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 $100eRT 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
Rc: continuously compounded rate
Rm: same rate with compounding m times per year
Rc=m ln(1+Rm/m)
Rm=m[exp(Rc/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
3.7Repo 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 whereF : 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 er T
3.9
Extension of the Gold Example
• For any investment asset that provides no income & has no storage costs
F = S er T (Equation 3.5, p.53)
3.10When an Investment Asset Provides an Income Which is
Known in Dollar Terms
F = (S – I )er 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 rf is the foreign risk-free interest rate
Futures on Currencies
F S r r Tf e ( )
(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 )er 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 = SecT
• For a consumption asset F S ec 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 ST back at maturity of the futures contract
• This shows that
F = E (ST )e(r–k )T(Equation 3.24, p.71)