Covered interest rate parity

The following common approximation is valid when S is not too volatile :


An example
In short, assume that

This would imply that one dollar invested in the US < one dollar converted into a foreign currency and invested abroad. Such an imbalance would give rise to an arbitrage opportunity, where in one could borrow at the lower effective interest rate in US, convert to the foreign currency and invest abroad.

The following rudimentary example demonstrates covered interest rate arbitrage (CIA). Consider the interest rate parity (IRP) equation,


Assume:

the 12-month interest rate in US is 5%, per annum
the 12-month interest rate in UK is 8%, per annum
the current spot exchange rate is 1.5 $/£
the forward exchange rate implied by a forward contract maturing 12 months in the future is 1.5 $/£.
Clearly, the UK has a higher interest rate than the US. Thus the basic idea of covered interest arbitrage is to borrow in the country with lower interest rate and invest in the country with higher interest rate. All else being equal this would help you make money riskless. Thus,

Per the LHS of the interest rate parity equation above, a dollar invested in the US at the end of the 12-month period will be,
$1 · (1 + 5%) = $1.05
Per the RHS of the interest rate parity equation above, a dollar invested in the UK (after conversion into £ and back into $ at the end of 12-months) at the end of the 12-month period will be,
$1 · (1.5/1.5)(1 + 8%) = $1.08
Thus one could carry out a covered interest rate (CIA) arbitrage as follows,

1.Borrow $1 from the US bank at 5% interest rate.
2.Convert $ into £ at current spot rate of 1.5$/£ giving 0.67£
3.Invest the 0.67£ in the UK for the 12 month period
4.Purchase a forward contract on the 1.5$/£ (i.e. cover your position against exchange rate fluctuations)
At the end of 12-months

1.0.67£ becomes 0.67£(1 + 8%) = 0.72£
2.Convert the 0.72£ back to $ at 1.5$/£, giving $1.08
3.Pay off the initially borrowed amount of $1 to the US bank with 5% interest, i.e $1.05
The resulting arbitrage profit is $1.08 − $1.05 = $0.03 or 3 cents per dollar.

Obviously, arbitrage opportunities of this magnitude would vanish very quickly.

In the above example, some combination of the following would occur to reestablish Covered Interest Parity and extinguish the arbitrage opportunity:

US interest rates will go up
Forward exchange rates will go down
Spot exchange rates will go up
UK interest rates will go down

Interest rate parity

Interest rate parity, or sometimes incorrectly known as International Fisher effect, is an economic concept, expressed as a basic algebraic identity that relates interest rates and exchange rates. The identity is theoretical, and usually follows from assumptions imposed in economic models. There is evidence to support as well as to refute the concept.

Interest rate parity is a non-arbitrage condition which says that the returns from borrowing in one currency, exchanging that currency for another currency and investing in interest-bearing instruments of the second currency, while simultaneously purchasing futures contracts to convert the currency back at the end of the holding period, should be equal to the returns from purchasing and holding similar interest-bearing instruments of the first currency. If the returns are different, an arbitrage transaction could, in theory, produce a risk-free return.

Looked at differently, interest rate parity says that the spot price and the forward or futures price of a currency incorporate any interest rate differentials between the two currencies.

Two versions of the identity are commonly presented in academic literature: covered interest rate parity and uncovered interest rate parity.

Cost of carry

The cost of carry is the cost of " carring " or holding a position. If long , the cost of carry is the cost of interest paid on a margin account. Conversely, if short, the cost of carry is the cost of paying dividends, or opportunity cost the cost of purchasing a particular security rather than an alternative. For most investments, the cost of carry generally refers to the risk-free interest rate that could be earned by investing currency in a theoretically safe investment vehicle such as a money market account minus any future cash-flows that are expected from holding an equivalent instrument with the same risk (generally expressed in percentage terms and called the convenience yield). Storage costs (generally expressed as a percentage of the spot price) should be added to the cost of carry for physical commodities such as corn, wheat, or gold.

The cost of carry model expresses the forward price (or, as an approximation, the futures price) as a function of the spot price and the cost of carry.


where F is the forward price, S is the spot price, e is the base of the natural logarithms, r is the risk-free interest rate, s is the storage cost, c is the convenience yield, and t is the time to delivery of the forward contract (expressed as a fraction of 1 year).

The same model in currency markets is known as interest rate parity.

For example, a US investor buying a Standard and Poor's 500 e-mini futures contract on the Chicago Mercantile Exchange could expect the cost of carry to be the prevailing risk-free interest rate (around 5% as of November, 2007) minus the expected dividends that one could earn from buying each of the stocks in the S&P 500 and receiving any dividends that they might pay, since the e-mini futures contract is a proxy for the underlying stocks in the S&P 500. Since the contract is a futures contract and settles at some forward date, the actual values of the dividends may not yet be known so the cost of carry must be estimated.

Carrying charge

A carrying charge is the cost of storing a physical commodity, such as grain or metals , over a period of time. The carrying charge includes insurance , storage and interest on the invested funds as well as other incidental costs. In interest rate futures markets, it refers to the differential between the yield on a cash instrument and the cost of the funds necessary to buy the instrument. Also referred to as cost of carry.

The interest expense on money borrowed to finance a margined securities position.

Why should a convenience yield exist?

Users of a consumption asset may obtain a benefit from physically holding the asset (as inventory ) prior to T (maturity) which is not obtained from the futures contract. These benefits include the ability to profit from temporary shortages, and the ability to keep a production process running.

One of the main reasons that it appears is due to availability of stocks and inventories of the commodity in question. Everyone who owns inventory has the choice between consumption today versus investment for the future. A rational investor will choose the outcome that is best.

When inventories are high, this suggests an expected relatively low scarcity of the commodity today versus some time in the future. Otherwise, the investor would not perceive that there is any benefit of holding onto inventory and therefore sell his stocks. Hence, expected future prices should be higher than they currently are. Futures or forward prices Ft,T of the asset should then be higher than the current spot price, St. From the above formula, this only tells us that r − c > 0.

The interesting line of reasoning comes when inventories are low. When inventories are low, we expect that scarcity now is greater than in the future. Unlike the previous case, the investor can not buy inventory to make up for demand today. In a sense, the investor wants to borrow inventory from the future but is unable. Therefore, we expect future prices to be lower than today and hence that Ft,T < St. This implies that r − c < 0.

Consequently, the convenience yield is inversely related to inventory levels.

Convenience yield

A convenience yield is an adjustment to the cost of carry in the non -arbitrage pricing formula for forward prices in markets with trading constraints.

Let Ft,T be the forward price of an asset with initial price St and maturity T. Suppose that r is the continuously compounded interest rate for one year. Then, the non-arbitrage pricing formula should be

Ft,T = Ster(T − t).

However, this relationship does not hold in most commodity markets, partly because of the inability of investors and speculators to short the underlying asset, St. Instead, there is a correction to the forward pricing formula given by the convenience yield c. Hence

Ft,T = Ste(r − c)(T − t).

This makes it possible for backwardation to be observable.

Example: A trader in derivatives market, has observed that the price of 6 month gold futures price is Rs.12,000 per 10 grams and the spot price is Rs.13,710 per 10 grams. The annualized borrowing rate is 12.5% and storage cost is negligible. In this regard, the convenience yield:



12000 = 13710 + (13710 x 0.125 x 6/12 - convenience yield)

CY = Rs. 2566.875; as a percentage of spot price = 18.72%

Known Risks

The 2008-2009 Icelandic financial crisis has among its origins the undisciplined use of the carry trade. The US dollar and the yen have been the currencies most heavily used in carry trade transactions since the 1990s. There is some substantial mathematical evidence in macroeconomics that larger economies have more immunity to the disruptive aspects of the carry trade mainly due to the sheer quantity of their existing currency compared to the limited amount used for FOREX carry trades.