IFT Notes for Level I CFA^{®} Program

The money duration of a bond is a measure of the price change in units of the currency in which the bond is denominated, given a change in annual yield to maturity.

Money Duration = AnnModDur x PV_{FULL}

ΔPV_{FULL}≈ -MoneyDur x Δyield

Consider a bond with a par value of $100 million. The current yield to maturity (YTM) is 5% and the full price is $102 per $100 par value. The annual modified duration of this bond is 3. the money duration can be calculated as the annual modified duration (3) multiplied by the full price ($102 million): 3 x $102 million = $306 million. If the YTM rises by 1% (100 bps) from 5% to 6% the decrease in value will be approximately $306 million x 1% = $3.06 million. If the YTM rises by 0.1% (10 bps), the decrease in value will be $306 million x 0.1% = $0.306 million.

An important measure which is related to money duration is the **price value of a basis point** (**PVBP**). The PVBP is an estimate of the change in the full price given a 1 bp change in the yield-to-maturity. The formal equation is given below.

PVBP =

where PV_ and PV_{+} are full prices calculated by decreasing and increasing the YTM by 1 basis point.

A quick way of calculating the price value of a basis point is to take the money duration and multiply by 0.0001. For example, if the money duration of a portfolio is $200,000 the price value of a basis point is $200,000 x 0.0001 = $20. (1 bp = 0.01% = 0.0001)

**Example 13: Calculating money duration of a bond**

A life insurance company holds a USD 1 million (par value) position in a bond that has a modified duration of 6.38. The full price of the bond is 102.32 per 100 of face value.

- Calculate the money duration for the bond.
- Using the money duration, estimate the loss for each 10 bps increase in the yield-to-maturity.

**Solution:**

- First calculate the full price of the bond: $1,000,000 x 102.32% = $1,023,200. The money duration for the bond is: 6.38 × $1,023,200 = $6,528,000.
- 10 bps corresponds to 0.10% = 0.0010. For each 10 bps increase in the yield-to-maturity, the loss is estimated to be: $6,528,000 × 0.0010 = $6,528.02.

** **

**Example 14: Calculating PVBP for a bond**

Consider a $100, five-year bond that pays coupons at a rate of 10% semi-annually. The YTM is 10% and it is priced at par. The modified duration of the bond is 3.81. Calculate the PVBP for the bond.

**Solution:**

Money duration = $100 x 3.81 = $381.00

PVBP = $381 x 0.0001 = $0.0381

The graph below shows the relationship between bond price and YTM. It shows the convexity for a traditional fixed-rate bond.

*Source: CFA Program Curriculum, Understanding Fixed-Income Risk and Return*

__Interpretation of the diagram__**:**

- Duration assumes there is a linear relationship between the change in a bond’s price and change in YTM. For instance, assume the YTM of a bond is 10% and it is priced at par (100). According to the duration measure, if the YTM increases to 11% the price moves down to a point on the straight line.
- Similarly, the price moves up to a point on the straight line if the YTM decreases.
- The curved line in the above exhibit plots the actual bond prices against YTM. So in reality, the bond prices do not move along a straight line but exhibit a convex relationship.
- For small changes in YTM, the linear approximation is a good representation for change in bond price. That is, the difference between the straight and curved line is not significant.
- In other words, modified duration is a good measure of the price volatility.
- However, for large changes in YTM or when the rate volatility is high, a linear approximation is not accurate and a convexity adjustment is needed
**.**

Here we need to factor in the convexity. The percentage change in the bond’s full price with convexity-adjustment is given by the following equation:

Change in the price of a full bond:

% ΔPV^{FULL }=

Expression in first braces: duration adjustment

Expression in second braces: convexity adjustment

Approximate convexity can be calculated using this formula:

Approx. Convexity =

where:

PV_ and PV_{+} = new full price when YTM is decreased and increased by the same amount

PV_{0 }= original full price

The change in the full price of the bond in units of currency given a change in YTM can be calculated using this formula:

**Convexity is good**

The following exhibit shows the price-yield curves for two bonds with the same YTM, price, and modified duration, and why greater convexity is good for an investor.

__Interpretation of the diagram__:

- Both the bonds have the same tangential line to their price-yield curves.
- When YTM decreases by the same amount, the more convex bond appreciates more in price.
- When YTM increases by the same amount, the more convex bond depreciates less in price than the less convex bond.
- The bond with greater convexity outperforms when interest rates go up/down.

The relationship between various bond parameters with convexity is the same as with duration.

For a fixed-rate bond,

- The lower the coupon rate, the greater the convexity.
- The lower the yield to maturity, the greater the convexity.
- The longer the time to maturity, the greater the convexity.
- The greater the dispersion of cash flow or cash payments spread over time, the greater the convexity.

**Effective Convexity**

For bonds whose cash flows were unpredictable, we used effective duration as a measure of interest rate risk. Similarly, we use effective convexity to measure the change in price for a change in benchmark yield curve for securities with uncertain cash flows. The effective convexity of a bond is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve. It is used for bonds with embedded options.

Effective Convexity =

Here is a summary of some important points related to bonds with embedded options:

- When the benchmark yield is high, prices of callable and non-callable bonds change almost similarly for interest rate changes.
- When the benchmark yield is low, call option becomes more valuable to the issuer.
- Callable bond exhibits negative convexity.
- Putable bonds always have positive convexity.

**Example 15: Calculating the full price and convexity-adjusted percentage price change of a bond**

A German bank holds a large position in a 6.50% annual coupon payment corporate bond that matures on 4 April 2029. The bond’s yield to maturity is 6.74% for settlement on 27 June 2014, stated as an effective annual rate. That settlement date is 83 days into the 360-day year using the 30/360 method of counting days.

- Calculate the full price of the bond per 100 of par value.
- Calculate the approximate modified duration and approximate convexity using a 1 bp increase and decrease in the yield to maturity.
- Calculate the estimated convexity-adjusted percentage price change resulting from a 100 bp increase in the yield to maturity.
- Compare the estimated percentage price change with the actual change, assuming the yield to maturity jumps to 7.74% on that settlement date.

**Solution: **

There are 15 years from the beginning of the current period on 4 April 2014 to maturity on 4 April 2029.

1. The full price of the bond is 99.2592 per 100 of par value.

FV = 100, I/Y = 6.74, PMT = 6.50, N = 15, CPT PV; PV = -97.777.

Full Price = = 99.2592.

2. PV_{+} = 99.1689

FV = 100, PMT = 6.5, I/Y = 6.75, N = 15, CPT PV; PV = -97.687.

PV_{+} == 99.1689.

PV_ = 99.3497.

FV = 100, I/Y = 6.73, PMT = 6.5, N = 15, CPT PV; PV = -97.869.

PV_ = = 99.3497.

ApproxModDur == 9.1075.

The approximate modified duration is 9.1075.

ApproxCon == 201.493

The approximate convexity is 201.493.

3. The convexity-adjusted percentage price drop resulting from a 100 bp increase in the yield-to-maturity is estimated to be -8.1% (-9.1075 + 1.00746).<

Modified duration alone estimates the percentage drop to be 9.1075%. The convexity adjustment adds 100.746 bps (0.5 × 201.493 × .01^{2} = 1.00746%).

4. The new full price if the yield-to-maturity goes from 6.74% to 7.74% on that settlement date is 90.7623. The actual percentage change in the bond price is -8.5603%. The convexity-adjusted estimate is -8.1%.

**Example 16: Calculating the approximate modified duration and approximate convexity**

The investment manager for a US defined-benefit pension scheme is considering two bonds about to be issued by a large life insurance company. The first is a 25-year, 5% semiannual coupon payment bond. The second is a 75-year, 5% semiannual coupon payment bond. Both bonds are expected to trade at par value at issuance.

Calculate the approximate modified duration and approximate convexity for each bond using a 5 bp increase and decrease in the annual yield-to-maturity.

**Solution:**

In the calculations, the yield per semiannual period goes up by 2.5 bps to 2.525% and down by 2.5 bps to 2.475%.

The 25-year bond has an approximate modified duration of 14.18.

PV_{+} : FV = 100, I/Y = 2.525, PMT = 2.5, N = 50, CPT PV, PV = -99.2945.

PV_{–} : FV = 100, I/Y = 2.475, PMT = 2.5, N = 50, CPT PV, PV = -100.7126.

ApproxModDur = = 14.18 and an approximate convexity of 284.

ApproxCon = = 284.

Similarly, the 75-year bond has an approximate modified duration of 19.51 and an approximate convexity of 708.