Bonds and Rates

Previously we looked at how bond performance compared to equity performance, as well as the potential tax advantages in the UK of investing in lower coupon gilts.

In order to consider the performance of a bond or bond index, we first need to understand how to price a bond.

Why Bond Pricing?

We will look at bond pricing, rather than stock pricing, as bonds by their nature lend themselves to a methodical model based approach for valuation. In the case of a fixed rate bond we know up front all of the coupon payment dates, the coupon amounts (or calculation formula), and the bond has a definite end date.¹

While we do intend to cover stock valuation in a future post, that topic requires a much more detailed analysis. Fixed rate bonds give us a more straightforward starting point for a clearer understanding the fundamentals of valuation.

In this post we’re going to focus on:

• the basics of bond pricing
• what a discount factor is
• how the price changes as rates change.

We’re going to ignore some of the more nuanced aspects such as complex curve construction, and default modelling, or any kind of complex structured or inflation linked bonds. We will cover these in later post.

Interest Rates and Discounting

A natural first attempt to price a bond is to take all of the future cash flows from the coupons and the redemption and sum them. Unfortunately this typically doesn’t work. In normal times² most investors want some kind of return on an investment that they make, so if they invest a dollar today then in a year’s time they will want to get back a dollar plus some return.

The interest paid compensates the lender for the risk that they took that the borrower wouldn’t pay back the loan, but also, crucially, the opportunity cost of that money, i.e. the chance to make money by say investing in a business that the lender has foregone by giving their money to the borrower for a year.

We can say that in order to lend $\$1$ today we are going to want to get back some greater amount $\$(1+r)$. An alternative, but equivalent, way of viewing this is to ask, how much am I willing to lend today to receive $\$1$ in a years time, this amount $\$y$, gives us a scale factor by which we can multiply any amount of dollars to be paid in 1 year’s time to get the value today. We’ll refer to this as the discount factor, or $df$.

Just as a dollar today isn’t worth the same as a dollar in a years, time, similarly a dollar in a years time isn’t worth the same as a dollar in a two years time. So we’ll have a different discount factor for each, let’s say $df_1$ and $df_2$.

This latter framing makes the valuation of future cashflows straightforward, $n$ dollars in one years time is worth $n*df_1$ dollars today, $n$ dollars in two years time is worth $n*df_2$ dollars today, and so on.

Bond Pricing

In our simple valuation we’re going to assume that all interest is paid annually and that $df_1 = 1/(1+r)$. This inverse relationship between the interest rate and the discount factor is why we see that as rates go up, bond prices go down.

Below we have a chart and table of the cashflows for a fixed rate bond paying a coupon of 20% annually (we’re taking a very high coupon rate for ese of visualisation, in current market conditions a UK government bond would be issued with a coupon of 3-4%).

Here we can vary the input rate in a simple discounting model to see the impact. We can observe how the discounted values (or present values) of the cashflows varies as we adjust the simple discounting rate.
Note: We are using discrete compounding of interest here, rather than continuous.

Relate

This is a relatively high level overview of how bonds are priced, and how rates impact bond prices. Using Relate we can go much further. We can automatically onboard and represent the definitions of traded bonds, handling day count conventions, schedule generation and resets in the case of floating rate bonds.

Market Data

In Relate we have a number of different ways to represent interest rate curves, and use them to price bonds. The simplest of these are a simple compounding curve, where we interest is considered to be applied continuously, and a discount curve, where we explicitly specify the value today of a dollar at specified times in the future (for any date that falls between two specified dates, we compute the discount factor via interpolation).

Below, we have generated the definition for the same example 20% coupon bond and cashflow table. Relate also allows us to include the bond in a portfolio, compute more detailed analytics (duration/tax adjusted yield), generate portfolio level cashflow monitoring. The bond representation looks as follows when exported to JSON:

{
    "accrualType": "Standard",
    "quoteType": "price",
    "schedules": [
      {
        "conventions": {
          "bdc": "NoAdjustment",
          "currency": "GBP",
          "dcc": "Act365L",
          "includeLeapDays": 0,
          "paymentCalendars": "LON",
          "roll": "NONE",
          "settleDays": 0,
          "tenor": "1Y"
        },
        "couponRate": 0.2,
        "maturity": "2030/02/05",
        "notional": 100,
        "paymentCcy": "GBP",
        "start": "2025/02/05",
        "stubType": "shortFront",
        "type": "Fixed"
      }
    ],
    "type": "Bond"
}

In the below screen capture we can see the relate server being called via excel, to replicate some of the simple calculations that we have shown above.

Pricing in Excel