stock market indices. A typical index is made up from the weight sum of a selection or basket of representative stocks.
Fixed-income securities
In lending money to a bank you may get to choose for how long you tie your money up and what kind of interest rate you receive. If you decide on a fixed-term deposit the bank ill offer to lock in a fixed rate of interest for the period of the deposit.
Interest payments
- fixed
- floating
Coupon-bearing bonds pay out a known amount every six months or year etc. This is the coupon and would often be a fixed rate of interest. At the end of your fixed term you get a final coupon and the return of the principal, the amount on which the interest was calculated. Interest rate swaps are an exchange of a fixed rate of interest for a floating rate of interest.
Inflation-proof bonds
index-linked bond
UK inflation: retail price index (RPI)
US inflation: consumer price index (CPI)
The coupons and principal of the index-linked bonds are related to the level of the inflation index.
Forwards and futures
A forward contract is an agreement where one party promises to buy an asset from another party at some specified time in the future and at some specified price. No money changes hands until the delivery date or maturity of the contract. The terms of the contract make it an obligation to buy the asset at the delivery date, there is no choice in the matter. This asset could be a stock, a commodity or a currency.
Futures contracts are usually traded through an exchange, which standardized the terms of the contracts.
Forwards and futures have two main uses, in speculation an din hedging.
A first example of no arbitrage
\[ F = S(t)e^{r (T - t)} \]
This is the relationship between the spot price and the forward price. It is a linear relationship the forward price is proportional to the spot price.
If this relationship is violated then there will be an arbitrage opportunity.
Further reading
