# Yield curve analysis

This document refers to Macrobond version 1.10 and later.

#### Overview

With the Yield curve analysis you can produce charts showing yield curves, where one axis shows the maturity length and the other the yield. It works like this:

- The input consists of a number of series with interest rates of different maturity lengths.
- You input the maturity lengths and type of rate (simple or effective). For some series there might be information in the database about this and it will then be filled in automatically.
- You input which dates you want to look at.
- The analysis will produce two time series for each yield curve: one series which contains the maturity lengths expressed as years and one series with the interest rate at the specified point in time for the corresponding maturity.
- If you plot these two series in a scatter chart and combine the dots with lines, you will get a yield curve with “years” on one axis and “Percent” on the other.
- In the analysis you can select to combine the lines with a natural cubic spline, which will give you a smooth yield curve. When this setting is turned on, a rate will be calculated for each month.
- You can calculate several curves at the same time. Either at different dates or with different settings or selection of series. If you select the special date called “Latest”, the last date where there are observations for the majority of the series is used.

A tip is to add two curves, where one calculates splines and one does not. You can then configure the chart to draw one of them as lines, without labels and the other as just markers, but including labels. It will then look something like this:

#### Settings

###### Output yield

All input rates will be converted to either simple annual interest rates or effective annual interest rates.

The relation between simple rates and effective rates is:

$1+\frac{{\text{r}}_{\text{simple}}}{\text{100}}\xb7\text{m}={\left(\text{1}+\frac{{\text{r}}_{\text{effective}}}{\text{100}}\right)}^{\text{m}}$where `m` is the maturity length expressed as the number of years.

###### Output unit

You can select to produce the discount factors instead of the interest rates. In this case the factors are calculated by one of these two expressions depending on the type of rate:

$\text{DF}={\left(\text{1}+\frac{{\text{r}}_{\text{effective}}}{\text{100}}\right)}^{-\text{m}}$ $\text{DF}=\frac{1}{1+\frac{{\text{r}}_{\text{simple}}}{\text{100}}\xb7\text{m}}$###### Use natural cubic spline

When the option for natural cubic splines is selected, the curve will not only consist of points corresponding to the maturity lengths, but there will be a value at least every month. Intermediate values will be calculated by creating a natural cubic spline based on the rates.

###### Spot rates

When “Spot rates” is selected, the rate at each point in time will be used.

###### Forward rates with constant maturity

When “Forward rates with constant maturity” is selected, you must also specify the length of the forward.

The calculation of the forward rate will then be done for each point on the curve by looking at the current rate and a future rate. If needed, the future rate is calculated by using a spline as described above.

###### Rates at a future time

With the option called “Rates at a future time” forward rates that start at the point of the maturity length that you provide are calculated. This will, in essence, give you a view of the yield curve at a future point in time relative the observations.

###### Forward rates between instruments

The rate at each point is the rate of a forward from the corresponding instrument to the next. The length of the forward will thus be the difference in maturity of the current instrument and the next.

#### Day count convention

When specifying maturity lengths, the 30/360 day count convention is used. There are 360 days per year, 12 months per year and 7 days per week.

Please note that the Count parameter is a decimal number and you can specify, for instance, 1.5 months.