# Inflation/Rates Correlation

I've been looking into a short piece of maths a colleague has written on pricing inflation with payment delays, and was hoping someone could confirm whether my understanding is correct, or if my colleague's calculation is invalid, then why. The setup: suppose, for example, suppose you have an inflation-linked cashflow that captures the level of inflation between 2014 - 2015, but this cashflow isn't paid until a year later in 2016.

Then the value of this cashflow is not necessarily just the expected 2014 $\rightarrow$ 2015 inflation discounted back from 2016, because you may have some correlation between interest rates and inflation. In essence, $\mathbb{E}(\text{I}_{2015} \cdot \text{DF}_{2016} ) \ne \mathbb{E}(\text{I}_{2015}) \cdot \mathbb{E}(\text{DF}_{2016} )$ if your discount factor and inflation are correlated.

This is not surprising - if inflation and interest rates are +100% correlated, then an increase in inflation is partially "hedged" by rates rising and your discount factors shrinking, and conversely if they're -100% correlated then a rise in inflation will be further compounded by lower discount rates. In either case, the expected value of your cashflow is nonlinear vs inflation, and you would expect it to be some function of correlation.

My aim therefore is to understand the magnitude of the "convexity adjustment" $\Delta$, where $\mathbb{E}(\text{I}_{2015} \cdot \text{DF}_{2016} ) = \Delta \cdot ( \mathbb{E}(\text{I}_{2015}) \cdot \mathbb{E}(\text{DF}_{2016} ))$. The maths my colleague wrote to value this adjustment is below. I am aware it's relatively crude, it doesn't need to be perfect for my purposes - I just want something in the right ballpark and better than making no adjustment at all. There are more sophisticated discussions of the maths such as this paper but I'd like to understand the simplified calculation first if possible.

Let $I_T$ = inflation fixing at time $T$, but paid on lagged date $T_L = T + D$. Write your discount factor from time $a$ to $b$, as seen at time $c$, as $\delta_c(a,b)$. Then a payment of $I_T$ at time $T_L$ is equivalent to $I_T \cdot \delta_{T}\,(T,T_L)$ at time $T$.

Letting $r$ be the short-term interest rate observed at $T$, we can say $\delta_{T}\,(T,T_L) = e^{-r D}$.

Suppose now that $I_T$ is lognormally distributed, variance $\sigma_I^2 T$, and $r$ is normally dist. with variance $\sigma_r^2 T$, and correlation is $\rho$. Then we have

$\mathbb{E}(I_T \cdot \delta_T(T,T_L)) = \exp{[-\rho \cdot \sigma_I \cdot \sigma_r \cdot D \cdot T]} \cdot \mathbb{E}(I_T) \cdot \delta_0(T,T_L) \,\,\,\, (*)$

and the first term is your convexity adjustment $\Delta$.

My first question is: is that all maths valid? I'm a little suspicious of saying $\delta_{T}\,(T,T_L) = e^{-r D}$ for a "short term rate" $r$ if $D$ is a long delay. It seems like by simply multiplying by $D$, you're simultaneously treating $r$ as fixed and not fixed. Or is the key point that once you reach time $T$ $r$ is observed and can be treated as "fixed"?

Assuming the maths is all ok, is $(*)$ just a consequence of the fact that for correlated normal R.V.s $X,\,Y$ we have $\mathbb{E}(e^X e^{-Y}) = \mathbb{E}(e^{X-Y})=$ and $X - Y \sim N(\mu_X - \mu_Y, \sigma_X^2 +\sigma_Y^2 - 2 \rho \sigma_X \sigma_Y)$? If I understand correctly think you just use that relation and substitute in for $X$ and $Y$.

Lastly, in this context would $\sigma_I$ and $\sigma_r$ both be normal vols? That's how I read the maths, but the numerical example I have suggests that $\sigma_I$ should be more like a lognormal order of magnitude (e.g. nearer to 10% than 1%). Many thanks for your help.