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David Compares Durations in CFA^{®} vs. FRM^{®}

I’d like to clarify duration terminology as it pertains to differences between the CFA and FRM. Our forum has hundreds of threads over 12+ years on duration concepts (it’s hard to say which links are the best at this point, but I’ll maybe come back and curate some best links). Our YouTube channel has an FRM P2.T4 that includes videos on DV01, hedging the DV01, effective duration, modified versus Macaulay duration, and an illustration of all three durations. There are many nuances and further explorations, but here my goal is only to clarify the top-level definitions.

I’ll use the simple example of a $100.00 face 20-year zero-coupon bond that currently yields (yield to maturity of) 6.0% per annum. If the yield is 6.0% per annum with continuous compounding, the price is $100.00*exp(-0.060*20) = $30.12. If the yield is 6.0% per annum with annual compounding, the price is $100.00/(1+0.060)^20 = $31.18. Unless otherwise specified, I will assume a continuous compound frequency. *Special note*: we so often price a bond given the yield (where CPT PV is the final calculator step) that it is easy to forget yield is not actually an input. Yield is the internal rate of return (IRR) assuming the current price. Yield does not determine price; price determines yield. Technical (non-fundamental) factors cause the price to fluctuate, therefore yield fluctuates.

**1. ∂P/∂y (or Δp/Δy)** is the *slope *of the tangent line at the selected yield. At 6.0% yield, the slope is -$602.39. How do I know that? Because dollar duration is the negated slope, so in this case dollar duration (DD) = P*D = $30.12 price * 20 years = $602.39. **Importantly**, the “y” in ∂P/∂y is yield and *yield is just one of several interest rate factors.*

**2. Dollar duration **(DD; aka, money duration in the CFA) is analytically the product of price and modified duration. Dollar duration (DD) = P*D = $30.12 * 20 = $602.39. Why is it so large? Because it’s the (negated) tangent line’s slope, so it has the typical first derivative interpretation: DD is the dollar change implied by one unit change in the yield, -∂P/∂y. One unit is 1.0 = 100.0% = 100 * 100 basis points (bps) per 1.0% = 10,000 basis points. So, DD is the dollar change implied by a 100.0% change in yield *if we use the straight tangent line which would be a silly thing to do*! Recall the constant references to limitations of duration as linear approximation. The linear approximation induces bias at only 5 or 10 or 20 basis points, so 10,000 basis points is literally “off the charts” and not directly meaningful. What is meaningful? The PVBP (aka, DV01) comes to our rescue with a meaningful re-scaling of the DD …

**3. Price value of basis point (aka, dollar value of ’01, DV01)** is the dollar duration ÷ 10,000. It’s the tangent line’s slope re-scaled from Δy=100.0% to Δy= 0.010% (one basis point). PVBP = P*D/10,000; in this example, PVBP = $30.12 * 20 / 10,000 = $0.06024. It is the dollar change implied by a one basis point *decline *in the yield. It is still a linear approximation, but much better because we zoomed in to a small change. In this way, the difference between the highly useful PVBP and the dollar duration is merely scale.

**4. Macaulay duration** is the bond’s * weighted average maturity *where the weights are each of the bond’s cash flow’s present value as a percentage of the bond’s price. Macaulay duration is tedious however it is reliable and it is

*analytical*. When we can compute the Macaulay duration, it is accurate; we don’t approximate by re-pricing the bond. A zero-coupon bond has a Macaulay duration equal to its maturity because it only has one cash flow (hence the popularity of the zero-coupon bond in exam questions, never mind the zero-coupon bond is a reliable primitive). Our 20-year zero-coupon bond has a Macaulay duration of 20.0 years.

**5. Modified duration **is the measure of interest rate risk. Modified duration is the *approximate *percentage change in bond price implied by a 1.0% (100 basis point) change in the yield. Just as ∂P/∂y refers to the tangent line’s slope which is “infected with price,” we divide by price to express the modified duration, **D(mod) = 1/P*∂P/∂y**. The key relationship between analytical modified and Macaulay duration is the following: modified duration = Macaulay duration / (1 + y/k) where k is the number of compound periods in the year; e.g., k = 1 for annual compounding, k = 2 for semiannual compounding and k = ∞ for continuous compounding. Importantly, if the compound frequency is continuous then a bond’s modified duration equals its Macaulay duration. Notice that T / (1 + y/∞) = T / (1 + 0) = T.

- If the 6.0% yield is annual compounded, our 20-year bond’s Macaulay duration is given by 20.0 / (1 + 6.0%) = 18.868 years.
- If the 6.0% yield is continuously compounded, our 20-year bond’s modified duration is 20.0 years.

**6. Effective duration** is an approximation of modified duration. Recall the modified duration is a linear approximation, but that’s because it is a function of the first derivative; otherwise, modified duration is an exact (analytical or functional) measure of the price sensitivity with respect to the interest rate factor that happens to most often be the yield. We can retrieve it easily whenever we can compute the Macaulay duration, which is the case for any vanilla bond. Otherwise (e.g., bond has an embedded option) we approximate the modified duration by calculating its effective duration. The *effective duration approximates the modified duration which itself is a linear approximation*. The effective duration is given by [P(-Δy) – P(+Δy)] / (2*Δy) * 1/P. I wrote it this way so you can see that it is essentially similar to ∂P/∂y*1/P where ∂P/∂y ≅ [P(-Δy) – P(+Δy)] / (2*Δy). I’ve observed that many candidates do not realize that the *formula for effective duration is simply slope*1/P*. Geometrically, it is the slope of the secant line that is near the tangent line! Secant’s slope approximates the tangent’s slope. If you grok the calculus here, I think you’ll agree that **this is all just one thing**! Now we can see how it’s not so different. But as you can visualize, there are an almost infinite variety of secants next to the tangent. We arbitrarily choose a nearby secant, but we’d prefer a small delta if the bond is vanilla (i.e., if the bond’s cash flows are invariant to rate changes). Although we do not need the effective duration for our example bond, we can compute it:

- If our arbitrary yield shock is 10 basis points such that Δy= 0.10%, then P(-Δy)= $100.00*exp(-5.90%*20)= $30.728, and P(+Δy)= $100.00*exp(-6.10%*20)= $29.523. Effective duration= ($30.728 – $29.523)/0.0020 *1/$30.12= 20.0013 years. Fine approximation!

**7. On the terminology (CFA versus FRM)**

- Interest rate factor: The FRM (informed by Tuckman) starts with a general
*interest rate factor*. This is typically the spot rate, forward rate, par rate, or yield. Importantly, the spot, forward and par rates are term structures or vectors; the par yield curve is a vector of par rates at various maturities, often at six-month or one-month intervals. Only the yield is a single (aka, scalar) value. - My above definition of the
*effective duration*is according to the FRM (and to me). The CFA sub-divides this effective duration into either*approximate modified duration*(if the interest rate factor is the yield) versus*effective duration*(if the non-vanilla nature of the bond requires a non-yield interest rate factor; i.e., a benchmark yield curve). Personally, I am not keen on this semantic approach because (i) both of these CFA formulas are approximating the modified duration and (ii) I prefer to reserve “effective” for its traditional connotation (e.g., effective convexity is analogous to effective duration), and (iii) we wouldn’t anyhow use an inappropriate factor (yield) for certain non-vanilla situations, so we don’t really need label-switches to guide us thusly! (the CFA’s formula for its approximate modified duration is essentially the same as its effective duration formula). To me, the CFA’s approach muddies the terms “approximate” and “effective” where the math gives us natural distinctions. Follow the math, I’d say!