What an exchange rate is

This subject is about owning things priced in other currencies. It has to start with the price of a currency, because that one number is quoted in a way that reverses the meaning of every sentence built on it.

The quoting problem

The RBI publishes a reference rate of 96.6149 rupees per US dollar.

That is the price of a dollar, in rupees. The dollar is the thing being priced; the rupee is the unit of measurement. Written as a ratio it is INR/USD, and the currency named second is the one being bought.

So the number going up means the rupee going down. A dollar costing 96.61 rupees today and 90 rupees a year ago means dollars got dearer, which means rupees got cheaper.

This reverses the intuition people arrive with, where a bigger number is better. It is worth fixing the habit now: when you see the INR/USD rate rise, read "the rupee weakened".

The same rate from the other side is $1/96.6149 = 0.010350$ dollars per rupee. Nobody quotes it that way for the rupee, but it is the honest form for checking your reasoning, because now a falling number obviously means a falling rupee.

Cross rates

The RBI quotes the pound at 127.8859 rupees. With the dollar at 96.6149, the implied pound-dollar rate is

$\frac{127.8859}{96.6149} = 1.3237 \text{ dollars per pound}$

That is a cross rate, derived rather than quoted, and it must hold or there is an arbitrage. If the three rates were inconsistent you could convert rupees to dollars to pounds and back to more rupees — the same no-free-lunch argument the Option pricing subject's put-call parity runs on, applied to currencies.

Which gives the first practical point of the subject. You do not have a dollar exposure or a pound exposure; you have an exposure to a set of relative prices, and a fund holding European companies gives you euro exposure whether its name mentions currency or not.

Nominal and real

The 96.6149 is a nominal rate — the price today, in today's money.

The real exchange rate adjusts for the two countries' price levels. If Indian prices rise 6% a year and American prices 2%, then the rupee must fall about 4% a year against the dollar simply to keep Indian goods equally competitive. A nominal depreciation of 4% in those circumstances is not a weakening at all — it is the exchange rate doing its job.

This matters for a long-horizon investor more than anything else in the chapter. Over decades, a currency of a higher-inflation economy tends to depreciate against one of a lower-inflation economy, roughly in line with the inflation gap. So some of what looks like currency gain on a foreign holding is the mirror of inflation you are also experiencing at home, and the Measuring your return subject's insistence on real rather than nominal returns applies with double force here.

The honest limit on that idea is that it holds loosely and over long periods. Over any few years, exchange rates move for reasons that have nothing to do with relative inflation — capital flows, rate differentials, commodity prices, politics — and those moves can be much larger than the inflation gap.

What moves it, briefly

Four things, in roughly descending order of how much they explain over an investor's horizon.

Interest rate differentials and capital flows. Money moves towards higher real returns, and the flow itself moves the rate. Chapter 5 makes this precise, because the differential is also exactly what hedging costs.

Inflation differentials, as above, over long horizons.

The trade and current account balance. A country importing more than it exports is a net seller of its own currency.

Commodity prices, for India specifically. India imports most of its crude, so a rise in oil prices is a rise in demand for dollars. The Financial institutions subject's chapter on inflation targeting makes the related point that fuel prices move Indian inflation through the same channel.

The RBI's position

The Financial institutions subject lists the RBI's jobs, and one of them is that it "manages the exchange rate, without a target for it."

Both halves of that are load-bearing. The RBI intervenes — buying and selling dollars from reserves to smooth movement — and it does not defend a level. An investor should expect the rupee's path to be smoother than a freely floating currency's and should not expect any particular level to be protected.

And there is a sharper way to see what the RBI does when the rupee is under pressure, which belongs in this chapter because it is the clearest evidence of the policy stance. The Liberalised Remittance Scheme limit — how much a resident can send abroad in a year — has moved like this:

Date	Limit (USD)
4 February 2004	25,000
20 December 2006	50,000
8 May 2007	1,00,000
26 September 2007	2,00,000
14 August 2013	75,000
3 June 2014	1,25,000
26 May 2015	2,50,000

Read the August 2013 row. The limit was cut by more than 60%, from 200,000 to 75,000, in a single step. The RBI's own description of the sequence is that it was "revised in stages consistent with prevailing macro and micro economic conditions."

August 2013 was a period of acute pressure on the rupee. So the permission to move money out of the currency was reduced at precisely the moment an investor would most have wanted to use it. That is not a criticism of the policy — restricting outflows is a standard response to currency stress — but it is a fact an investor should plan around, and chapter 3 returns to it as the main structural risk of the whole subject.

Working the problem

Rate now 96.6149; a year ago 90.

Step 1 — write it both ways.

	A year ago	Now
Rupees per dollar	90	96.6149
Dollars per rupee	0.011111	0.010350

Step 2 — the rupee fell. Dollars per rupee went down, which is unambiguous.

Step 3 — the two percentages.

From the dollar's side: a dollar went from 90 rupees to 96.6149, so

$\frac{96.6149}{90} - 1 = +7.35\%$

The dollar appreciated 7.35% against the rupee.

From the rupee's side: a rupee went from 0.011111 dollars to 0.010350, so

$\frac{0.010350}{0.011111} - 1 = -6.85\%$

The rupee depreciated 6.85% against the dollar.

Step 4 — why they differ, which is the point of the exercise. 7.35% and 6.85% describe the identical event. They differ because a percentage change depends on what is in the denominator, and the two calculations use different denominators.

The relationship is exact:

$\frac{1}{1 + 0.0735} - 1 = -0.0685$

A gain of $x$ on one side is a loss of $x/(1+x)$ on the other, which is always smaller in size. The gap widens as the move gets larger: a currency that halves against another means the other doubled — a −50% and a +100% describing one event.

Why this is not pedantry. Headlines and fund factsheets quote currency moves from whichever side makes the story larger, and the difference is real money on a large holding. When someone tells you a currency moved by a percentage, the first question is which currency they measured it in.

And the practical habit it produces. Before any calculation in this subject, write the rate in the direction that puts your currency in the numerator — rupees per unit of foreign currency — because that is the direction in which your returns are actually measured. Chapter 2 does nothing else.

The point

An exchange rate quoted as 96.6149 rupees per dollar prices the dollar in rupees, so the number rising means the rupee falling — the reverse of most people's intuition. Rates for three currencies imply a cross rate that must hold or there is an arbitrage, which means an exposure is to a set of relative prices rather than to one currency. Nominal rates drift with inflation differentials over long horizons, so part of an apparent currency gain mirrors inflation you are also living through; over shorter periods, rate differentials, flows and commodity prices dominate. The RBI manages the rate without targeting a level — and when the rupee was under pressure in August 2013 it cut the amount residents could send abroad from USD 200,000 to USD 75,000, which is the risk the rest of this subject has to plan around.