Bond Prices and Yields

Author

Affiliation

Mingze Gao, PhD

Macquarie University

Published

August 21, 2023

In the last post we introduce features of a bond. Now, let’s look at how to price a plain vanilla bond and examine the relation between bond price and yield.

Price of a bond

First, what’s the fair price of a bond?

For the investor, a bond represents a series of cashflows to receive in the future. So its price is naturally the total present value of all cashflows from the bond, including coupon payments (if any) and the repayment of principal (bond face value).

Therefore, the following equation holds true universally at all time $t$ :

${Bond Price}_{t} = {PV}_{t} (future coupons) + {PV}_{t} (face value)$

A bond’s price at time $t$ is the present value as at time $t$ of all coupons to receive in the future, plus the present value as at time $t$ of the bond face value. Personally, I’d call this fundamental.

Before we move on

Remember that all the complications we will see later are just results of uncertainties about the PVs, which ultimately are determined by cashflows and discount rates. Whenever you’re lost, pause and think about how they are affected, and then reason about how bond price may be affected.

I like examples. Suppose we are to issue a 10-year bond with a $10,000 face value that pays 5% annual coupon in arrears¹, with 8% discount rate², what would be the price today $t = 0$ at which we can sell the bond to investors?

¹ “in arrears” means that the payment is made at the end of the period. So in this case, the bond pays coupon annually at the end of the year, not any time sooner.

² We assume that this is an effective annual rate, and is the same for all 10 years in the future.

The chart below shows the result. Specifically, the blue bars indicate the bond’s cashflows, the overlaying red bars indicate their present values as at time $t = 0$ .

viewof maturity = Inputs.range(
  [1, 30], 
  {value: 15, step: 1, label: "Maturity (years):"}
)
viewof discountRate = Inputs.range(
    [0.01, 0.2],
    {value: 0.1, step: 0.01, label:"Discount rate:"}
)
viewof couponRate = Inputs.range(
    [0.01, 0.2],
    {value: 0.1, step: 0.01, label:"Coupon rate:"}
)
viewof couponFreq = Inputs.radio(
    ["Annual", "Semiannual"],
    {value: "Annual", label:"Coupon frequency:"}
)
a = {
  const f = 100;
  let c, m, r;
  if (couponFreq == "Semiannual") {
    c = couponRate / 2;
    m = 2 * maturity;
    r = discountRate / 2;
  } else {
    c = couponRate;
    m = maturity;
    r = discountRate;
  }
  const cashflows = {"Time": [], "Cashflow": [], "PV": []};
  let cf = f * c;
  for (let i = 1; i < m+1; ++i) {
    cashflows["Time"].push(i);
    if (i == m) cf = f * (1+c);
    cashflows["Cashflow"].push(cf);
    cashflows["PV"].push(cf / (1+r)**i);
  }
  return cashflows;
}

data = transpose(a)
Plot.plot({
  x: {padding: 0.4},
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.barY(data, {x: "Time", y: "Cashflow", fill: "blue", dx: -2, dy: -2}),
    Plot.barY(data, {x: "Time", y: "PV", fill: "red", dx: 2, dy: 2}),
  ]
})

maturity = 15

discountRate = 0.1

couponRate = 0.1

couponFreq = "Annual"

a = Object {Time: Array(15), Cashflow: Array(15), PV: Array(15)}

data = Array(15) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object]

Try it out

Change the parameters, see what happens to the bond price!

The formula

The initial price $P_{t = 0}$ of a plain vanilla $N$ -year bond with face value $F$ , annual coupon $C$ , at a constant discount rate $r$ ³, is given by

³ This discount rate $r$ is assumed to be an effective rate for a period of time between two coupons, i.e., effective annual rate in this case.

$P_{t = 0} = \underset{sum of coupons’ PVs}{\underset{⏟}{\sum_{τ = 1}^{N} \frac{C}{(1 + r)^{τ}}}} + \underset{face value’s PV}{\underset{⏟}{\frac{F}{(1 + r)^{N}}}}$

Note that this is only the initial price when the bond is issued at time $t = 0$ .

Price over time

Question

So, other things equal, how does bond price changes over time as we approaches the maturity date?

We need a better formula that can let $t$ take values other than 0. Recall the rationale that the price is nothing but sum of all PVs of future payments.

A slightly improved formula

At time $t$ , which is exactly $n$ years till maturity, the price, $P_{t}$ , of a plain vanilla bond with face value $F$ , annual coupon $C$ , at a constant discount rate $r$ , is given by

$P_{t} = \underset{sum of coupons’ PVs}{\underset{⏟}{\sum_{τ = 1}^{n} \frac{C}{(1 + r)^{τ}}}} + \underset{face value’s PV}{\underset{⏟}{\frac{F}{(1 + r)^{n}}}}$

From only $P_{t = 0}$ to ${P_{t}}$ is a major improvement!⁴

⁴ However, here $t$ can only be positive integers as we assume $t$ is exactly $n$ years before the bond’s maturity. See the next question.

⁵ Note that at maturity, $n = 0$ such that the price $P_{t} = F$ .

Let me show you another graph. Note that in this graph, each bar represents the bond price as at a point in time.⁵

viewof maturity2 = Inputs.range(
  [1, 30], 
  {value: 15, step: 1, label: "Maturity (years):"}
)
viewof discountRate2 = Inputs.range(
    [0.01, 0.2],
    {value: 0.10, step: 0.01, label:"Discount rate:"}
)
viewof couponRate2 = Inputs.range(
    [0.01, 0.2],
    {value: 0.05, step: 0.01, label:"Coupon rate:"}
)
viewof couponFreq2 = Inputs.radio(
    ["Annual", "Semiannual"],
    {value: "Annual", label:"Coupon frequency:"}
)
b = {
  const f = 100;
  let c, m, r;
  if (couponFreq2 == "Semiannual") {
    c = couponRate2 / 2;
    m = 2 * maturity2;
    r = discountRate2 / 2;
  } else {
    c = couponRate2;
    m = maturity2;
    r = discountRate2;
  }
  function pv(c, f, t, r) {
    return c * (1 - (1+r)**(-(m-t))) / r + f / (1+r)**(m-t)
  }
  const cashflows = {"Time": [], "PV": []};
  let coupon = f * c;
  for (let i = 0; i < m+1; ++i) {
    cashflows["Time"].push(i);
    cashflows["PV"].push(pv(coupon, f, i, r));
  }
  return cashflows;
}

data2 = transpose(b)
Plot.plot({
  caption: "Assume $100 bond, coupons paid in arrears and effective annual discount rate.",
  x: {padding: 0.4},
  grid: true,
  marks: [
    Plot.ruleY([0]),
    Plot.ruleY([100]),
    Plot.barY(data2, {x: "Time", y: "PV", fill: "blue"}),
  ]
})

maturity2 = 15

discountRate2 = 0.1

couponRate2 = 0.05

couponFreq2 = "Annual"

b = Object {Time: Array(16), PV: Array(16)}

data2 = Array(16) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object]

Assume $100 bond, coupons paid in arrears and effective annual discount rate.

Change the discount rate to be lower than coupon rate, what do you find?

It’s not difficult to find that, as it approaches the maturity, the bond price approaches the face value, regardless of whether the bond is traded at a premium or discount.⁶

⁶ A bond is traded at a premium (discount) if its price is above (below) face value.

Confused?

Recall earlier we said that the longer the maturity, the lower the bond price. This is true because we are talking about the initial price at issue. For example, other things equal, the price of a 30-year bond is lower than the price of a 10-year bond.

Here, time is changing. There is a bond of a given maturity (e.g., 30 years), and we study how its price changes over time as we get close to the 30-year mark.

A more improved formula

But we can still do better!

Question

So far we have been assuming the next coupon is exactly one period (e.g., year) in the future, or in other words, the last coupon has just been paid. What if this is not the case? What if the next annual coupon is in 2 months, not in 12 months, from now?

When the coupon payment date does not align with the time at which we compute the bond price, only a simple adjustment is required.

The basic idea is, since next coupon and all future payments are closer than what’s assumed in computation, we have over-discounted the bond value. So, we can correct it by “growing” the undervalued price for the time since last coupon payment.

$P_{t} = \underset{bond price right after last coupon}{\underset{⏟}{[\sum_{τ = 1}^{n} \frac{C}{(1 + r)^{τ}} + \frac{F}{(1 + r)^{n}}]}} \times (1 + r)^{\frac{days since last coupon}{days between coupons}}$

As such, we can now derive a continuous path for the bond price since issue to maturity, assuming other things equal. This is shown in the next chart as a blue line.⁷

⁷ In the last one, I deliberately use bar chart to indicate discreteness.

Price: dirty and clean

Attention!

Now imagine you are to buy a bond immediately before it matures. What would be the price according to the formula and the chart above?

No matter how much coupon the bond pays, the price (indicated by the the last bar) is the face value of the bond $10,000. After the purchase, however, you will immediately receive a total payment of bond face value and the last coupon payment, which surely is greater than $10,000.

Apparently, you need to pay more than the price described by the formula to the seller.

In fact, a bondholder starts to accumulate accrued interest⁸ the moment their own the bond. Even though they may sell the bond right before a coupon payment, but given that they have been holding the bond for almost entire the time until selling just before the next coupon payment, they should be given compensation for not receiving the next coupon, which will be paid to the buyer.

⁸ Accrued interest: payoffs entitled to the investor but not yet paid.

Further, we generalize this idea to bond transactions any time between coupon payments – the buyer should compensate the seller additionally a coupon payment proportional to the time that the seller has been holding since last coupon payment relative to the time between two coupon payments.

viewof maturity3 = Inputs.range(
  [1, 30], 
  {value: 15, step: 1, label: "Maturity (years):"}
)
viewof discountRate3 = Inputs.range(
    [0.01, 0.2],
    {value: 0.10, step: 0.01, label:"Discount rate:"}
)
viewof couponRate3 = Inputs.range(
    [0.01, 0.2],
    {value: 0.05, step: 0.01, label:"Coupon rate:"}
)
viewof couponFreq3 = Inputs.radio(
    ["Annual", "Semiannual"],
    {value: "Annual", label:"Coupon frequency:"}
)
c = {
  const f = 100;
  let c, m, r;
  if (couponFreq3 == "Semiannual") {
    c = couponRate3 / 2;
    m = 2 * maturity3;
    r = discountRate3 / 2;
  } else {
    c = couponRate3;
    m = maturity3;
    r = discountRate3;
  }
  let scale = 300 / m;
  function pv(c, f, t, r) {
    return c * (1 - (1+r)**(-(m-t))) / r + f / (1+r)**(m-t)
  }
  const cashflows = {"Time": [], "Clean": [], "Dirty": []};
  let coupon = f * c;
  for (let i = 0; i < m * scale + 1; ++i) {
    let t = parseInt(i / scale, 10);
    cashflows["Time"].push(i);
    let clean = pv(coupon, f, i/scale, r);
    let dirty = clean + coupon * (i-t*scale)/scale;
    cashflows["Clean"].push(clean);
    cashflows["Dirty"].push(dirty);
  }
  return cashflows;
}

data3 = transpose(c)
Plot.plot({
  caption: "Assume $100 bond, coupons paid in arrears and effective annual discount rate.",
  x: {padding: 0.4, 
        transform: (x) => (x/(300/maturity3)),
        domain: [0, maturity3],
        label: "Year"},
  grid: true,
  marks: [
    Plot.ruleY([0, 100]),
    Plot.ruleX([0]),
    Plot.lineY(data3, {x: "Time", y: "Clean", stroke: "blue"}),
    Plot.lineY(data3, {x: "Time", y: "Dirty", stroke: "red"}),
  ]
})

maturity3 = 15

discountRate3 = 0.1

couponRate3 = 0.05

couponFreq3 = "Annual"

c = Object {Time: Array(301), Clean: Array(301), Dirty: Array(301)}

data3 = Array(301) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, …]

Assume $100 bond, coupons paid in arrears and effective annual discount rate.

The chart above shows the continuous path of bond price described by the formula in blue and the price including the additional compensation, i.e., the accrued interest. We names these two prices “clean price” and “dirty price”, respectively.

The clean price is the price suggested by the formula. In reality, this is the price that market participants deal with more often, e.g., in financial reporting, portfolio management, etc.
The dirty price is the clean price plus accrued interest. This is the actual settlement price for bond transactions.

And the accrued interest is given by

$coupon \times \frac{time since last coupon}{time between coupons}$

which periodically increases and resets.

Day count convention

For Treasury bonds, we use “actual/actual” convention.
For others, typically “30/360” convention.

Price and yield (to maturity)

So far, we have not yet spent a single word on a bond’s “yield”, but instead been using “discount rate”. What’s the difference?

The answer is straightforward. A bond consists of a series of cashflows in the future, each of which may be discounted at different rates. For example, the coupon in 1 year may be discounted at a 5% discount rate, the coupon in 2 years may be discounted at 6%, and so on.

It turns out that, at any time $t$ , while a bond can have multiple future payments each discounted at different rates ${r_{τ}}$ , we can always find a single discount rate $y$ which, when applied to all future payments, leads to the same bond price at the time:

$P_{t} = \underset{each discounted at varying rates}{\underset{⏟}{\sum_{τ = 1}^{n} \frac{C}{(1 + r_{τ})^{τ}} + \frac{F}{(1 + r_{n})^{n}}}} = \underset{all discounted at the same rate}{\underset{⏟}{\sum_{τ = 1}^{n} \frac{C}{(1 + y)^{τ}} + \frac{F}{(1 + y)^{n}}}}$

This single discount rate $y$ is called the yield to maturity, or yield, of the bond at time $t$ .

If we plot bond prices against yields, at a given time, it’s easy to see that they have a one-to-one mapping and a inverse non-linear relationship.

viewof maturity4 = Inputs.range(
  [1, 30], 
  {value: 15, step: 1, label: "Maturity (years):"}
)
viewof couponRate4 = Inputs.range(
    [0.01, 0.2],
    {value: 0.05, step: 0.01, label:"Coupon rate:"}
)
d = {
  const f = 100;
  let c, m;
  c = couponRate4;
  m = maturity4;
  function pv(c, f, t, r) {
    return c * (1 - (1+r)**(-t)) / r + f / (1+r)**(t)
  }
  const prices = {"YTM": [], "Price": []};
  let coupon = f * c;
  for (let ytm = 0.01; ytm < 20; ytm++) {
    let price = pv(coupon, f, m, ytm/100);
    prices["YTM"].push(ytm);
    prices["Price"].push(price);
  }
  return prices;
}

data4 = transpose(d)
Plot.plot({
  caption: "Assume $100 bond, annual coupons paid in arrears and effective annual discount rate.",
  x: {padding: 0.4, label: "YTM (%)"},
  grid: true,
  marks: [
    Plot.ruleY([0, 100]),
    Plot.ruleX([0]),
    Plot.lineY(data4, {x: "YTM", y: "Price", stroke: "blue"}),
  ]
})

maturity4 = 15

couponRate4 = 0.05

d = Object {YTM: Array(20), Price: Array(20)}

data4 = Array(20) [Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object, Object]

Assume $100 bond, annual coupons paid in arrears and effective annual discount rate.

There are many interesting features about the bond price-yield relationship.

The one-to-one mapping between bond price and yield implies that we can always compute the other when given either of them. So, knowing either price or yield is sufficient when dealing with bonds.⁹

⁹ Assuming other bond features are known, such as coupon rate, frequency, etc., which are public information.

The inverse relationship suggests that the higher the yield, the lower the bond price.

The non-linearity suggests that the sensitivity of bond price to yield is not static. In fact, we can tell from the graph that the curve is convex. This convexity will be of great importance in later studies of bond risk.

Other yields

When we talk about a bond’s yield, we usually refer to the yield to maturity. But there can be some other yield measures:

Portfolio of bonds

The yield of a portfolio of bonds is NOT a weighted average of individual bonds’ yields because the bonds are not homogeneous.

Trouble maker: floating-rate bonds

We cannot easily compute the yield of a floater, simply because the values of reference rate in the future are unknown. Instead, we can use some spread measures that describe the yield in excess of the reference rate. The most popular measure of yield spread for a floating-rate bond is discount margin.

As its name suggests, discount margin basically captures the “discount rate in excess of reference rate”.

Suppose the bond market price is $P_{t}$ , reference rate is assumed constant at $R$ , the discount margin $D M$ is the one that solves the equation below:

$P_{t} = \sum_{τ = 1}^{n} \frac{C}{(1 + R + D M)^{τ}} + \frac{F}{(1 + R + D M)^{n}}$

Note that here the coupon payment $C$ is determined by the reference rate $R$ and the quoted margin.¹⁰

¹⁰ Quoted margin: a floating-rate bond’s coupon rate is determined by a reference rate plus its quoted margin.