In this section we will look at several standard market measurements used in the market for bond product, excluding value at risk (VaR). In particular the manner in which the following measurement is calculated will be shown.

- Duration or Macaulay Duration
- Modified Duration
- First derivative of bond value with respect to yield
- PVBP01
- Convexity

Market risks of bonds are typically measured by duration. The most common is Macaulay duration, or just duration, which is the weighted average of the cash flow and measured in terms of number of years.

Table 4-2 will be used and modified as illustration to calculate Macaulay duration. Each cash flow of the MGS will be discounted and and weighted by the tenor of the cash flow occurrence. It is then totalled and divided by the present value of the MGS. The detail calculation is shown in Table 4-7 using the yield of 2.9600%.

Duration is calculated to be 1.9567 and interpreted as the bonds having an average cash flow tenor of 1.9567 years. Although duration is interpreted as weighted average tenor of the cash flow of the MGS, it is often used to measure its sensitivity. It can also be interpreted as percentage change in the value of the bond when the yield changes by 1%. If the YTM moves to 3.90% for instance, the value of the bonds will be 9,820,438.16, a 1.9054% change - good approximation by the duration method. Reader can use our Bond Pricing to verify the numbers.

For a 1bp change or 0.01%, the expected change in value using the duration will be 0.019567. At the yield of 2.97%, a 1bp change, value of the bond becomes 9,997,686.52, a change of -1,927.85 or -0.01930% - a closer estimate by duration.

Modified duration is another method for estimating a bond price change. It is closely related to Macaulay duration, due to the fact that it is -Macaulay duration divided by \(1 + \frac{\textrm{yield}}{\textrm{frequency}}\). Given our bond is paying semiannual coupon, at the yield of 2.96%, the duration is simply \( -\frac{1.9567} {\left(1 + \frac{0.0296}{2}\right)}\) = -1.9282. Using the previously calculated value in the above paragraph, modified duration is a much better estimate than macaulay duration in estimating a price change from a small move in the yield.

Take note of the -ve sign. The reason is that price and yield do not have a linear relationship. It inversely related to each other and that is the reason why when we increase the yield, the price becomes lower. Of course, there is a layman explanation to it - as yield goes up, investors will ask for compensation in terms of lower price for the bond, for a given coupon.

In mathematical terms, this refers to the first derivative of the bond value with respect to yield. We are not going to represent the formula here, but our Bond Pricing calculates the derivative and the number is very close to modified duration suggesting an approximation with modified duration is sufficient, in most cases.

Readers are recommended to simulate a bond with various coupons, yield and maturity to see how far modified duration diverges from its first derivatives using our calculator. Our calculation of the derivative is using Sympy simply by feeding the bond equation into it.

This is another common measure used to estimate a bond sensitivity in dollar and cents, and is the short name for present value of one basis point. In fact, it is common for most interest rate products as the number can simply be added to obtain the portfolio PVBP01. We cant do that with durations, both Macaulay and modified. A portfolio duration is the weigthed average of durations accross various interest rate product and converting it to dollar and cents would add additional steps.

In fact, when we calculated the actual percentage change for a 1bp move up and compare against duration numbers, we have actually calculated the PVBP01. The change in the value of the bond after the 1bps move is the PVBP01. By dividing it with the value of the bond, we are actually using forward differential method to estimate the first derivative of the bond value with respect to yield. If we were shifting the rates down by 1 bp instead, then it is referred to as backward differential method. The average of the two is known as central differential method.

This is the second derivative of bond value with respect to yield. Recall that when we revalue the bond at the yield of 3.90%, the percentage change is quite far from duration.

Our calculation of convexity is utilising by Sympy and provided in the Bond Pricing page.

Most reading material on FRN pricing or valuation will stop at something like our Equation 4-3. We extended our pricing/valuation methodology to include FRN with margin above/below the reference rate. At the same time introduced the concept on investor's required spread which can be different from the margin and included that element in our pricing equation. From the pricing equation, we can calculate the PVBP01 for a FRN.

Unlike a fixed rate bond, however, there is no yield to be input in the valuation model. There were two elements in the model:

- Coupon Cash Flows - we used the forward rate from the reference curve, add the margin and calculate the coupon interest based on the day vount convention
- Discounting - we used the forward rate from the reference curve, add the spread, convert to a forward discount factor and lastly convert to a normal discount factor to present value the cash flows