# CFA Level III: Interest Rate Parity

Hello everyone,

Today I’m gonna talk about some economic concepts that were mentioned at least since Level II and that are quite useful in the whole curriculum and in finance in general when it comes to dealing with currency management.

# Covered Interest Rate Parity

The idea is quite simple, we will compute the forward exchange rate between two currencies using an arbitrage argument, say EUR and USD. The spot exchange rate is denoted $S_{\text{EUR}/\text{USD}}$: it corresponds to the number of euros you get today for 1 US dollar. Furthermore, the risk-free rate in USD is denoted $R_{\text{USD}}$ and the risk-free rate in EUR is denoted $R_{\text{EUR}}$. The question is, after some time $T$, how many euros will I get for 1 US dollar? This rate is called the forward rate and is denoted $F_{\text{EUR}/\text{USD}}$.

Well, you can price that quite easily using an arbitrage argument! The idea is simple:

• I’m going to borrow today $\frac{1}{1+R_\text{USD}}$ USD which means that at time $T$, I will have to pay back 1 USD.
• Then I’m going to convert what I just borrowed in EUR, which gives me $S_{\text{EUR}/\text{USD}} \cdot \frac{1}{1+R_\text{USD}}$ euros
• I then invest this at the risk-free rate in EUR, and I get at time $T$ $(1+R_\text{EUR}) S_{\text{EUR}/\text{USD}} \cdot \frac{1}{1+R_\text{USD}}$ euros

What I wrote above means that:

$$F_{\text{EUR}/\text{USD}}= S_{\text{EUR}/\text{USD}} \cdot \frac{1+R_\text{EUR}}{1+R_\text{USD}}$$

That’s as simple as it sounds, we have determined the forward price of US dollars in euros at time $T$.

A very easy way of remembering the formula above is noticing that the rate in the numerator and in the denominator are from the same currency as is shown in the rate label: EUR/USD.

Also, recall from this post that in this case (EUR/USD), the US dollar is the asset being priced in euros; the US dollar is an asset like anything else.

Finally, we understand from the formula above that:

$$R_\text{EUR} > R_\text{USD} \implies F_{\text{EUR}/\text{USD}}> S_{\text{EUR}/\text{USD}}$$

This is very useful because you very often have to say what currency is trading at premium or at discount in another currency. First I used to always get that wrong. In fact, it’s very easy.  The currency being traded is the one in the denominator of the label, here USD. Then, if the forward price is higher (lower) than the spot price, it is of course trading at premium (discount).

So, if we say that the USD is trading at premium in EUR, it means that we can have more EUR in the forward market than in the spot market for 1 USD.

# Uncovered interest rate parity

This is in a sense an extension of the covered interest rate parity we just discussed which says that:

$$\mathbb{E}({S_{\text{EUR}/\text{USD}}}_T)= {S_{\text{EUR}/\text{USD}}}_0 \cdot \frac{1+R_\text{EUR}}{1+R_\text{USD}}$$

Notice that here the implication is different than previously, because we say that we expect that the spot rate at time $T$ will be equal to the forward price at today. This comes to says that the currency that has the higher (lower) interest rate is expected to depreciate (appreciate):

$$R_\text{EUR} > R_\text{USD} \implies \mathbb{E}({S_{\text{EUR}/\text{USD}}}_T)> {S_{\text{EUR}/\text{USD}}}_0$$

Indeed, a higher spot rate in the future means that you would get more euros for the same amount of US dollars which means that the euro has depreciated!

A lot of traders disagree with that statement, and this comes to a very famous trading strategy called the carry trade. The idea is really simple as well: these traders do not think that the currency with the higher interest rate will depreciate. They hence short the currency with lower interest rate and invest in the currency with higher interest rate. In the curriculum, they say that this strategy tends to work most of the time, generating positive income. However, they say that, when for some reason, the interest rate goes in the expected direction, they tend to do so very violently and that it can lead to very large losses.

That’s it for today, I hope this little post will help you in mastering this concept, which is key to a lot of different topics at different levels of the CFA curriculum.

Cheers,

Jeremie

# CFA Level II: Economics, Exchange Rates Basics

Good evening everyone,

My weekly task is to go through the Economics part of the Level II curriculum. I was a bit afraid of it, because it was clearly my week point at the Level I and because I think that this topic covers a lot of material compared to its allocated number of questions.

In this level, the first challenge is to take into account the bid-ask spread for currency exchange rates. Just as we saw for security markets at Level I, exchange rate do not value a single “value”. That is, you cannot buy and sell a currency at the same price instantaneously. This is because you need to go through a dealer who has to make money for providing liquidity: this economical gain is provided by the bid-ask spread.

Let’s go back to the basics by looking at the exchange rate $\frac{CHF}{EUR}$. The currency in the denominator is the base currency; it is the asset being traded. The currency in the numerator is the price currency; it is the currency used to price the underlying asset which is in this case another currency. This is exactly like if you were trading a stock $S$. The price in CHF could be see as the $\frac{CHF}{S}$ “exchange rate”, i.e. the number of CHF being offered for one unit of $S$. Now as mentioned before, exchange are quoted with bid and ask prices:

$$\frac{CHF}{EUR} = 1.21 \quad – \quad 1.22$$

This means that $\frac{CHF}{EUR}_\text{bid}$ is 1.21 and $\frac{CHF}{EUR}_\text{ask}$ is 1.22. Again, you are trading the base currency: here Euros.

• The bid price is the highest price you can sell it for to the dealer.
• The ask price is the lowest price you can buy it for to the dealer.

If you want to make sure you got it right, just make sure  you can’t instantaneously buy the base currency at a given price (which you believe to be the ask) and sell it at a higher price (which you believe to be the bid). In this example, you can buy a EUR for 1,22 CHF and sell it instantaneously for 1.21 CHF making a loss of 1.22-1.21=-0.01 CHF. In fact, the loss can be seen as the price of liquidity which is the service provided by the dealer for which he has to be compensated. So the lower value is the bid, the higher value is the ask (also called the offer).

Recall from Level I that you could convert exchange rates by doing:

$$\frac{CHF}{EUR} = \frac{1}{\frac{EUR}{CHF}}$$

This is simple algebra and it works fine as long as you don’t have the bid-ask spread to take into account. The problem is that at Level II, you do. To invert the exchange rate with this higher level of complexity, you have to learn the following formula:

$$\frac{EUR}{CHF}_\text{bid} = \frac{1}{\frac{CHF}{EUR}_\text{ask}}$$

This might look complicated at first, but I got something in my bag to help you learning it. Look do the following steps:

• Define what you want on the left-hand side of the equation (currency in the numerator, currency in the denominator, bid or ask).

$$\frac{A}{B}_\text{side}$$

• On the right-hand side of the equation, write the inverse function:

$$\frac{A}{B}_\text{side}=\frac{1}{\cdot}$$

• On the right-hand side, replace the $\cdot$ by the inverted exchange rate:

$$\frac{A}{B}_\text{side}=\frac{1}{\frac{B}{A}_\cdot}$$

• Finally, replace the remaining dot on the right-hand side by the opposite side:

$$\frac{A}{B}_\text{side}=\frac{1}{\frac{B}{A}_\text{opp. side}}$$

Let’s take show how this work using our base example:

$$\frac{EUR}{CHF}_\text{bid}=\frac{1}{\frac{CHF}{EUR}_\text{ask}}$$

Simple. You can simply apply this method interchangeably to suit your needs. Actually, you might wonder what you need that to compute cross rates, which will be the subject of another post. Until that, grasp the concepts presented here and stay tuned on this blog!