3.1 The Determination of Forward & Futures Prices

3.2 Compounding Frequency The compounding frequency used for an interest rate is the unit of measurement The difference between quarterly & annual compounding is analogous to the difference between miles & kilometers

3.3 Continuous Compounding In the limit as we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $100e RT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $100e -RT at time zero when the continuously compounded discount rate is R

3.4 Conversion Formulas Define R c : continuously compounded rate R m : same rate with compounding m times per year R c =m ln(1+R m /m) R m =m[exp(R c /m)-1]

3.5 Short Selling Short selling involves selling securities you do not own Your broker borrows the securities from another client & sells them in the market in the usual way

3.6 Short Selling (continued) At some stage you must buy the securities back so they can be replaced in the account of the client You must pay dividends & other benefits the owner of the securities receives

3.7 Repo Rate The repo rate is the relevant rate of interest for many arbitrageurs A repo (repurchase agreement) is an agreement where one financial institution sells securities to another financial institution & agrees to buy them back later at a slightly higher price The difference between the selling price & the buying price is the interest earned

3.8 Gold Example For gold F = S (1 + r ) T where F : forward price, S : spot price, & r : T -year rate of interest (This assumes no storage costs ) If r is compounded continuously instead of annually F = S e r T

3.9 Extension of the Gold Example For any investment asset that provides no income & has no storage costs F = S e r T (Equation 3.5, p.53)

3.10 When an Investment Asset Provides an Income Which is Known in Dollar Terms F = (S – I )e r T (Equation 3.6, p.56) where I is the present value of the income

3.11 When an Investment Asset Provides a Known Dividend Yield F = S e (r–q )T (Equation 3.7, p.57) The asset is assumed to provide a return during time  t equal to qS  t where q is the dividend yield and S is the asset price

3.12 Valuing a Forward Contract Suppose that K : delivery price in a forward contract & F : forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F – K )e –r T (Equation 3.8, p.58) Similarly, the value of a short forward contract is (K – F )e –r T

3.13 Forward vs Futures Prices Forward & futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: - strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price - strong negative correlation implies the reverse

3.14 Stock Index Can be viewed as an investment asset paying a continuous dividend yield The futures price & spot price relationship is therefore F = S e (r–q )T (Equation 3.12, p. 62) where q is the dividend yield on the portfolio represented by the index

3.15 Stock Index (continued) For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio The Nikkei index viewed as a dollar number does not represent an investment asset

3.16 Index Arbitrage When F>Se (r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F<Se (r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

3.17 Index Arbitrage (continued) Index arbitrage involves simultaneous trades in futures & many different stocks Very often a computer is used to generate the trades Occasionally (eg “Black Monday”) simultaneous trades are not possible & the theoretical no-arbitrage relationship between F & S may not hold

3.18 A foreign currency is analogous to a security providing a continuous dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if r f is the foreign risk-free interest rate Futures on Currencies (Equation 3.13, p.64)

3.19 Futures on Consumption Assets F  S e (r+u )T (Equation 3.20, p.68) where u is the storage cost per unit time as a percent of the asset value. Alternatively, F  (S+U )e r T (Equation 3.19, p.68) where U is the present value of the storage costs.

3.20 The Cost of Carry The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F = Se cT For a consumption assetF  S e c T The convenience yield on the consumption asset, y, is defined so that F = S e (c–y )T (Equation 3.23, p.70)

3.21 Futures Prices & Expected Future Spot Prices Suppose k is the expected return required by investors on an asset We can invest F e –r T now to get S T back at maturity of the futures contract This shows that F = E (S T )e (r–k )T (Equation 3.24, p.71)

3.22 Futures Prices & Future Spot Prices (continued) If the asset has - no systematic risk, then k = r, & F is an unbiased estimate of S T - positive systematic risk, then k > r, & F < E (S T ) - negative systematic risk, then k E (S T )