Time value, rates and linear products

4 lessons · 8 equations. Each lesson below gives its formulas and key rules; open the lesson for the full explanation.

Forwards, futures and the cost of carry

The no-arbitrage forward

F_0 = (S_0 - D)e^{rT}

Spot, less any income the asset throws off, carried forward at the financing rate.

Commodities: storage and convenience yield

F_0 = S_0\,e^{(r + u - y)T}

Holding a physical commodity costs financing $r$ and storage $u$ but earns a convenience yield $y$ — the benefit of having it on hand when supply is tight. When $y$ exceeds $r + u$ the curve is backwardated. Unlike a financial asset, $y$ is not observable directly: it is backed out of the curve.

Rate futures versus forwards

f_{\text{fwd}} \approx f_{\text{fut}} - \tfrac{1}{2}\sigma^2 T_1 T_2

Because a rate future is margined daily, and gains are received when rates are high and can be reinvested at high rates, a short rate-future position is worth more than the equivalent forward. So futures-implied rates sit above forward rates, by an amount that grows with volatility and roughly with the square of maturity.

Remember

$F_0 = (S_0 - D)e^{rT}$ , enforced by cash and carry rather than by anyone’s forecast.

Discounting: compounding conventions and present value

Converting between them

\left(1 + \frac{r_m}{m}\right)^{m} = e^{r_c} \quad \Rightarrow \quad r_c = m\ln\!\left(1 + \frac{r_m}{m}\right)

Two rates are equivalent when they produce the same value after a year. The continuously compounded rate is always the *lowest* of the equivalent set, because it compounds most often.

A growing annuity

PV = \frac{C}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^n\right]

Payments that start at $C$ in one year and grow at rate $g$ for $n$ years. It is a growing perpetuity minus the same perpetuity started $n$ years later, the same subtraction that gives the ordinary annuity.

Remember

Simple $1+rT$ , compound $(1+r/m)^{mT}$ , continuous $e^{rT}$ .

Bonds: yield, duration and convexity

Duration

D_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}, \qquad \frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y

Modified duration is the percentage price change per unit change in yield. Macaulay duration is the weighted average time to the cash flows, and the two differ by a factor of $(1 + y/m)$ .

Convexity

\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}C\,(\Delta y)^2

The second-order term. It is positive for an ordinary bond, so duration alone always *overstates* the loss from a rise and understates the gain from a fall.

Remember

Price falls as yield rises, convexly.

Swaps, FRAs and interest parity

Covered interest parity

F = S\,\frac{1 + r_d T}{1 + r_f T}

The forward exchange rate is pinned by the two interest rates, because borrowing at home, converting, depositing abroad and selling the proceeds forward must return the domestic rate.

Remember

A payer swap is short a fixed bond and long a floating one.

Detailed formula cards

The no-arbitrage forward