Five relations connect spot rates, forward rates, interest rates and inflation. Textbooks present them as a list. They are one system, and solving them together from a single data set is what makes the differences between them visible.
Take one set of inputs and hold it for the rest of the section. Spot is ₹84.5000 per dollar. The one-year rupee interest rate is 6.75% and the one-year dollar rate is 4.35%, both quoted annually. Expected one-year inflation is 4.60% in India and 2.40% in the United States. Every number below comes from those five.
How do covered and uncovered interest parity work?
Covered interest parity is the only member of the family enforced by arbitrage. Two riskless ways to hold rupees for a year must pay the same, so the forward is pinned:
F = S × (1 + i_INR) / (1 + i_USD) = 84.5000 × 1.0675 / 1.0435 = ₹86.443460
The forward premium on the dollar is 86.443460 / 84.5000 − 1 = 2.299952%, and the market would quote that as 194.3460 paise of forward points. The crude approximation, the interest differential of 2.40 percentage points, overstates it by 0.100048 percentage points, which is 10 basis points of error on a relation that trades to within one or two. Use the ratio.
Advertisement
Notice which currency ends up at a forward discount. The rupee pays 240 basis points more than the dollar and trades to weaken by almost exactly that much. Higher yield and forward discount are the same fact seen twice, and any pitch that offers the yield without mentioning the discount has left out the price.
The arbitrage that does the pinning is the funding trade run in reverse against a mispriced quote, and it is worth executing once on paper. A dealer quoting the one-year forward at 85.9000, some 54.3460 paise below parity, hands anyone who borrows $1,000,000 a locked ₹567,100, which is 0.671124% of the rupees invested, with no open position at any moment. Practice problem P3 runs the five steps.
Uncovered interest parity takes the same arithmetic and removes the forward contract. It claims the expected future spot equals the covered forward, so the expected rupee depreciation is 2.299952% and the expected spot in a year is ₹86.443460. Identical number, entirely different epistemic status. Covered interest parity is an identity a machine enforces. Uncovered interest parity is a hypothesis about what people expect, and nothing enforces a hypothesis.
What does purchasing power parity say?
Relative purchasing power parity says the exchange rate drifts with the inflation differential:
E[%ΔS] = (1 + π_INR) / (1 + π_US) − 1 = 1.0460 / 1.0240 − 1 = 2.148438%
giving an expected spot of ₹86.315430. That is 12.803 paise stronger for the rupee than the covered forward implies, on the same day, from the same five inputs. The gap is not an error. It is information, and the reconciliation at the end of this run of five says exactly what it contains.
Absolute purchasing power parity is the level version and it is far more demanding: an identical basket should cost the same everywhere once converted. A basket at ₹4,200 in India and $58.00 in the United States implies a rate of ₹72.413793 per dollar, so a market rate of 84.5000 sits 16.690476% above it and the rupee is 14.303203% cheaper than the basket says. Both numbers describe the same gap and differ by the appreciation-versus-depreciation asymmetry, so say which one you quoted.
Absolute purchasing power parity fails badly in levels and it fails for a reason worth knowing. Most of a consumption basket is non-traded (rent, haircuts, school fees, a doctor's time), and non-traded prices track local wages rather than world prices. Poorer countries have lower wages, so their price levels are lower, so their currencies look permanently undervalued on this measure. That is the Balassa-Samuelson effect, and it means a 14% "undervaluation" against a basket is the normal condition of a middle-income currency rather than a trade signal.
How does the Fisher effect fit in?
The Fisher effect decomposes a nominal rate into a real rate and expected inflation. Exactly, rather than by subtraction:
(1 + r) = (1 + i) / (1 + π)
The Indian real rate is 1.0675 / 1.0460 − 1 = 2.055449%. The US real rate is 1.0435 / 1.0240 − 1 = 1.904297%. The gap is 0.151152 percentage points. Subtraction would have given 2.15% and 1.95%, a gap of 0.20 points, which is a third too wide.
The international Fisher effect is the claim that real rates are equal across countries, so the whole nominal differential is an inflation differential and the currency should depreciate by it. On these inputs it does not hold: the nominal gap is 2.40 percentage points, the inflation gap is 2.20 points, and 0.20 points of real-rate difference is left over.
How do the five conditions fit together?
Now the reconciliation, which is the point of solving all five together. Divide the covered forward's 2.299952% by relative purchasing power parity's 2.148438% and you get a wedge of 0.148328%. Compute the real-rate wedge directly, (1.02055449)/(1.01904297) − 1, and you get 0.148328%. The two agree to fifteen decimal places because they are algebraically the same quantity. The forward premium is the purchasing-power drift multiplied by the real-rate wedge, and once you can produce that identity on demand the five relations stop being a list to memorise. A forward premium above the inflation differential says the domestic real rate is higher, and says nothing else.
Here is how the five rank by how tightly they bind, which is the ranking an analyst actually uses:
| Relation | Enforced by | Typical residual | Use it for |
|---|---|---|---|
| Triangular arbitrage | machines, milliseconds | inside the spread | never a trade, always a data check |
| Covered interest parity | a hedged funding trade | a few basis points, wider in stress | pricing a forward, reading a funding basis |
| Uncovered interest parity | nothing | large and persistent | a null hypothesis to test, never a forecast |
| Relative purchasing power parity | goods arbitrage, slowly | large for years, mean-reverting over decades | long-horizon real-rate work |
| Absolute purchasing power parity | goods arbitrage, badly | 15% to 50% for middle-income currencies | comparing price levels, not predicting rates |