The average duration, or duration, of bonds
Duration is a measure of risk, by calculating it you can find out how long it will take on average to get your invested capital back.
Where do you come across the concept of duration?
The duration occurs for securities that have maturity and your principal will be repaid to you at maturity or till that time. That is, they are typically debt securities or forms of investment based on them. These include:
- bond
- government securities
- bond fund: here the information you need is included not only in the description of the fund, but also in its name. After all, you can meet e.g. with a short bond fund and a long bond fund. Of these, short-term and long-term bonds predominate.
Accordingly, you can use the average time in the following situations, among others:
- when choosing a particular security: the question is whether it is right for you in terms of risks.
- when compiling bond-containing portfolios: the average time of your portfolio will be the weighted average of the average time of the bonds in it, based on this logic, e.g. bond funds as well.
- matching assets and liabilities: if the average duration of liabilities (eg loans) is in sync with the average duration of the assets financing it, maintaining liquidity is less of a headache.
If you are more interested in, read about bond mathematics 🙂 .
What does duration mean?
Duration expresses the sensitivity of a bond price to changes in interest rates. In other words, the duration of a bond measures the movement of the bond price for every 1% change in interest rates.
The duration is somewhat related to, but different from the actual remaining maturity. The main question here is how long you have to wait to get your capital back. Accordingly, this value will be affected by:
- the actual remaining term
- the interest rate of the bond (if the interest rate is lower, the duration will be less affected)
- the expected market interest rate corresponding to the maturity and the degree of risk
- the method of repaying the bond principal: if you do not get 100% back at the very end, but in a different way, it will affect the rate of interest payment as well as the duration.
However, it is associated with investment risk. The more time you have until you see the capital invested again, the greater your risk.
Example of application
You have a bond that has a maturity of 5 years, pays you 10% per annum, and returns the principal at maturity. The price and interest payments on the bonds are usually set as a percentage of face value, and I will follow this below. Now, for the sake of easier understanding, let’s look at the nominal value of the bond at HUF 100.000.
Then in the first year you will only get back 10%, ie HUF 10,000, which is less than your investment. At the very end of the term, you received the 5 × 10%, ie HUF 50.000, and you also received back the capital of HUF 100.000. And with that, you exceeded the target, you got more than what was your investment.
If the question is when the 100.000 forints will be (in theory) yours, then it will happen somewhere between the 4th and the 5th year.
Duration calculation
The formula for the calculation
You can see that in order to be able to determine the duration, you need to know a few things. Let’s take a look at:
n, I, C: you will know the bond description at least at the formula level, so this is not a big challenge
r: the expected market interest rate is a function of maturity and risk level, which consists of the risk-free rate of return and the risk premium. Since this 1 factor will be a little different for everyone, apart from interest payments and the passage of time, this factor will change the exchange rate the most.
P: denotes the market price of the bond, but it is easier and more accurate if you do not count on the current market price, but on the fair price of the bond.
Calculation of duration
It can be broken down into the following steps:
- After writing down the cash flows of the bond, we determine their present value, adding them up get the price of the bond.
- We calculate what percentage of the exchange rate each cash flow represents.
- Substituting these percentages into the duration formula gives the duration.
Determining the price (P)
The duration affects the price of the bond. The price that can be calculated from the parameters of the bond (yields, time remaining, etc.) and the expected yield on the market is called the fair price. The actual price will deviate somewhat from this, but it will be around that.
As the interest payment and redemption scheme of the bonds is known at least at the formula level, the price of the bonds can be estimated, which is why their price is much more stable compared to the shares.
The meaning of fair price
The present value of future payments, that is, the present value of interest and principal repayments discounted to risk-adjusted returns.
Determining the fair price
In this case, the cash flows are discounted, and the formula will be the same as already known in the present value calculation. Accordingly, we calculate the present value of the cash flows due in each year.
Continuing the example above, bond parameters are as follows:
Interest payment: 10%
Term: 5 years
Expected return: 5%
P= 10%/(1+0,05)1 + 10%/(1+0,05)2 + 10%/(1+0,05)3 + 10%/(1+0,05)4 + (10% + 100%)/(1+0,05)5= 121,65%
Currently the fair price of our bond is 121.65% of the face value.
Calculation of ratios
The price of the bond has just been set, so we can now focus on the second step.
| Year | The present value of the cash flow | The ratio of the present value to the exchange rate |
|---|---|---|
| 1. | 10%/(1+0,05)1 = 9,52% | 9,52% / 121,65% = 7,82% |
| 2. | 10%/(1+0,05)2 = 9,07% | 9,07% / 121,65% = 7,46% |
| 3. | 10%/(1+0,05)3 = 8,64% | 8,64% / 121,65% = 7,10% |
| 4. | 10%/(1+0,05)4 = 8,23% | 8,23% / 121,65% = 6,76% |
| 5. | (10%+100%)/(1+0,05)5 = 86,19% | 86,19% / 121,65% = 70,85% |
Determination of duration
Based on the formula, we just have to replace it
DUR = 1*7,82% + 2*7,46% + 3*7,10% + 4*6,76% + 5*70,85% = 4,25 years
The duration of the above bond is 4.25 years.
Factors affecting duration
Length of term
The present value of a later payment is more sensitive to changes in yields, so a longer maturity means a higher duration.
The interest rate of the bond
The lower the interest rate, the lower the weighting in the calculation, as the higher the weight of repayment of the principal due later in the term.
Change in expected return
The higher the expected return, the higher the discount factor, thus modifying the average time.
The modified duration
As long as the average time shows how long it takes to get your capital back, i.e. how long or at risk, the modified duration gives an answer to how sensitive the bond price is to a 1% change in the interest rate.
In other words, the modified duration shows the volatility of the bond price.
Calculation:
DURmod = DUR / (1+r)
For the above 10% bond with a duration of 4.25 years, this is:
DURmod = 4,25 / (1+0,1) = 3,86
That is, each 1% change in market yield level causes a 3.86% change in the price of this bond.
Rule of thumb
According to an easy-to-apply rule of thumb, the price of bonds will change by multiplying their duration by the change in market interest rates. Roughly like this:
| Interest rate change% | Duration | ||||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | |
| 1 | 1 | 2 | 3 | 4 | 5 |
| 2 | 2 | 4 | 6 | 8 | 10 |
| 3 | 3 | 6 | 9 | 12 | 15 |
| 4 | 4 | 8 | 12 | 16 | 20 |
| 5 | 5 | 10 | 15 | 20 | 25 |
What is the difference between beta and duration?
The part dealing with the risk of investments was about beta, which we defined then and there if the market moves by 1%, then how much the exchange rate of the given investment reacts to it, and now we said the same about the duration of a 1% change. what the exchange rate reacts. The two, however, are not the same, they show the same, only in different areas.
The beta
It can be interpreted for stocks or equity mutual funds and looks at price movements relative to a selected market. The price of a stock is affected by a lot of things, so you wouldn’t even be able to calculate the duration here. (Yes, beta can be calculated, not just “told”.)
The duration
The price of bonds is relatively stable, with far fewer factors affecting its price due to the known interest rate and the debt ratio rather than ownership. The two most important of these are the present value of the actual cash flow and the return environment. If I wanted to articulate the selected market here, I would not say it would be called the Dow Jones index, but “bonds with a given cash flow in a given yield environment”. (Don’t search, Google won’t recognize this category either)
Factors affecting the price of a bond
The following follows from what has been said so far.
Change in bond repayment
Here, the problem is not caused by a change according to a predetermined rhythm, after all, investors can prepare for it from the moment of issue, but it is the unexpected change that causes a significant exchange rate movement. This is why this rarely changes, mostly due to the issuer’s near-bankruptcy situation. The eventual occurrence of such, on the other hand, has a very serious impact on the price of the bond.
Length of term
Over time, on the one hand, the issuer pays more and more interest, which will have an effect on the exchange rate, and on the other hand, we have learned from the discount formula that time has an exponential role. As the remaining time decreases, its impact becomes less and less important.
The price of bonds will decline over time (until they reach face value) if the bond’s own interest rate is above the market environment. If they are unable to meet the expected return, their exchange rate will rise to face value towards the end of the term.
Changes in the yield environment
By keeping the level of inflation in check, the central bank may raise the base rate, as a result of which everything else will change. An increase in market yields leads to a fall in exchange rates, but the price of bonds does not react equally:
- those with a fixed interest rate: have stronger reaction, being that the level of interest is known there until the end of the term
- variable interest rate: these are fixed to an external factor, which is often the level of inflation, either directly or indirectly, as their interest rate also moves, so they are less affected by changes in the market yield environment
Questions:
- Based on the above, where can you benefit from the duration in your own finances?
- Based on the effect of exchange rate changes, what risk category would you classify the bonds for?
- How would you choose an investment vehicle for yourself?
If you have any questions, please contact us so we can help you.