This is a post which serves as a follow-up question to Nature of short VIX strategies. I am trying to understand where the change in value in a synthetic strategy constructed from futures comes from:
- Roll-over;
- Roll yield.
Below I display a simplified framework to try to quantify these different factors. We will assume deterministic rates hence forward and futures prices coincide.
Model
We are interested in a non-traded asset $X_t$. We assume there is a liquid forward market for asset $X_t$ for some maturities $\{T_i:i = 1, \dots, n\}$. The price at time $t_0$ of a forward of maturity $T_i$ is given by $F(t_0,T_i)$ and its value at $t>t_0$ by:
$$ V_F(t,t_0,T_i)=X_t-F(t_0,T_i)B(t,T_i) $$
where $B(t,T_i)$ is the value of a zero-coupon bond maturing in $T_i$. We assume a constant rate $r_F$ implied by the asset's forward term structure:
$$ dB(t,T_i) = r_FB(t,T_i)dt $$
Importantly, we assume that $r_F>0$ hence the term structure of the asset is in contango, meaning:
$$ \forall \ t_0<T_i<T_j<T_n \ , \quad X_{t_0}<F(t_0,T_i)<F(t_0,T_j)$$
Let us consider a strategy $S_t$ which gives a short forward exposure to asset $X_t$ with a horizon of $T_S$ $-$ e.g. 30 days $-$ through forwards. We assume the strategy is initiated at a time $t_0$ and that there is a traded forward maturity $T_1$ such that:
$$ T_S=T_1-t_0$$
Hence at inception the strategy is constructed by shorting a quantity $w^{(1)}_{t_0}$ of forwards with price $F(t_0,T_1)$ and its value is:
$$ S_{t_0} = w^{(1)}_{t_0}V_F(t_0,t_0,T_1)$$
Note that $w^{(1)}_{t_0}\leq 0$ because we are short the asset.
Roll-over
Now let us consider some time $dt$ passes. The effective maturity of the strategy is now lower than $T_S$. We then need to trade in the "next" forward with maturity $T_2$ and:
- hold $w^{(1)}_{t_0+dt} \leq 0$ units of forwards of maturity $T_1$; and
- hold $w^{(2)}_{t_0+dt} \leq 0$ units of forwards of maturity $T_2$
so as to keep an effective strategy maturity of $T_S$. The value of the strategy after rebalancing at $t_0+dt$ is:
$$S_{t_0+dt}=w^{(1)}_{t_0+dt}V_F(t_0+dt,t_0,T_1)+w^{(2)}_{t_0+dt}V_F(t_0+dt,t_0+dt,T_2)$$
Now, we require:
- the effective maturity of the strategy to be equal to $T_S$;
- the strategy needs to be self-financing;
Because we have 2 unknowns, namely the 2 weights for each forward, and 2 constraints we can determine values for $w^{(1)}_{t_0+dt}$ and $w^{(2)}_{t_0+dt}$. Because the strategy is self-financing, the roll-over has no impact on its value.
[Edit 21/04/2020] More intuitively, note that also that forwards/futures are entered at 0 cost: while you are scaling-down your front-end position, you are entering into offsetting positions at 0 cost, while the shift to the next-month future is also done at 0 cost, thus the process of rolling over does not generate by itself any PnL.
Roll yield
Replacing $t_0+dt$ by $t_1$ to avoid (mathematical) confusion, because the strategy is self-financing we have (note that $F(t_k,T_i)$ is contractually defined at inception hence fixed throughout the trade):
$$ \begin{align} dS_t & = w^{(1)}_{t}dV_F(t,t_0,T_1) + w^{(2)}_{t}dV_F(t,t_1,T_2) \\[3pt] & = w^{(1)}_{t}(dX_t-r_FF(t_0,T_1)B(t,T_1)dt) + w^{(2)}_{t}(dX_t-r_FF(t_1,T_2)B(t,T_2)dt) \end{align} $$
Ignoring the asset's fluctuations, we define the roll yield as:
$$ -r_Fw^{(1)}_{t}F(t_0,T_1)B(t,T_1)dt - r_Fw^{(2)}_{t}F(t_1,T_2)B(t,T_2)dt \geq 0 $$
The last inequality stems from the fact that $w^{(i)}_t\leq 0$ because the strategy is short.
Conclusion
From the analysis above I conclude the following:
- The value of strategies, such as ETFs or ETNs, that offer synthetic exposure to some asset through the forward/futures market is not impacted by forward/futures roll-over because they need to be self-financing. Indeed, if they were not self-financing they might require the investor to pour in additional cash throughout the life of the trade which is not the case for ETFs or ETNs.
- The value of these strategies is positively (or negatively, depending on the term structure and whether they are long or short) impacted by the roll yield. For example, in a contangoed market a short forward exposure generates a positive roll yield.
Are the conclusions above correct? Have I missed some important factor in my analysis, or is there something incorrect?
