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CFA Level III: Implementation Shortfall

Good evening,

A quick post tonight to discuss a topic of Trading, Rebalancing and Monitoring part of the Level III curriculum called Implementation Shortfall. The reason why I chose to do this is because it took me some time to overcome the naming conventions of the CFA institute, which are, with all due respect, very counterintuitive in my opinion.

The idea is very simple: you would like to be able to measure the quality of the execution of a trade compared to an ideal execution.

From what I’ve seen in mock exams and exercises, they always give you a little story like the following one:

  • At some point, the investment manager decides to buy 10 Manchester United stocks, which trades at 20.
  • This is called the benchmark price (BP), for some reason.
  • Then (usually the following day), a limit order is placed in the market, say at 19.95 and is not executed at all. The market closes on that day at 20.10. Too bad.
  • You pay 0.05 per share of commission.
  • The following day, the order is revised at like 20.15 and 8 stocks (i.e not the whole 10) are filled at that price, and the market closes at 20.20.

What happens there? Well, assume you are able to magically implement your trading ideas instantly at no cost: this is called the paper portfolio. What is your profit at the end of the story?

  • You buy 10 shares at 20 for 200.
  • At the end of the story, your stocks are worth 20.20 each, which gives you a total of 202.
  • You earned 202 – 200 = 2

In the real world, it did not work out that way:

  • You bought 8 stocks at 20.15 for 161.2
  • You pay 0.4 in commission
  • At the end of the story, your stocks are worth 20.20 each, which gives 161.6
  • You earned 161.6 – 161.2 – 0.4 = 0

The implementation shortfall is defined as follows:

$$\frac{\text{paper portfolio gain}-\text{real portfolio gain}}{\text{paper portfolio investment}}=\frac{2}{200}=1.0\%$$

This means that 1.0% of the potential investment was lost (or, more precisely, not won)  in the implementation, due to different frictions.

The CFA Institute then provides a way to split this difference in different components.

First the explicit costs, which consists in all the obvious transaction costs that are expressed in the trade:

$$\frac{\text{commission}}{\text{paper portfolio investment}}=\frac{0.4}{200}=0.2\%$$

That’s fine. But then comes the bizarre naming conventions.

Some extra costs come from the fact between the moment when the investment manager decides to buy the stock and the day when the order is partially filled, the market moved.

The slippage or delay costs is the difference between the benchmark price and the closing price, the day before the execution day (which is called, poorly the decision price, I don’t understand why) divided by the benchmark price, times the percentage of the order that was filled. In order case we have:

$$\frac{20.10-20.00}{20} \cdot \frac{8}{10}=0.4\%$$

It is the portfolio of the implementation shortfall that was lost because of the delay between the time the manager saw the opportunity and the day the trade was partially executed.

Then, the realized loss is the difference between the execution price and the closing price the previous day (so-called decision price), divided by the benchmark price times the percentage of the order that was filled:

$$\frac{20.15-20.10}{20} \cdot \frac{8}{10}=0.2\%$$

This is what was lost during the execution day.

Finally the missed trade opportunity cost is the difference between the price at the end of the story and the benchmark price divided by the benchmark price time the proportion of the order that was not filled:

$$\frac{20.20-20.00}{20} \cdot \frac{2}{10}=0.2\%$$

This is what was lost by not being executed.

If you sum all the components, you get 0.2% + 0.4% + 0.2% + 0.2% = 1.0%, the total implementation shortfall.

So you are able to see that, in this example, the main component of the implementation shortfall is the delay between the trade idea and the trade execution day. The limit order at 19.95 was too ambitious and resulted in a loss.

Notice also that all the examples I saw are examples where the market goes in the trade direction (i.e. market goes up after a buy decision). It could be possible that the market goes adversely, which would result in a negative implementation shortfall… i.e a gain.

That’s all for today.

I’ll be back soon with more.



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.



CFA Level III, here I come!

Good evening everyone!

I know it’s been a long time since I last posted on the site and I would like to apologize for not having been adding new content since the end of May.

Quite frankly the reason why I’ve been away from the blog from a long time is twofold. First, I am planning my wedding for the end of the year and it is taking me most of my spare time – for those of you who are married, you know what I mean. Second, I had a very bad feeling when I came out of the Level II exam and I was actually quite disappointed. So, I decided to take a break and to focus on the wedding before heading to my summer holidays and to wait for the results. In the end, once it’s done, there is nothing you can do about it.

Tuesday, late in the afternoon, I finally got the crucial CFA e-mail, which noticed me that I had passed the Level II along with 43% of that level’s candidates. I literally jumped out of my chair as if I had scored a goal in the champions league final. Let’s face it, I most certainly had a bit of luck here.

Looking at my “detailed” results (which, as you well know, I’m not allowed to share), I then realized that my work had paid off. Although I did not post about the accounting part because I feel I am not expert enough to really publish something about it, I really paid a lot of attention to this section of the curriculum. And it paid off. As a matter of fact, there is so much weight on this topic in the exam that you can definitely get yourself in a nice position by being confident on the classic exercises. I did practice a lot using Schweser’s QBank and practice exam. In the real exam, I felt accounting questions were not that vicious and hence I managed to score high there. So, for those of you who are taking the Level II next June, spend a lot of time on accounting if you’re not very good at it. The material is quite huge, but as your practice it will start making sens and quite surprisingly there is not much to learn by heart in a way: it is more logic that it might seem at first glance.

Free cash flows are also a major topic of the Level II. Again, when you look at the formulas you might get scared at first glance. However, you should realize that by learning the main formula, you can derive all the other without learning them by heart. Again, even this basic formula might seem quite complicated at the beginning, but it really make sens once you get used to it and I found it quite easy to apply once you understand the rationale behind it.

Economics and Alternative Investments were the most complicated parts of the curriculum to master in my opinion. In fact, I believe that both these topics require a lot of material to be learnt by heart if you’re not practicing them in your daily job. And I hate learning stuff by heart. Derivatives are of the same kind, but with my quantitative finance academic background, I already knew all the material. Again, if you’re not familiar to derivatives, you will have to grasp a lot of concepts which take time to get accustomed to.

All in all, I would say that the basic strategy is really not to give up on any topic of the curriculum. Of course, there will be part of the material you will not be comfortable with on exam day, but you definitely should have an idea of what’s going on everywhere. If you can’t master something, at least grasp the global picture. This is key because it will most of the time allow you to discard one of the three possible answers. Once this is done, even if you choose randomly between the remaining two answers, your probably of success drastically goes up to 50% instead of the initial 33%. Plus, don’t forget that item sets are composed of questions of various difficulties. Hence, being able to score on the easy questions is critical as it can considerably improve your overall result. Remember that if you get battered in a particular topic, it probably won’t matter what you’ve done in the rest, you will fail the exam; the passing criteria are not public and I suspect there is a minimum score for each of the topics.

To sum up, I’m really happy to be done with this Level II because I heard it was the most difficult of the three for candidates with my profile. I must say that I won’t be disappointed of not having Economics and Financial Accounting in the Level III material, but I will have to work extra hard on my writing skills as the Level III’s morning session consists in writing small essays… not in question sets.

I’ll be back soon with more posts on something else than the CFA!



CFA Level II: Valuating Bonds with Embedded Options

Good evening everybody,

Tonight’s focus will be dedicated to valuing bonds with embedded options. I chose this topic because it involves binomial trees, which is an important an very testable point of the Level II curriculum.  Binomial trees are encountered in two different topics: Fixed Income and Derivatives. Although the concept is the same for the two instances,  the main difference is the fact that the interest rate tree is given, whereas you have to construct the tree for stock values yourself. Besides, in interest rate trees, the probability of going to up or down nodes is 0.5. For options on stocks, you have to determine the risk-neutral probability value.

For simple bonds with no embedded options, in each node, you write:

  • The current price of the bond for the remaining payments (including face value)
  • The coupon at that node
  • The interest rate (more specifically the forward rate) at that not (which is given).

You use the interest rate tree using backward induction, i.e. you start by the node at the right of the tree (the ones the further away from now) where you know the price: the face value. It’s the only place where you know what the price will be. At a given node before the final layer, you have to compute the price of the bond which is given by computing the average (because the probability of ups and downs is 0.5) of the present value of the prices at the two following nodes. To compute the present value, you use the forward rate given at the current nodes. Therefore, there are no forward rates given for nodes at the final layer. Using this process sequentially for each node from the right to the left of the tree, you end up computing the initial node’s value: the price of the bond today.

Because the interest rate trees are built to be arbitrage trees, they are computed such that the benchmark security’s price you get using the tree is the market price.

So if you compute the price of a bond which is different from the benchmark, you will get a different value. This is where spreads come into play. Nominal spreads are computed using YTM. You compute the constant yield implied by the market price of the bond and its benchmark, and you compute the difference between them. However, this is not a good measure as it assumes the yield curve is flat. The Z-spread, for zero-volatility spread, is the spread that is added to each of the term structure’s rate to make its price equal to the benchmark’s price. You can do exactly this by adding the Z-spread to each of the nodes’ rates (you use a value to add to each node’s rate until you find the right value, by trial and error).

For bonds with embedded options, you do exactly the same as for the option-free bond, but the price at the nodes where the call can be exercised you compute the price using the option’s criteria. For example, for a bond callable at 100 in 2 years, if you compute that the price at the second node is 101.5, then you have to use the price which is equal to $\min(101.5,100)=100$.

Recall the following for bonds with embedded options:

$$V_\text{callable}=V_\text{option-free bond} – V_\text{call}$$

$$V_\text{putable}=V_\text{option-free bond} + V_\text{put}$$

The thing is, the market value of the bond with the embedded option will often be different than is theoretical, arbitrage-free, value.  The reason for this comes mainly from the fact that the interest rate tree assumes some interest rate volatility. Remember the following very important facts:

  • Option-free bonds prices are unaffected by interest rate volatility, they are priced as of today and that’s it.
  • The price of the embedded option (call or put) is positively related to interest rate volatility.

When we make the valuation of the bond with the embedded option, we essentially compute the value of the cash flow assuming the interest rate volatility. In a sense, we price the option-feature of the bond, assuming the interest rate volatility of the tree. However, market can change its expectation on the volatility of the bond, and hence, the price is not the same anymore, because although the option-free bond value has not changed, the value of the embedded option is different.

Similarly to the Z-spread, if you find the value you have to add to each node’s rate to get the same value from the tree as the market value, you get the OAS, the option-adjusted spread, which is the spread of the bond “removing” its option feature. You have to use this  spread to compare bonds with embedded option with bonds without embedded options (such as their relative benchmark) or even bonds with embedded options with each other. The price of the option is relative to the volatility of interest rate, not to the credit risk and liquidity risk of the bond; it shall hence be remove to compare inherent quality of the fixed income security.

That’s it for this post.

CFA Level II: Forward Markets and Contracts

Good evening,

A few days away from the exam, I am taking a bit of time to post the main picture of some topics on the curriculum, which I think can be simply explained. This post is dedicated to forwards.

The rationale behind forwards is very simple. Assume there is an asset $S$ which is worth $S_0$ today. You want to enter a contract with somebody to agree to buy the asset at time $T$ at the forward price $FP$. This might seem a bit complicated at first glance for people not familiar with finance mainly because you do not know what $S_T$ (the price of $S$ at time $T$) will be. Well, the truth in the matter is that… it doesn’t matter. Indeed, there is a way you can replicate the action of buying a stock at time $T$ by doing the following:

  1. At time $t=0$:
    1. Borrow $S_0$ at the current rate $R$
    2. Buy the asset at $S_0$
  2. At time $t=T$:
    1. Repay what you borrowed at time 0 with interest for $S_0 \cdot (1+R)^T$
    2. Keep the stock at the value $S_T$.

So, the net investment at $t=0$ is 0, and at $t=T$, you have the stock and you pay $S_0 \cdot (1+R)^T$. So, this is exactly exactly the same thing as buying the stock forward. Hence, you can deduce that:

$$FP=S_0 \cdot (1+R)^T$$

The results for the law of one price, and we call that an arbitrage argument because if the price was any different from stated above, the you could make instant risk-free profit by doing the strategy previously stated (or its opposite).

For example, assume a stock is worth $S_0=100\$$ today, that the interest  rate $R=10\%$, and that you want to buy the stock forward in 1 year. Then, $FP=100 \cdot.(1+10\%)=110$.

If somebody is willing to buy it forward for 115, then enter the forward contract as a seller, thus agreeing selling $S$ for 115 in a year. Borrow 100 today, buy the stock, hold the stock, repay your loan plus interest in a year for 110 and give the stock to the counterparty for the agreed 115. You get a free lunch of 115-110=5.

If somebody wants to sell the stock forward for 105, you should agree to enter the contract as a buyer. Sell the stock short for 100 today, invest the proceeds for the interest rate today, collect the interest invested of 110 in a year, and buy back the stock as agreed for 105. You make a free lunch of 110-105=5.

That’s it. It’s easy. The forward price is the price today invested at the interest rate. What we just did implies that the value today of a forward contract is 0, by definition. However, the value of the contract will evolve between time 0 and expiration T. Clearly at expiration, the value of the contract is given by:


When time $t$ is between 0 and T, we get the following result:

$$V_t = S_t – \frac{FP}{(1+R)^{T-t}}$$

This is quite logical and you can always check that the value at $t=0$ is 0:

$$V_0=S_0 – \frac{FP}{(1+R)^{T-0}}=S_0 – \frac{S_0 \cdot (1+R)^T}{(1+R)^T}=0$$

And for $t=T$

$$V_T=S_T – \frac{FP}{(1+R)^{T-T}}=S_T – FP$$

Notice that this is the value for the long side, i.e. for the person agreeing on buying the asset at expiration for the forward price. Because derivatives are zero-sum games, the value of the short side is the opposite of the value of the long side.

That’s it. I’ll come back with variants later.