Modelling roll-over and roll yield in a forward strategy

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:

1. the effective maturity of the strategy to be equal to $T_S$;
2. 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.

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?

• Hi Daneel. Not sure I follow everything. i) Why consider a non-traded asset? ii) I assume this asset does not generate capital distributions and that you could write - if it was traded + AOA - that: $V_F(t,t_0,T_i) = B(t,T_i) ( F(t,T_i) - F(t_0,T_i) )$ showing that $V_F(t_0,t_0,T_i) = 0$ for any $i$ ? To generate an exposure of effective maturity $T_S$ while entering a forward agreement at $T_1 > T_S$ how do you pick the weight $w_{t_0}^{(1)}$ ? By that I mean that as time passes and $T_1$ becomes smaller than $T_S$ you can then simply let $F_1$ expire and just enter $F_2$ with the same rule? – Quantuple Mar 8 '18 at 14:30
• @Quantuple i) I am thinking of VIX; ii) indeed, no capital distributions for simplicity; you can write that, with deterministic rates we have $X_t=E_t^Q[B(t,T_i)X_{T_i}]=B(t,T_i)E_t^Q[X_{T_i}]=B(t,T_i)F(t,T_i)$ where $F$ can be a future or a forward; finally in my simplified reasoning and changing $t_0$ by $T_0$, we can assume that for all $i \in \mathbb{N}$ we have $T_{i+1}-T_i=T_S$ so that at $t=T_i$ you let the $F_i$ future/forward expire and you are fully allocated in the $F_{i+1}$ future/forward, then start rebalancing with $F_{i+2}$ as the effective maturity goes below $T_S$ and so on. – Daneel Olivaw Mar 8 '18 at 15:35
• i bis) The idea is that the forward term structure of the asset, being non-traded, can be "dislocated" from the spot value of the asset, as it seems it is the case with VIX (as far as I know at least). – Daneel Olivaw Mar 8 '18 at 15:37
• @Quantuple The "non-traded" assumption was motivated by the fact that I am interested in the VIX, but we can also assume it is traded, on top of my head I don't know if that changes much. My main concern is with the analysis of roll-over and roll yield, namely that 1) roll-over is value-neutral only if the strategy is self-financing, and 2) how roll yield is generated. – Daneel Olivaw Mar 8 '18 at 15:45
• @Quantuple: Would you like to take a look at my question quant.stackexchange.com/q/38719/6686? Thank you. – Hans Mar 11 '18 at 0:36