Annuity Formula - How it Works

One formula I wanted to have a look at (just in time for both the accounting exam tomorrow and the finance exam on Friday) is the annuity formula. While the math looks rather convoluted, I wanted to strip it down to it's parts to understand how (and why) it works.

First let's look at a few things. Assume that you've already explained how a perpetuity formula works (without growth), you know that the value of a perpetuity is:

(Assume it goes forever beyond period 7).

PV = CF / r

Where:

• PV is the present value
• CF is the cash flow per period
• r is the rate per period

The next question I would propose is this, what is the value of the perpetuity in period n at time 0? Well, it would be:

Well it would be the same as the PV's value at time n, discounted back to 0. Since the cash flows at time n would look the same as now, the PV at time n should be the same as the PV now.

PV @ n = PV / (1+r)^n

= CF / [r x (1+r)^n]

Now the last question, what is th value (both of the cash flows and the PV) of the perpetuity now minus the perpetuity at time n? Well, if you draw a diagram, the answer is an annuity from 0 to n. And the math shows the same:
(Note this graph is merely the first graph minus the second graph in the same way the math is the first PV minus the second.)

PV - PV @ n = PV - PV / (1+r)^n
= PV (1 - 1/(1+r)^n)
= CF (1 - 1/(1+r)^n) / r

This is the annuity formula for a cash flow CF, to period n at discount rate r, which is much easier than doing a DCF for each of the cash flows (imagine doing a DCF for 30 even cash flows mechanically).

This is a slight variation on the question that Kent Womack presented to us at our review session in the ROM and also highlights how the formula for annuities is constructed.

Venture Capital - Burn Rate

In preparing for our accounting exam (final exams for this quarter start next week with the rest of the university), I wanted to have a quick look at burn rate. One of the concepts taught in our class is cash flows of companies in different stages. An idea I wanted to look into a bit deeper was the idea of cash flows for start up companies.

Cash flow is the primary metric of financial health. For a start up company, there is a lot of money going out the door, but often little money coming in. Revenues are low or non-existant, R&D (and expenses in general) can often be high, and working capital and CAPEX are growing.

Although I've seen different "interpretations" of burn rate it is essentially an FCF which is negative. An operational definition is how much cash is going out the door excluding what is being replaced through financing activities (CFF). I would say that CFO less CAPEX is generally a good proxy of where burn rate is. The assumption is that other sources of cash flows (selling assets, raising cash through financing etc) are not guaranteed and also not sustainable.

The next important measure of financial health is the actual cash and equivalents account. Between the two values, burn rate and cash, you can approximate how long the company will survive without additional financing activities (cash / burn rate per quarter = approximate longevity in quarters).

The goal, of course, is the hockey stick shaped recovery: eventually investing enough to develop a revolutionary product or service that causes revenue and profits to go through the roof (and provided dramatic long term IRR).

Enterprise Value - Cash, how much is too much?

One interesting note made by our professor, Heather Ann Irwin, was for calculating EV.

I've learned the two basic ways to calculate EV (again in theory they should work out to be the same):

EV = EBITDA x Multiple
or
EV = Market Capitalization + Net Debt

Where
Market Capitalization = Share Price x # of Shares
Net Debt = Long Term Debt + Short Term Debt - Cash and Equivalents

I've been told there are more sophisticated versions of EV which include:
+ Preferred Shares
+ Minority Interest

And that the reason we use EV is to look a the company's value assessed under a capital structure neutral scenario (because the capital structure will change when you acquire it).

What I wanted to focus on is the Net Debt component, specifically cash and equivalents. Heather Ann Irwin mentioned a version of the formula which uses "excess cash" instead of cash. I had an idea what she meant but I asked her to clarify. She confirmed my perspective of her idea:

You subtract cash from net debt (and from EV) because cash has a special relationship as a highly liquid asset, so it is often seen as different from other working capital accounts (such as AR, Inv or Prepaid Exp). However, a company still requires some cash to run. The assumption of removing cash from net debt implies that you are acquiring the company for it's "raw" value. Leaving the cash in the company's EV is like buying cash with cash.

However, Heather Ann Irwin proposed that instead of "cash and equivalents" we should use "excess cash". This subtle difference is rather interesting. Yes, cash has a special relationship and is therefore different than other current accounts and working capital, however, you still need SOME cash in order to operate and maintain liquidity. But you don't want ALL the cash (you don't want to have to raise more funds than you need or else you risk screwing up your WACC when you try to raise too much funds). So rather than cutting out all the cash (or none at all) she is suggesting that you remove the excess cash, cash that is not necessary.

In other works, there is some cash which should be treated like working capital because it is actively employed in keeping the company running. The other "cash and equivalents" which are relatively stagnant should be excluded from the EV calculation. When I asked her what constitutes "excess cash" and how would you determine it, she had a great answer: Look at the liquidity ratios and do comps analysis. I'd have to think that it would also be prudent to look at the cash cycle.

I guess this is one of those subtle points that would probably come up in the negotiation of an M&A deal if someone was thinking of acquiring a company. It would probably come up as a point of discussion in terms of the strategic nature of the acquisition and the target capital structure after the deal was done.

Terminal Valuations - Theory and Practice

Yesterday, we had another Capital Markets Technical Prep session. We were looking at valuation methods including DCF, multiples, book value and precedent transactions.

In DCF, we talked about how to value a company's terminal value and discussed how in theory the values should be the same. This a concept I talked about at the Analyst Exchange when I was giving my lecture on geometric series (the math behind DCF's perpetuity formula). In the video, I briefly mentioned how our Hedge Fund Manager commented how it was a coincidence that the numbers were so similar. In theory, as our professor, Heather Ann Irwin mentioned, they should be the same and I just wanted to have a quick look at what the implication is.

There are two methods for valuing a companies terminal value are using a perpetuity method and EBITDA multiples.

The first method, the Terminal Value calculation using a perpetuity formula is: TV = FCF / (WACC - g).

The second method, the TV using EBITDA multiples is: TV = EBITDA x Multiple.

However, if in theory, they are supposed to be the same:

FCF / (WACC - g) = EBITDA x Multiple

I wanted to express the multiple in terms of the perpetuity formula so I rearranged the equation to get:

Multiple = (FCF/EBITDA) / (WACC - G)

This is exactly the point I was trying to make in my lecture in New York when I said that the two formulas and methods were related (except I forgot to highlight the "correction factor" between FCF and EBITDA which is essentially the same as a cash flow to operating profit margin - a factor which adjusts for the difference between FCF and EBITDA - just because EBITDA is often a proxy for FCF doesn't mean it's exact).

Another way of looking at this is as a mathematical proof for why comps valuation works. From an Integrative Thinking perspective, it is essentially looking at two different models for valuation which are looking at the same object and producing different results. Even though in theory both models look at the identical object, they will produce different values, yet I think this is a good integrative solution for understanding what is salient and causal in both models (and how they are related despite their differences).

In this way, if you could have perfect information, assuming that other analysts did comprehensive DCF, you could take a similar company and use the comps multiples to value that company.

Revisiting History - "Pricing the upside derivative"

Our finance professor, Kent Womack, was just describing the model for pricing derivatives and it is almost exactly what I suggested the best method for pricing options would be based on my intuition in January (and was the topic of my Peter Godsoe Scholarship Award in Financial Engineering). He even asked the same questions I was looking into when I was studying for the CFA exam regarding the profit profiles for different put options and call options and why you would enter into different positions.

In his slide, he even mentions the idea of the distribution of stock prices influencing the expected outcome. I think the only difference in our models is that he used a distribution called Geometric Brownian Motion. It's similar to my normal distribution assumption, however, it accounts for an upward drift (which I didn't account for). However, I wonder if it can be approximated with a shifted normal distribution (mean greater than zero).

Also, to actually determine his option price prof. Womack used simulations whereas my model was based more on mathematical calculations. I'm sure both methods to determine the price are acceptable as it's more the model for describing the final underlying price that is more important.

Another improvement from his model is the inclusion of the time value of money as he discounts the future gains back to the present value.

Bova @ the ROM

Due to construction, renovation and expansion at Rotman being disruptive, we've moved our classes to the Royal Ontario Museum theater.

Our accounting professor, Francesco Bova, had the honour of being the first prof to teach us in the ROM. He gave us a warm morning welcome (shown above). It was another rare class where all four sections were together.

At the end of class, Francesco thanked the PSO and the class reps: Myself, Amit, Thi and Katie. Anita McGahan had also done the same in our last Strategy class. What classy profs.

Kingsford Charcoal Case

For our Managing Customer Value class, myself and Jasmine were warm called to do a presentation on the brand of Kingsford Charcoal case and present an overview of the situation. Vincent and Yijun provided an analysis of the price strategy analysis. Finally Harsh and Kim represented agency 1 and Irina, Mainak and Gang presented as agency 2.

Our professor asked us (Jasmine and I) to ask questions of the two agency groups which was a bit of a unique experience. I drew on the classes' conversation to try to ask intellegent questions to understand how they were positioning Kingsford charcoal relative to it's competitors, what their vision was for differentiating our product and how they planned to build an advertising strategy to drive consumer behaviour.

Harsh and Kim had picked mediums which we thought were in line with the BBQ experience. Agency two, with Irina, Mainak and Gang, showed that they understood Kingsford's position as a market leader and planned to exploit it by emphasizing the advantages of charcoal. However, we felt that their advertising strategy wasn't as focused.

In the end, we had to choose which group we prefered and we went with Agency 1 (Harsh and Kim).

It was an intersting exercise as Jasmine and I had to have a discussion outside of the class while the class came up with it's own selection. I'm told they came up with the same decision, albeit possibly based on different criteria.

Dubai World in Trouble

I've had this article (or variants) sent to me from many different people who are concerned / aware of what is happening in Dubai.

We had just done our Dubai presentation on market entry (our project was done on Sugar Mountain and their positioning with international infrastructure to source confectionaries from around the world) and were looking at different possible locations.

We had discovered Dubai World in our research and commented on how monsterously large it was. It actually dwarfs other famous projects from the same real estate development company, Nakheel, such as the famous World project, which creates small artificial islands on the coast in the shape of the earth.

An interesting point, the financing involve is actually islamicly based, as the bonds are sukuks. But there are interesting implications when it involves default and unwinding financial positions which have islamic components. It will be an interesting lesson in understanding not only islamic financial instruments, but also a pragmatic lesson in how distressed islamic investment instruments.

I wonder how liquidation would work. One of the general tenants (as I understand it) of islamic finance is that there are not many recourses for default, however, bonds do have an equity component so I wonder if the bonds will just naturally "convert" if the bond (sukuk) holders do not agree to delaying / suspending payments.

Strategy with Anita McGahan, Fleck Atrium Part II

In another rare treat, Anita ran another tutorial based on our most recent quiz in the Atrium earlier today (around lunch time). She discussed the answers and rational for the quiz questions as well as what she was looking for in good responses.

This time the tutorial was only for two of the four sections at Rotman. The other two courses are having their tutorial now.

Next week, due to construction in extending the Rotman facilities, we've been told that some of our classes will be in the Royal Ontario Museum (ROM). That plans to be exciting, not only to also be in a large class with all our 265 classmates, but also to be in such an interesting "class room". I hope we don't have to pay additional "admission".

Companies Investing in Securities

Our professor, Francesco Bova, has the honour / misfortune of having us for our Friday afternoon Financial Accounting class. It's usually a good time (yes, you read that correctly).

Just now, he was asking us what types of securities would be considered as being classed as "held to maturity". He jokingly hinted that the only securities with maturities are bonds. The best question of the day:

"Does a zero coupon bond have a maturity?"