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- How to calculate bond yield and risk

To evaluate the risk of a bond, if it is worth investing and how much, we must consider effective net yield at maturity, volatility and modified duration, to find out the annual return on investment and the changes in the price of the security based on interest rates.

In a previous bond guide, I described the main factors that determine bond yield and risk. But **how do we calculate how much a bond earns?** How to understand if the securities that the bank or financial advisor advises us, or that we find online, are advantageous in terms of **risk and return**?

Two fundamental concepts for choosing a bond must be explained immediately. First of all, remember that the **quotation prices** normally refer to the so-called “**clean price**“, ie the price at which only the capital is quoted. The “**tel quel price**” or “**dirty price**” instead is the price including the interest accrued in the period considered. Next, **volatility** and **duration** should be explained to evaluate the risk of a bond.

## How to calculate the yield of a bond

The most concise value to consider when choosing a fixed income security is the **net effective interest rate at maturity** (or **YTM**: **effective yield to maturity**), defined as that rate which, under a compound interest regime, equals the dirty price of a fixed rate security (initial amount invested) to the present value (value to date) of the cash flows (coupons and repayment of capital at maturity) to which that title gives right.

The effective yield to maturity is calculated by counting all the coupons to be collected and the redemption price (VR) of the security itself (for convenience, in the formula we write r instead of YTM). However, it is not necessary to calculate bonds actual yield, you can find the value published in the specialized press or ask your broker.

## How to calculate the risk of a bond: volatility and duration

Now that we know how to evaluate the return on a fixed income security, let’s see how to calculate its risk and introduce the concept of bond volatility first, the sensitivity of its price to changes in market rates, then the concept of duration, a measure of this sensitivity with respect to changes in interest rates.

### What is the volatility of a bond and how is it calculated

Bond volatility is the absolute value of the percentage change in the price of the security for a given change in the effective net interest rate at maturity, attributable to changes in interest rates.

Imagine a 5-year government bond, with a redemption value of 100 and a coupon of 4% per annum; if the 5-year rates were exactly 4%, then the market price of the bond would be exactly 100. If the rates immediately after purchase rose to 6%, what would be the new value of the government bond? Its price will drop by about 10, as for 5 years we will have 2 points less than the new market rate (we bought it when they were at 4%).

In this example the volatility of the security is around 5%, that is, 10/100/2 = 5%. **The bond’s volatility depends on**:

- time: the longer the bond maturity, the greater the volatility; imagine the previous example with a 10-year deadline: the price drop would have been 20 points
- coupons: the higher and the more frequent they are, the less volatile the security

### What is the duration of a bond and how is it calculated

Bond duration indicates its residual financial life and may not coincide with its effective residual life.

**Definition of bonds duration**: number of years within which the capital invested is recovered, also calculating coupons. Normally a longer duration indicates a greater financial risk and this means if there is a change in interest rates, there will be a change in the bond price: this change will be greater the higher the duration.

The mathematical formula that links the change in the price of the security to the change in interest rates is:

ΔP/P= -DUR/(1+r)* Δr

The ratio DUR / (1 + r) is the **modified duration** (and r is the YTM as in the formula above).

For example, consider a security with price = 100 and YTM = 5%, with a modified duration = 5. If the TRES requested by the market rose from 5% to 5.5%, the price of the security would drop to 97.5. (ΔP = -2.5).

**The duration of zero coupon securities** is equal to their residual life, because the capital is repaid in full on maturity. **The duration of coupon securities** is less than the maturity, because part of the present value of the security derives from the flow of interest. **The duration of variable coupon securities** is very low (close to 0) and calculated assuming that each index corresponds to a reinvestment of all capital at the “new variable rate” and therefore the risk (duration) exists in fact only for the time elapsed between one indexing and another. This is because the yield to maturity of the variable coupon securities cannot be calculated, as the cash flows generated in the future by the coupons of these securities are not known.