# Level 1 CFA® Exam:

Duration & Convexity - Advanced

We distinguish between:

- Macaulay duration,
- approximate Macaulay duration,
- modified duration,
- approximate modified duration,
- money duration,
- price value of a basis point, and
- effective duration.

An important distinction and one to remember is between:

- yield duration, and
- curve duration.

Yield duration measures interest rate risk using a change in the bond’s YTM. On the other hand, curve duration measures interest rate risk using a change in the benchmark yield curve.

The yield duration:

- Macaulay duration,
- modified duration,
- approximate modified duration, and
- approximate Macaulay duration.

An example of the curve duration is effective duration.

In the previous lesson, we saw how to compute the Macaulay duration and modified duration. Now, it is time we had a look at the approximate modified duration and approximate Macaulay duration.

Last time, we learned how to compute the modified duration assuming that we know the Macaulay duration. However, not always do we know the Macaulay duration.

If we don’t know the Macaulay duration, we can calculate the approximate modified duration using the following formula that you can use in your level 1 CFA exam:

\(AMD = \frac{PV_{-} - PV_{+}}{2\times \Delta r\times PV_{0}}\)

- \(AMD\) - approximate modified duration
- \(PV_{+}\) - price when the yield is increased
- \(PV_{-}\) - price when the yield is reduced
- \(PV_{0}\) - original price
- \(r\) - yield-to-maturity

After we calculate the approximate modified duration, we can compute the approximate Macaulay duration:

\(ApproxMacDur = AMD\times (1+r)\)

- \(ApproxMacDur\) - approximate Macaulay duration
- \(AMD\) - approximate modified duration
- \(r\) - yield-to-maturity

An option-free bond issued by Corpo, Inc. has 11 years until maturity. The coupon rate amounts to 8.2% and coupons are paid annually. The price of the bond and its par value are equal to USD 48,000 and USD 50,000, respectively. What is the value of the approximate modified duration and approximate Macaulay duration?

(...)

(...)

The money duration equals the annual modified duration multiplied by the current full price of the bond:

\(MoneyDur = AnnModDur\times PV^{Full}\)

- \(MoneyDur\) - money duration
- \(AnnModDur\) - annual modified duration
- \(PV^{Full}\) - full price of the bond

So, if we use money duration to measure the interest rate risk, we will get a change in dollars or some other currency, and not the percent change in the bond price:

\(\Delta PV^{Full} \approx -MoneyDur\times \Delta r\)

- \(PV^{Full}\) - estimated percentage price change of a bond
- \(MoneyDur\) - money duration
- \(r\) - yield-to-maturity

The annual modified duration equals 5.7 and the full price of the bond is equal to USD 145,000. What is the value of money duration? How can you interpret it?

(...)

The price value of a basis point (PVBP) is calculated using the following formula to be used in your level 1 CFA exam:

\(PVBP = \frac{PV_{-} - PV_{+}}{2}\)

- \(PVBP\) - price value of the basis point
- \(PV_{-}\) - price when the yield-to-maturity is reduced
- \(PV_{+}\) - price when the yield-to-maturity is increased

As you can see the formula for the price value of a basis point is similar to the formula for the approximate modified duration. The difference is that we don’t use a change in the YTM times the current bond price in the denominator. Thus, we can interpret the price value of a basis point (PVBP) as the money duration for the bond assuming that the change in the YTM is equal to 1 basis point.

In the previous lesson, we learned the formula for the approximate convexity statistic. Today you will learn how to calculate:

- the effective convexity, and
- money convexity.

### Effective Convexity

To calculate the effective convexity, we will use a formula similar to the formula for the approximate convexity:

\(EC = \frac{PV_{-} + PV_{+} - 2\times PV_{0}}{(\Delta c)^{2}\times PV_{0}}\)

- \(EC\) - effective convexity
- \(PV_{-}\) - new price when the benchmark curve is shifted downward
- \(PV_{+}\) - new price when the benchmark curve is shifted upward
- \(PV_{0}\) - initial price
- \(c\) - benchmark yield curve

Note, however, that as in the case of the effective duration, we use the benchmark yield curve and not yield-to-maturity for the bond. Similarly to the effective duration, the effective convexity is used to measure interest risk for bonds with embedded options, like:

- callable bonds,
- putable bonds, and
- mortgage-backed bonds.

### Money Convexity

Money convexity is equal to annual convexity multiplied by the bond's full price:

\(C_{money}=C_{approx}\times PV^{Full}\)

We can use money convexity together with money duration when we want to compute the change in bond price (in USD) as the result of the change in interest rates. The following formula can come in handy then:

\(\Delta PV^{Full} \approx -MoneyDur\times \Delta r + \frac{1}{2}\times MoneyCon\times (\Delta r)^{2}\)

- \(PV^{Full}\) - price of a bond
- \(MoneyDur\) - money duration
- \(r\) - yield-to-maturity
- \(MoneyCon\) - money convexity

- Yield duration measures interest rate risk using a change in the bond’s YTM.
- Curve duration measures interest rate risk using a change in the benchmark yield curve.
- Examples of yield duration include: Macaulay duration, modified duration, approximate modified duration, and approximate Macaulay duration.
- An example of the curve duration is effective duration.
- We use the effective duration for bonds with embedded options.
- If we use money duration to measure the interest rate risk, we will get a change in dollars or some other currency, and not the percent change in the bond price.
- We can interpret the price value of a basis point (PVBP) as the money duration for the bond assuming that the change in the YTM is equal to 1 basis point.
- The effective convexity is used to measure interest risk for bonds with embedded options.
- Money convexity is equal to annual convexity multiplied by the bond's full price.