## Monday, July 25, 2016

### Scopes and scales: the present value formula

I'm not entirely sure if this conversation broke off from the discussion of NGDP futures markets, but Nick Rowe put up a post about the difficulties of calculating the present value of currency. This represents another good example of why you need to be careful about scales and scope.

The basic present value formula of a coupon $C$ with interest rate $r$ at time $T$ is

$$PV(C, r, T) = \frac{C}{(1 + r)^{T}}$$

First, note that an interest rate is actually a time scale. The units of $r$ are percent per time period, e.g. percent per annum. Therefore we can rewrite $r$ as a time scale $r = 1/\tau$ where $\tau$ has units of time (representing, say, the e-folding time if you think about continuous compounding).

Second, this formula comes from looking at a finite non-zero interest rate over a finite period of time. You can see this because the formula breaks if you decide to take $T$ or $\tau$ to infinity in a different order:

Paul Romer had this problem with Robert Lucas: the limit doesn't converge uniformly. Romer would call taking the limits as $r = 1/\tau \rightarrow 0$ and $T \rightarrow \infty$ in a particular order "mathiness". And I think mathiness here is an indicator that you need to worry about scope. Just think about it -- why would you calculate the "present value" of coupon that had a zero interest rate? It's like figuring out how many stable nuclei decay (see previous link).

The present value formula does not apply to a zero interest rate coupon. It is out of scope.

There are only two sensible limits of the present value formula: $T/\tau = r T \gg 1$ and $T/\tau = r T \ll 1$. This means either $T \rightarrow \infty$ or $\tau \rightarrow \infty$ -- not both. If you want to take both to infinity at the same time, you have to introduce another parameter $a$ because then you can let $T = \tau/a$ and take the limit

$$\lim_{T \rightarrow \infty} \frac{C}{(1 + a/T)^{T}} = C e^{-a}$$

The present value can be anything between $C$ and zero. You could introduce other models for $T = T(\tau)$, but that's the point: the present value becomes model dependent. (That's what Nick Rowe does to resolve this "paradox" -- effectively introducing some combination of nominal growth and inflation.)

That also brings up another point: the present value formula doesn't have any explicit model dependence, but it does have an implicit model dependence! It critically depends on being near a macroeconomic equilibrium (David Glasner's macrofoundations). For example, it's possible the value of a corporate bond is closer to zero because there's going to be a major recession where the company defaults. Correlated plans fail to be simultaneously viable, and someone has to take a haircut.

Basically, the scope of the present value formula is near a macroeconomic equilibrium and non-zero interest rates.

1. Jason: I don't want to take Tau to infinity. The interest rate stays fixed at 5% *per year* (or whatever). I only want to take T to infinity.

And the Coupon C doesn't just get paid once. It gets paid each and every year. It's \$5 paid every January 1st (or whatever) until year T.

Or maybe I misunderstand you (I usually do).

This wasn't really related to the NGDP futures argument (except that "finance people" are the ones on the other side of both arguments).

1. Hi Nick,

Then maybe I misunderstood -- however I thought the question was about currency that has a zero rate of return. There are two places where you do look at r = 0% per annum:

"If we stick r(t) = 0% for all t in the Present Value calculation above ..."

and

"Just use the liquidity demand function ... to figure out the P(0) at which the ratio of market capitalisation to NGDP gives us an r=0%."

That seems to indicate you do want to take r ≡ 1/τ to zero, or alternatively τ to infinity. If it is only in those quotes, then those quotes are the only part of your post to which this post is germane.

If the interest rate is 5% per year, then you're only taking one limit (rT >> 1) and it is fine.

2. If the interest rate is 5% so the coupon is not zero, are we still talking about money?

1. Nope! And that's the point ... the PV formula is out of scope for money.