A common question asked by FRM candidates (and people who are considering whether to sit for the FRM exam) is, where can I find an introduction to the math? Although the FRM has many qualitative and conceptual topics, it also contains a lot of formulas and numerical illustrations. The CFA, by way of comparison, is broader but the FRM goes deeper. In large part that is due to the nature of risk. For example, in the CFA, (the last time I checked) you need to be able to value a stock option according to the two common option pricing models (Black-Scholes and binomial). The FRM requires you to understand both of these models as a sort of prerequisite: to understand the risk of an option, we start with the valuation of the option then proceed to risk.
After hosting this forum for well over a decade, you can imagine that I’ve learned so much about the candidate’s introduction to quantitative risk. In fact, most of my expertise derives from the joyful interaction with learners; so many mistakes and imprecisions have helped me sharpen the saw, if you will! It’s a good time to summarize the quantitative foundation that I am calling an Introduction to the Quantitative Foundation of Risk. I hope you find it helpful.
1. Present Value (PV)
A common action for us in finance is to estimate the present value (PV) of an asset or portfolio. This present value (PV) we tend to call a theoretical price. The PV is a theoretical price because we do not expect the theoretical price to equal the observed (aka, traded) price. Maybe our theoretical price of some REIT or a bond or a stock is $43.00 but we observe the price is trading right now at $39.00 on an exchange. Then we can say “this REIT is trading cheap”, as in, it’s trading below its theoretical price which is a present value; alternatively if the traded price bids up to the $47.00 we can say “this thing is trading rich.” Notice we can say that because we have two different definitions of price, theoretical versus observed. (A asset’s theoretical price could also be called its intrinsic value, but we’ll avoid intrinsic because we want to use intrinsic for an option’s immediate value–as opposed to its time value–and intrinsic value connotes an obvious value that we can immediately realize. Whereas our asset might not be liquid such that we may not be able to immediately realize its theoretical value.).
A mentor once said to me, all of finance is basically bond math. I’m not sure I agree but bonds are a great place to start because the theoretical price of a bond is its discounted present value (where the “discounted” is superfluous, to me anyway). We want to price a bond, and while there are many different types of prices, unless we say otherwise we generally mean that we want today’s price or the price right now; i.e., the present value. We can either observe a (traded) price in a marketplace (or exchange), or we will estimate the theoretical price by computing its present value. Before we do it, let’s make a philosophical observation: the traded price is objective (we all see the same bid/ask spread, more or less) but the theoretical price is subjective. My theoretical price will be different than yours, but we will observe the same traded price.
Say Acme is an industrial company that wants to borrow cash, so Acme issues a bond with a face value of $1,000. Like many professions, finance sometimes invents multiple synonyms for the same thing, presumably to make the people inside the profession feel special because they have fancy words when simple words will suffice (That’s a joke. Or was it?). Face value has literally four synonyms, which seems confusing to me. We prefer face value for a trivial reason: the face value is a future value and our calculators have a [FV] keystroke. It’s a good tip because when you see “face value” you know that amount is entered via the [FV] keystroke. But it’s also somewhat useful to realize this $1,000 is also called a par value, the principal, or even redemption value (as each of these synonyms has meaning that is helpful in other contexts).
In this example, the investor is a lender who gives Acme cash but expects to be repaid in the future. To keep it simple, we’ll assume this bond pays no coupon interest along the way: it is a zero-coupon bond. Maybe is a 5-year bond which means that it will mature in five years and, at that time, the lender expects to be repaid the redemption value (aka, face value) of $1,000. An exam question about a bond will either need to mention its maturity (e.g., five years), or less commonly but more realistically, will need to provide the bond’s maturity date (e.g., matures on 2031-09-01 so is a bond with ten years to maturity if today is 2021-09-01).
But the $1,000 face value is a future value, to be repaid in five years (in our illustration). How much cash does the lender give the company today? In formal terms, we’re asking, what is the bond’s theoretical price? This is the same as asking, what is the bond’s present value (PV)? To estimate the PV we need the discount rate. Let’s say our discount rate is 8.0% per annum with annual compounding. Yay, we’ve got what we need. The theoretical price of this 5-year zero-coupon bond is given by:
$1,000 / (1 + 8.0%)^5 = $1,000 * (1 + 8.0%)^(-5) = $680.58
By our estimation, omitting fees and frictions, the lender will give Acme $680.58 today and expect to be repaid $1,000 (the face value) in five years. If we increase the discount rate to 9.0%, the theoretical price (PV) declines:
$1,000 / (1 + 9.0%)^5 = $1,000 * (1 + 9.0%)^(-5) = $649.93
And we just illustrated duration risk: the risk that the bond price will go down because the interest rate goes up. The lender, who is long the bond, is maybe a bit sad: she’s earning 8.0% but the market rate (an opportunity cost) is now higher. The borrower, Acme who is short the bond, is maybe a bit happy: they are suddenly paying a below-market interest rate. If the interest instead drops from 8.0% to 7.0%, then the bond price increases from $680.58 to:
$1,000 / (1 + 7.0%)^5 = $1,000 / (1 + 7.0%)^(-5) = $712.99
Now the lender is happy and Acme is sad. In fact, if this bond were like a mortgage (with its prepayment option) and had an embedded call option, then Acme might call the loan and “refinance” the borrowing (with a new bond) at the lower rate.
So our present value (PV) is linked to our future (FV) by a maturity (in this case five years) and a discount rate. Before we finish, I’d like to make three technical points if only because they build our foundation for future ideas.
First, the discount rate is an interest rate but in risk “interest rate” or “interest rate factor” is a bucket that contains many different types and flavors of rates; included in the big bucket are spot rates (a fundamental building block) and our discount rate is a spot rate. The reason I mention this to risk students is so that we start to realize the term “interest rate” is by itself not very specific. Second, I specified the discount rate (aka, spot rate) in a specific way: “8.0% per annum with annual compounding.” A well-written question should be thusly specific. The per annum (aka, annualized) is to distinguish from a different periodicity such as 8.0% per month. However, here is an important convention: interest rate inputs and outputs should (almost) always be annualized. This avoids confusion, please trust me. In this way, an inferior but acceptable assumption is “8.0% with annual compounding”, because we can safely assume the 8.0% is per annum.
The reader (or candidate) should not be left to wonder, is this per month or per quarter? We can assume it is per annum (per year). We serve the rate annualized, and let the reader slice the rate into months or quarters to their own taste. And, when finished the candidate will want to annualize the output. For example, if this were a coupon rate instead of discount rate, we still do not default (pun intended) to saying “4.0% every six months.” Instead we say “8.0% per annum with semi-annual compounding” or, if we are comfortable, “semi-annual 8.0%” or even “s.a. 8.0%”. Due to convention, we know that’s not 8.0% twice a year, rather: it is an annualized or per annum 8.0%, which is 4.0% every six months. This 8.0% is also called the stated or nominal rate, but it’s not an effective rate. In summary, unless we have a good reason otherwise, in which case it should be clearly articulated, stated or nominal rates should be annualized as both inputs and outputs.
Third, the discount rate assumption of “8.0% per annum” is incomplete until we specify the compound frequency. Why? Here is the theoretical bond price if the discount rate, instead, is assumed to be 8.0% per annum with semi-annual compounding:
$1,000 / (1 + 8.0%/2)^(2*5) = $1,000 * (1 + 8.0%/2)^-(2*5) = $675.56
Although the nominal rate is 8.0% in both cases, the bond discounted with semi-annual compound frequency is theoretically cheaper. Put another way, these are two slightly different discount rates. The $680.58 and $657.56 bonds both share the same 8.0% nominal rate but their effective rates differ. The bond discounted at 8.0% per annum with semi-annual compounding has a lower present value (PV) than the bond discounted at 8.0% per annum with annual compounding. To recap this third point: a nominal rate is annualized (per annum) but the nominal rate is an incomplete specification until we also provide the compound frequency.