FINE 332. 4.2 Interest Rates and Bond Valuation

Open the spreadsheet below and answer the questions provided. Make sure to go to four (4) significant digits (e.g. 3.4576%). the Corporate Bonds Template is the question file.

Question 1 – Use the coupon, par and purchase price given to show the yield to maturity (YTM)
Cash Flows
Purch.
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Coupon
Par
YTM
5.12%
1,000.00
(802)
Question 2 – Use the coupon, interest rate and par given to calculate the bond’s price
0.5
Coupon
Par
Rate
Bond price
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
4.75%
1,000.00
4.45%
Question 3 – Use the coupon, par and purchase price given to show the yield to maturity (YTM)
Cash Flows
Purch.
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Coupon
Par
YTM
5.66%
1,000.00
(711)
Question 4 – Use the coupon, interest rate and par given to calculate the bond’s price
1.0
Coupon
Par
Rate
Bond price
5.78%
1,000.00
5.23%
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
h Flows
4.5
5.0
5.5
6.0
6.5
7.0
7.5
5.0
5.5
Cash Flows
6.0
6.5
7.0
7.5
8.0
9.0
10.0
11.0
13.0
14.0
15.0
11.0
Cash Flows
12.0
13.0
14.0
15.0
16.0
8.5
9.0
9.5
10.0
10.5
11.0
17.0
18.0
19.0
20.0
21.0
22.0
h Flows
10.0
12.0
11.5
12.0
12.5
23.0
24.0
25.0
Question 5 – Use the coupon, par and purchase price given to show the yield to maturity (YTM)
Cash Flows
Purch.
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Coupon
Par
YTM
0.00%
1,000.00
(509)
Question 6 – Use the coupon, interest rate and par given to calculate the bond’s price
0.5
Coupon
Par
Rate
Bond price
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.00%
1,000.00
6.15%
Question 7 – Use the coupon, par and purchase price given to show the yield to maturity (YTM)
Cash Flows
Purch.
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Coupon
Par
YTM
0.00%
1,000.00
(303)
Question 8 – Use the coupon, interest rate and par given to calculate the bond’s price
1.0
Coupon
Par
Rate
Bond price
0.00%
1,000.00
4.75%
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
rity (YTM)
Cash Flows
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
7.5
8.0
8.5
5.0
5.5
6.0
Cash Flows
6.5
7.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
12.0
Cash Flows
13.0
14.0
15.0
16.0
17.0
9.0
9.5
10.0
10.5
11.0
18.0
19.0
20.0
21.0
22.0
rity (YTM)
Cash Flows
10.0
11.0
11.5
12.0
12.5
13.0
23.0
24.0
25.0
26.0
Non-Callable
Callable
YTM
Coupon
Par
5.44%
1,000.00
Purch.
0.5
1.0
1.5
2.0
Non-Callable
Callable
Notes:
The callable bond is callable at the end of year 4 at a price of $909.0000 including coupon.
The bond price is $902.1500.
Cash Flows
2.5
Cash Flows
3.0
3.5
4.0
4.5
5.0
5.5
6.0
a. YTM
A
YTM
Coupon
Par
Cash Flows:
Purchase price
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
B
b. Duration
A
c. New
B
New Prices are
4.85%
1,000.00
4.430%
1,000.00
(767.00)
(788.00)
c. New Prices if rates go to
A
B
when rates go to
5.000%
No content – Intentionally left blank
Corporate Bonds: Interest Rates, Prices and Yields
Alfonso F. Canella Higuera
October 19, 2021
“I used to think that if there was reincarnation, I wanted to come back as the president or the pope
or as a .400 baseball hitter. But now I would like to come back as the bond market. You can intimidate
everybody.” James Carville
Bonds are, quite simply, how corporations and governments (federal, state, municipal, and
agencies) borrow money from investors. While there are other bonds, we will concern ourselves
here with corporate bonds.
Most corporate bonds pay interest in the form of a coupon. The coupon is usually a per
cent of par (which is usually $1,000). So, a 5.00% coupon, 2-year bond had the following
characteristics:
• the coupon interest paid is 5.00% x $1,000 x 0.5 = $25; it is paid every six months over
2 years (unless stated otherwise)
• the final payment at the end of year 2 is $1,025; this contains the coupon interest plus
the $1,000 principal owed on the bond (the par)
The coupon per cent is usually the market rate of interest that the issuer is expected to pay
given that issuer’s credit rating at the time of issuance – in this case, 5.00%. So, the way the math
works, when the market interest rate equals the coupon rate, the bond sells at par ($1,000). This
is because the coupon rate equals the discount rate (the market rate of interest) being used to
present value the coupon payment stream and the final par value payment.
Note that the coupon rate and the market interest rate (that is, the discount rate applied to
value the bond) are almost never the same once the bond is issued and is trading in the markets.
The coupon rate is set by the issuer. The market interest rate is set by the bond markets and it is
constantly fluctuating.
Bond Prices
Let’s show the formula for pricing the bond described above:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 =
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
+
+
+
. 𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎
(𝟏𝟏 + ( 𝟐𝟐 )𝟏𝟏 (𝟏𝟏 + ( 𝟐𝟐 )𝟐𝟐 (𝟏𝟏 + ( 𝟐𝟐 )𝟑𝟑 (𝟏𝟏 + ( 𝟐𝟐 )𝟒𝟒
With Excel, the above formula can be solved easily using the NPV function:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝑵𝑵𝑵𝑵𝑵𝑵(
𝟎𝟎. 𝟎𝟎𝟎𝟎
, 𝟐𝟐𝟐𝟐, 𝟐𝟐𝟐𝟐, 𝟐𝟐𝟐𝟐, 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏)
𝟐𝟐
The second formula is much easier to solve than the first and explains why finance shops
use Excel as the front-end tool for their work. Mind you, in a finance shop and using Excel, you
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would have separate cells for the discount rate (0.05 in decimal form), the par value ($1,000), and
the coupon rate (0.05). That way, if you have to value more than one bond with different values
for any of these three components, you can just insert the new values, and the answer will calculate
automatically. [In most finance shops that work with bonds, though, the analysts have bond
valuation programs, such as Bloomberg, that display bond values without the need to do the math.]
Note at this juncture that the timing of the cash flows and the timing of the discount rate
MUST match! So, if the cash flows are semi-annual, the discount rate, which is usually an annual
rate, must be divided by two to make it semi-annual. Most bonds pay semi-annually so you should
expect bonds to be valued using formulas along the lines of that just shown.
Similarly, if the bond were paying coupons annually, the discount rate to be applied to it is
the annual discount rate (that is, the discount rate without it being divided by two.)
So, at all times in finance, you must match the periodicity of the discount rate to the
periodicity of the cash flows. This goes for the NPV and the IRR calculations used in discounted
cash flows and capital budgeting. If the cash flows are monthly, then the discount rate, if you are
doing a present value using the NPV function, must be monthly. If you run the IRR on monthly
cash flows, then you must multiply the result from the IRR function by 12. In doing this, you are
making it annual.
Now, let’s say that interest rates, which were 5.00% when the bond was issued, have
dropped to 4.95%. This means that the discount rate used to price the bond in the pricing formula
above must change. The updated formula is now:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
=
+
+
+
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 𝟐𝟐
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
(𝟏𝟏 + ( 𝟐𝟐 )𝟏𝟏 (𝟏𝟏 + (
)
(𝟏𝟏 + ( 𝟐𝟐 )𝟑𝟑 (𝟏𝟏 + ( 𝟐𝟐 )𝟒𝟒
𝟐𝟐
Note that the coupon did NOT change as it is fixed. Only the discount rate being applied
to the cash flows changes. This result underscores a key relationship between bond prices and
interest rates – namely that they are inversely related.
So, if interest rates fall, bond prices rise; conversely, if interest rates rise, bond prices
fall. That’s it. Sometimes, the news reports these movements as “bond yields have risen” or “bond
yields have fallen”. In the first case, when yields (another word for interest rates) have risen, it
means that bond prices have fallen. So, if you hold some investments in a bond fund, most likely
that portion of your portfolio’s value has fallen.
Conversely, when the news report that “bond prices have fallen” what they are implicitly
saying is that interest rates rose that day. Similarly, when they say “bond prices rose”, they mean
that interest rates fell that day.
Note that in the U.S., the benchmark bond when you hear news reports is the 10-year
Treasury. As of October 19, 2021, the 10-year Treasury closed at a yield of 1.663%. The price
progression for this bond is made evident in this screen shot of the Wall Street Journal’s Market’s
section:
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Note here the term everything else equal. By this I mean that if you leave everything else
the same (credit rating, coupon rates, etc.), the price of a bond will rise if interest rates fall. You
may find yourself feeling dizzy, so just keep this in mind: if interest rates rise, bond values fall;
if rates fall, bond values rise. Bond fund managers try to guess the direction and amount of
these changes to determine which bonds to sell and which to buy to maximize gains and
minimize losses.
Conversely, if rates rise, bond prices will fall. The formula above is recalculated for a new
interest rate environment of 5.05% (instead of the old 5.00% environment):
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
=
+
+
+
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
(𝟏𝟏 + ( 𝟐𝟐 )𝟏𝟏 (𝟏𝟏 + ( 𝟐𝟐 )𝟐𝟐 (𝟏𝟏 + ( 𝟐𝟐 )𝟑𝟑 (𝟏𝟏 + ( 𝟐𝟐 )𝟒𝟒
You may think that the difference of less than one dollar is picayune but in the practice it
can be very material. Take the bond changing in price from $1,000 to $999.0601, this difference
represents a drop of 0.09399%. Now let’s say that you are the fund manager for a $600 million
bond fund, this drop represents a loss of $563,956.46 – a loss large enough to give even the
strongest a dose of heartburn!
So, bottom line, you buy a 2-year bond that pays a 5.0% coupon; if you hold it to maturity,
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you will get 5.0% and nothing more! This is called the yield-to-maturity (YTM). In effect, if you
hold to maturity, you collect the coupon as interest – that’s it. Thus, the coupon IS the interest.
This said, bond fund managers usually do not hold to maturity. Instead, they try to assess
where interest rates will go and will buy and sell bonds before they mature. Bond markets in the
U.S. are fairly liquid so buying and selling is not very hard. The key, as they say, is to know
WHEN to sell. A bond fund manager that sells before interest rates rise will do well, then, as
he/she will sell before the price of the bond drops. Conversely, a bond manager that buys up long
term bonds before interest rates drop will do well.
So, to recap, to calculate the price of a bond, you calculate the NPV of the coupon
payments (and the final $1,000 par payment) using the discount rate that the market has set
for that bond based on its credit quality, time to maturity, and liquidity. Just make sure that
the periodicity of the discount rate matches that of the coupons. So, if the coupons are semiannual, the discount rate must be semi-annual (the annual rate divided by two.)
Bond Yields
Now that we know how to calculate the price of a bond, let’s look at how to calculate the
yield on a bond. To calculate yield, we must know the current price and the coupons; the yield is
the discount rate that when applied to the coupons gives the price we have. For this example, we
will assume that the 5.0% coupon, 2-year bond is selling for $966.7900. The yield is X and we
have to solve for it. Note that we have four cash flows as the payments are made semi-annually:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕 =
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟐𝟐
$𝟐𝟐𝟓𝟓
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
+
+
+
𝑿𝑿
𝑿𝑿
𝑿𝑿
𝑿𝑿
(𝟏𝟏 + ( 𝟐𝟐 )𝟏𝟏 (𝟏𝟏 + ( 𝟐𝟐 )𝟐𝟐 (𝟏𝟏 + ( 𝟐𝟐 )𝟑𝟑 (𝟏𝟏 + ( 𝟐𝟐 )𝟒𝟒
𝑿𝑿 = 𝟔𝟔. 𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖%
Having solved X, we note that the discount rate (or interest rate, you choose the wording)
that applies to this bond is 6.8041%. So, if you buy this bond at $966.79 and hold it to maturity,
you will receive an annual interest rate of 6.8041%. Had you paid $1,000 for the same bond, you’d
be receiving 5.0%. At maturity, the bond is worth $1,000 so the difference between 5.0% and
6.8041% is the gain of $33.21 over the two years for the bond.
It is very important that you understand that only if you buy and hold will you get the yield
that you calculated. If you sell early, you will likely get a different yield. Again, most bond fund
managers will sell early because they think they can time the market and maximize their gains by
selling bonds at the most opportune time.
Continuing with calculating yield, the easiest way to calculate yield using Excel is to use
the IRR function. For the bond above, the formula would be:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀 = 𝟔𝟔. 𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖% = 𝑰𝑰𝑰𝑰𝑰𝑰(−𝟗𝟗𝟗𝟗𝟗𝟗. 𝟕𝟕𝟕𝟕, 𝟐𝟐𝟐𝟐, 𝟐𝟐𝟐𝟐, 𝟐𝟐𝟐𝟐, 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏) 𝒙𝒙 𝟐𝟐
Here, you must note the following:
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•
•
•
the initial cash flow is negative because you are buying the bond with cash
the cash flows thereafter are the coupon payments made to the investor (you), hence
positive cash flows go to the investor every six months including the final coupon and
par value at maturity at the end of year 2
the IRR result is multiplied by 2 to make it annual; this is because the cash flows are
semi-annual thus making the IRR’s output semi-annual
Up to now, we covered bonds paying semi-annual coupons. The calculations are made a
lot easier when one is valuing bonds paying annual coupons. So, for the same bond we have been
examining, namely a 5.0% coupon, 2-year bond, we can draw up the bond’s price formula this
time using yearly coupon payments.
Note that the annual coupon is just the 5.0% coupon times the $1,000 par, which is $50.
At maturity, the $50 coupon is paid along with the $1,000 par:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 =
$𝟓𝟓𝟓𝟓
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
+
(𝟏𝟏 + (. 𝟎𝟎𝟎𝟎)𝟏𝟏 (𝟏𝟏 + (. 𝟎𝟎𝟎𝟎)𝟐𝟐
With Excel, the above formula can be solved easily using the NPV function:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝑵𝑵𝑵𝑵𝑵𝑵(. 𝟎𝟎𝟎𝟎, 𝟓𝟓𝟓𝟓, 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏)
Let’s say, as we did previously, that interest rates rise to 5.05%. The coupons will stay the
same but the discount rate now changes. For the annual coupon paying bond, we now have:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 =
$𝟓𝟓𝟓𝟓
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
+
𝟏𝟏
(𝟏𝟏 + (. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏 + (. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)𝟐𝟐
If you look back, the bond price for the semi-annual 5.0% coupon-bearing bond with a 2year maturity is $999.0601. The timing of the coupon payments is different between the semiannual bond and the annual bond and this is reflected in the price. The difference is minute – all
of .001089%. But, as we said before, if you are managing a $600M bond fund, this tiny per cent
becomes $6,534.43. It may be a small loss but no one likes losses and even small amounts, if they
repeat over time, can become sizable losses.
Yields will also be different between the semi-annual bond and the annual bond. Using
the same price that we used for the semi-annual bond example, we reconfigure the formula for the
annual bond’s yield:
𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕 =
$𝟓𝟓𝟓𝟓
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
+
𝟏𝟏
(𝟏𝟏 + (𝑿𝑿)
(𝟏𝟏 + (𝑿𝑿)𝟐𝟐
𝑿𝑿 = 𝟔𝟔. 𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖%
Given a yield of 6.8326%, we note here that the semi-annual bond’s yield is 6.8041%. So,
same price, coupon, and maturity, but one bond paying semi-annually and another paying annually
makes a big difference!
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Zero-Coupon Bonds
Up to now, we have looked at bonds that pay a coupon. Now, we will look at bonds that
do not pay a coupon. The bonds are called zero-coupon bonds (zeros, for short). These bonds
pay the $1,000 par at maturity ONLY. Because that final payment may be far out in the future,
these bonds tend to have much lower prices than coupon-paying bonds.
Let’s take a look at the bond price formula for a 2-year zero in an environment where that
zero should be paying a 5.0% interest rate:
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
𝒁𝒁𝒁𝒁𝒁𝒁𝒁𝒁 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪 𝑩𝑩𝑩𝑩𝒏𝒏𝒅𝒅 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 =
(𝟏𝟏 + (. 𝟎𝟎𝟎𝟎)𝟐𝟐
Using Excel and the NPV function, one can also obtain the zero’s price:
𝒁𝒁𝒁𝒁𝒁𝒁𝒁𝒁 𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪 𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝑵𝑵𝑵𝑵𝑵𝑵(. 𝟎𝟎𝟎𝟎, 𝟎𝟎, 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏)
Note that the first cash flow in the NPV’s argument, after the discount rate, must be a zero.
If it is blank, you will get the wrong price.
So, in comparison, the price of the zero is $907.0295 whereas the price of its couponpaying brethren is $1,000 for either the semi-annual or annual coupon payments. The difference
between the $1,000 par value paid at maturity and the price now is $92.9705. This is the amount
that the zero will appreciate over the next two years until, at maturity, it reaches $1,000.
Mind you, the path that the zero’s price will take from now until maturity two years from
now will meander and the variability of this path from the expected will be due to changes in
interest rates over the next two years. So, only two things are certain for this zero, its price today
is $907.0295 and its price at maturity will be $1,000. That’s it.
While one can estimate what these prices will be along the way, once can never be certain.
At this juncture, there are only probabilities of being correct. It is important to understand that
bond prices are constantly changing because interest rates are also constantly changing. That is
why it is far harder to price bonds than stocks. For stocks, whose prices hinge on the present value
of future cash flows, the main driver of prices is the long-term growth rate of the stock’s cash flows
and these are usually stable over time. For bonds, it is much harder to ascertain the changes to
interest rates as rates change constantly over a short span of time.
In effect, stock price forecasts are changed, at most, every quarter or earlier if there are
material news affecting the firm. Bond prices change constantly in a given day as interest rates
fluctuate constantly during the day. No matter how minute these changes in interest rates are, they
change bond prices and, for a large enough bond fund, even these small changes can turn into very
large dollar numbers.
Parenthetically, stock prices also fluctuate every day but those fluctuations are based purely
on the buying and selling of shares of stock. These trades affect the price of stocks. However, the
underlying fundamentals of stock pricing don’t much change over time.
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Duration
At this moment, you may ask yourself: what is the point of a zero coupon bond? It is a
good question and the answer is that zeros play an important role in bond fund management. This
is because bond fund managers track their bond portfolio’s duration.
Duration is crucial to any bond fund manager because it quantifies the sensitivity of a bond
or a bond portfolio to changes in interest rates. Remember that bond prices and interest rates
follow an inverse relationship. If interest rates rise, bond prices fall. Conversely, if interest rates
fall, bond prices rise. How much bond prices rise and fall is dependent on two key factors:
•
•
the change in interest rates, normally presented as basis points (bps). The best way to
explain a basis point is to give an example – the difference between 5.00% and 5.10%
is 10 basis points; in effect, a basis point (pronounced “bip”) is one 100th of one per
cent – so 10 “bips” is one 10th of one percent
the portfolio’s duration. The higher the duration, the larger the swing in prices.
Conversely, the lower the duration, the lower the swing
So, if you were a bond fund manager and you are predicting that interest rates will fall, you
will want to maximize your portfolio’s gains if those interest rates fall. Since you can’t control
how much the rates will change by (in bps), you control what you can – namely your portfolio’s
overall duration by increasing its overall duration.
The way you increase duration very rapidly is to sell low duration bonds (usually shortterm, coupon-paying bonds) and/or buy high duration bonds (basically long-term zeros, such as
10-year, 20-year or 30-year zeros). The duration of a 10-year zero is 10; that of a 20-year zero is
20 and so on – their durations are easy to determine as they are the number of years to maturity.
For a coupon-paying bond, it’s a bit more complicated but very doable using Bloomberg terminals
or spreadsheets.
Let’s take a look at the durations of four sample bonds. For this example, we will use The
Walt Disney Company’s “Sleeping Beauty” bonds. These were 100-year, 7.55% coupon-paying
bonds issued in 1993 with a face value of $300M. The 7.55% yield was 95 basis points higher
than the 30-year Treasury bond. Note that Disney reserved the right to call this bond in 30 years
for $1,030.20.
While we will discuss callable bonds and the valuation further on, it is worthwhile to note
at this moment that Disney was being very clever when it issued the Sleeping Beauty bonds:
•
•
•
if interest rates fell, Disney could call the bonds after 30 years to take advantage of the
new lower rates
if interest rates stayed level or rose, Disney would do nothing and continue paying
7.55%
Disney was being advised by Merrill-Lynch, which knew the market was receptive to
such long-term bonds so this wasn’t a leap into the void. For example, Coca-Cola
issued a non-callable 100-year bond yielding 7.455% at the time
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The company also issued “Napping Beauty” bonds, a 10-year, 7.55% coupon paying bond
issued at the same time. For comparison purposes, we will compare these two bonds to a 30-year,
7.55% coupon paying bond and a 30-year zero coupon bond. Ready?
The table below shows the four bonds and their calculated durations as of 1993:
Bond
Sleeping Beauty
Napping Beauty
30-year, 7.55% coupon
30-year zero
Maturing
in (yrs.)
100
10
30
30
Duration
14.2
7.4
12.6
30.0
One would think that the 100-year Sleeping Beauty bonds would have a much longer
duration than either the 30-year coupon bond or the 30-year zero. They don’t. In fact, they have
less than half the duration value of a 30-year zero. So, the impact of the Sleeping Beauty bonds
on a portfolio’s overall duration is much more limited than one would think. As a result, they are
not a poor choice for a medium to long term bond portfolio manager.
Now, let’s get a handle on duration in a way that we can grasp a bit more intuitively.
Duration is the mid-point at which the present value of the payments from issuance to the duration
year equal the present value of the payments from the duration year to maturity.
Underlying this is the expectation that when the investor reaches the time of duration (say
14.2 years out for the Sleeping Beauty bonds), the investor will sell the bond and collect the present
value of those payments to be made from year 14.2 to 100. So, for the Sleeping Beauty bonds, by
year 14.2, the investor will have received half of the present value that they will get from that bond.
Over the following 85.8 years, they will receive the other half.
Why are the remaining years so seemingly unimportant? This is because the later coupon
payments are discounted very heavily in the present value calculation. Let’s take a look at, say,
the formula for the present values of the payments made in the last 10 years of the Sleeping Beauty
bonds; note that these present values are as of the date of issuance!:
𝑷𝑷𝑷𝑷 𝒐𝒐𝒐𝒐 𝒚𝒚𝒚𝒚𝒚𝒚 𝟗𝟗𝟏𝟏 − 𝟏𝟏𝟏𝟏𝟏𝟏 = $𝟎𝟎. 𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕
$𝟕𝟕. 𝟓𝟓𝟓𝟓
$𝟕𝟕. 𝟓𝟓𝟓𝟓
$𝟕𝟕. 𝟓𝟓𝟓𝟓
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎. 𝟓𝟓𝟓𝟓
=
+ ⋯+
+
+
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎)𝟏𝟏𝟏𝟏𝟏𝟏
So, in today’s dollars, the last 10 years of payments that Disney makes on the Sleeping
Beauty bonds are worth less than one dollar! Those high exponents in the denominator sure make
for taking value down to size.
This brings us to an important point in finance. Cash flows beyond the first 10 years don’t
add that much value to the present value. Cash flows beyond 20 years barely bring value. So,
when investors are looking for returns, they aim for getting those within the first 10 years. That’s
when their present value is at its highest and where the numbers are reasonably close to the
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forecast. Thankfully, the mathematics help in that they discount later cash flows more heavily
than earlier cash flows and thus makes the decision-making process more accurate.
The four bonds compared previously have different present values depending on interest
rates. Remember, as interest rates fall, bond prices rise. As interest rates rise, bond prices fall.
Their rise and fall are NOT linear as they are governed by power (exponential) functions. This
means that as rates diverge further away from the 7.55% base interest rate, the present values of
these bonds will diverge markedly.
This divergence is made visible in the graph below:
PV of the Four Bonds as a function of Interest Rates
4,000
3,500
3,000
2,500
Sleeping
2,000
1,500
30 Year
Napping
1,000
Zero
500
0
2%
4%
6%
7.55%
9%
11%
13%
Interest Rate
15%
17%
19%
Note how the prices of the three bonds paying the 7.55% coupons all meet at the 7.55%
mark. This is because they are all valued at par ($1,000) when the market interest rate (the
discount rate) of 7.55% exactly matches the coupon rate! The zero does not count here because it
is not paying a 7.55% coupon.
What the table shows is that the price volatility of the Sleeping Beauty bonds is visibly
higher than that of the Napping Beauty bonds. That should be expected as the duration of the
Sleeping Beauty bonds is so much higher than that of the Napping Beauty bonds.
The zero coupon bonds are even more volatile in price but don’t show it so visibly. This
is all due to the scale used in the graph and not due to the numbers themselves. To show you that
this is the case, we compared the prices between the 2% PV and the 20% PV in the table below
for the four bonds:
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Bond
Sleeping Beauty
Napping Beauty
30-year, 7.55% coupon
30-year zero
% Difference between
Prices @2% and @20% interest
799%
213%
490%
13,005%
Now you see it? The reason why the zero coupon did not show that large a visible change
in price is that its value at a 20% interest rate is about $4, whereas the value of the Sleeping Beauty
bond at a 20% rate is $378. So, if you want price volatility, the way to go is the zero. This is why
bond fund managers trade these as follows:
•
•
if expecting interest rates to fall, buy zeros to increase duration and maximize the gain
from the appreciation of the bond portfolio
if expecting interest rates to rise, sell zeros to decrease duration and minimize the loss
from the depreciation of the bond portfolio
Calculating Duration (the hard way)
Let’s say we have a bond with the following characteristics:
•
•
7.5 years to maturity
4.4% semi-annual coupon
•
•
$1,000 par value
Current price = $902.0000
The cash flows are as follows:
YTM
Coupon
Par
Cash Flows:
Purchase price
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
6.0445%
4.400%
1,000.00
(902.00)
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
22.00
1,022.00
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The yield-to-maturity (YTM) shown above is calculated using the formula:
𝒀𝒀𝒀𝒀𝒀𝒀 = 𝟔𝟔. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎% = 𝑰𝑰𝑰𝑰𝑰𝑰(𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄: 𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚 𝟕𝟕. 𝟓𝟓 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄) 𝒙𝒙 𝟐𝟐
By using the IRR function, we quickly calculate the YTM for this bond and all we have to
do is reference the cell for the first cash flow and the cell of the last cash flow – by putting a colon
between them inside the IRR function we tell Excel that we want it to include all the cash flow
cells between these in the calculation. Since the payments are semi-annual, we multiply by 2 to
make the yield an annual one.
To calculate the duration for these payments, we start with the first payment, made six
months out. The formula for the duration of this payment is:
($𝟐𝟐𝟐𝟐 𝒙𝒙 𝟎𝟎. 𝟓𝟓)
𝟏𝟏𝟏𝟏𝟏𝟏 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 (𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚 𝟎𝟎. 𝟓𝟓) 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 =
(𝟏𝟏 + 𝒀𝒀𝒀𝒀𝒀𝒀)𝟎𝟎.𝟓𝟓
Similarly, the 2nd payments duration is calculated as:
𝟐𝟐𝟐𝟐𝟐𝟐 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 (𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚 𝟏𝟏. 𝟎𝟎) 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 =
($𝟐𝟐𝟐𝟐 𝒙𝒙 𝟏𝟏. 𝟎𝟎)
(𝟏𝟏 + 𝒀𝒀𝒀𝒀𝒀𝒀)𝟏𝟏.𝟎𝟎
𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 (𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚 𝟕𝟕. 𝟓𝟓) 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 =
($𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎 𝒙𝒙 𝟕𝟕. 𝟓𝟓)
(𝟏𝟏 + 𝒀𝒀𝒀𝒀𝒀𝒀)𝟕𝟕.𝟓𝟓
You calculate the interim payments the same way until the final payment at the end of year 7.5:
Once these are all done, you calculate duration as follows:
𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 =
∑𝟕𝟕.𝟓𝟓
𝟎𝟎.𝟓𝟓 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫
𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽𝑽 (𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷)
The numerator is the sum of the durations of all the payments from year 0.5 to 7.5. This sum is
divided by the absolute value of the bond price ($902.0000 in this case) to get the bond’s duration:
Purchase price (902.00) Duration
0.5
22.00
10.68
1.0
22.00
20.75
1.5
22.00
30.22
2.0
22.00
39.13
2.5
22.00
47.49
3.0
22.00
55.35
3.5
22.00
62.70
4.0
22.00
69.59
4.5
22.00
76.02
5.0
22.00
82.03
5.5
22.00
87.62
6.0
22.00
92.82
6.5
22.00
97.65
7.0
22.00
102.12
7.5 1,022.00 4,935.74
6.44
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So, this bond’s duration is 6.44. What, then, does this mean?
It means that in the period from issuance to 6.44 years out, the investor will receive half
the present value of the bond’s payments. The remaining half comes in the 1.06 years that follow.
This is a bit lop sided towards the latter years of the bond because the early payments are too small
in comparison to the final payment of $1,022 at maturity. So, in a way, this coupon-paying bond
is exhibiting characteristics of a zero coupon bond if only because its payments are tilted towards
the ending months.
The duration we have been examining up to now is called Macaulay duration. For the
purposes of calculating the change to a portfolio’s value, this duration must be converted to
modified duration as follows:
𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 =
𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫
𝒀𝒀𝒀𝒀𝒀𝒀
(𝟏𝟏 + 𝟐𝟐 )
In this case, the YTM above is the required yield on the bond. Let’s say that our bond fund
manager thinks that interest rates will drop by 10 basis points (bps). This means that the 6.0445%
YTM we calculated originally will now be 5.9445%. The modified duration is:
𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 =
The modified duration is also defined as:
𝟔𝟔. 𝟒𝟒𝟒𝟒
= 𝟔𝟔. 𝟐𝟐𝟐𝟐
. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
(𝟏𝟏 +
)
𝟐𝟐
𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 = 𝟔𝟔. 𝟐𝟐𝟐𝟐 =
𝚫𝚫𝚫𝚫 𝟏𝟏
𝒙𝒙
𝚫𝚫𝚫𝚫 𝑷𝑷
where:
• ΔP = change in price
• Δy = change in yield (10 bps = 6.0445% – 5.9445%)
• P = price of the bond ($902.0000)
Solving for ΔP:
𝟔𝟔. 𝟐𝟐𝟐𝟐 𝒙𝒙 $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 𝒙𝒙 (𝟔𝟔. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎% − 𝟓𝟓. 𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗%) = 𝜟𝜟𝜟𝜟 = $𝟓𝟓. 𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔
This means that the new bond price is $902.0000 + $5.6422 = $907.6422; the value of the
bond thus went up by 0.6255%. If the bond portfolio’s size were $600M, the dollar gain because
of the 10 bps change in interest rate would be $3,753,123.8792.
Note at this moment that a duration-only based calculation of the change in a portfolio’s
value is an APPROXIMATION. This is because the duration formula misses the natural convexity
of a bond’s present value depending on interest rates. You saw this convexity in the graph of the
four bonds that was shown previously.
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The graph below isolates the price arc (green line) of a generic bond and shows the
duration-based calculation (white line). The error in price based only on duration is highlighted.
In the practice, bond shops will calculate the convexity error and make the price calculations much
more exact.
Convexity
Actual Price
Price
Error in price based only on Duration
p*
Tangent Line at y* (estimated price)
y1
y2 y* y3
y4
Yield
Callable Bonds
Corporations, at times, will issue bonds that can be called – that is, the issuer can
obligate a bond investor to tender the bond back to the issuer at an agreed price. This, as you have
already seen, was what Disney proposed with its Sleeping Beauty bonds.
The prices and yields of callable bonds always assume that the bond will be called at the
agreed upon price at the earliest date that is allowed in the bond agreement. Because of this,
callable bonds are valued using a yield-to-call (YTC) methodology, which assumes the worst case.
Let’s say that we have a 5-year, 6.0% annual coupon bond with a call option in year 3 at
$900. If the bond’s associated interest rate is 5.50%, what should be its price?
𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑 =
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟗𝟗𝟗𝟗𝟗𝟗
+
+
𝟏𝟏
𝟐𝟐
(𝟏𝟏+. 𝟎𝟎𝟓𝟓𝟓𝟓)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)𝟑𝟑
In this case, the bond is not assumed to reach maturity in year 5. It will be called at the call
price (plus the coupon but this varies based on the bond’s agreement) at the end of year 3. So, the
final cash flow is the call price plus the coupon as of the end of year 3. Had the bond been allowed
to continue and mature at the end of year 5, its price should be:
𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟗𝟗𝟗𝟗𝟗𝟗
=
+
+
+
+
𝟏𝟏
𝟐𝟐
𝟑𝟑
𝟒𝟒
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)
(𝟏𝟏+. 𝟎𝟎𝟎𝟎𝟎𝟎)𝟓𝟓
So, in effect, the callability option of the bond drops its price by $16.5097, or 1.7474%.
This difference is the premium that is charged by bond investors that have to account for the
callability option, one which favors the issuer.
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The actual yield to call can be calculated if one has the bond price already in hand and the
projected stream of payments. Let’s assume that the bond shown above is selling for $933.5000.
What is its YTC?
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟗𝟗𝟗𝟗𝟗𝟗
𝒀𝒀𝒀𝒀𝒀𝒀 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓 =
+
+
𝟏𝟏
𝟐𝟐
(𝟏𝟏 + 𝑿𝑿)
(𝟏𝟏 + 𝑿𝑿)
(𝟏𝟏 + 𝑿𝑿)𝟑𝟑
Solving for X, we get that the YTC is 5.2923%. Now, let’s assume that only a portion of
the bond issued is called and a portion is not. Because no bond investor can be sure that their bond
will or will not be called, they will pay only $933.5000.
What happens if the bond investor that paid this price does not have the bond called and,
instead, receives payments over the five years. What is that investor’s YTM?
𝒀𝒀𝒀𝒀𝒀𝒀 = $𝟗𝟗𝟗𝟗𝟗𝟗. 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓 =
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟔𝟔𝟔𝟔
$𝟏𝟏, 𝟎𝟎𝟎𝟎𝟎𝟎
$𝟔𝟔𝟔𝟔
+
+
+
+
𝟏𝟏
𝟐𝟐
𝟑𝟑
𝟒𝟒
(𝟏𝟏 + 𝑿𝑿)
(𝟏𝟏 + 𝑿𝑿)
(𝟏𝟏 + 𝑿𝑿)
(𝟏𝟏 + 𝑿𝑿)
(𝟏𝟏 + 𝑿𝑿)𝟓𝟓
Solving for X, we find that the YTM for this investor is 7.6502%. So, the lucky investor
that does not have the bond called gets an extra 2.3579% in yield.
Corporate Bond Issuances
Corporate debt, in the form of bonds, usually have seniority when it comes to any potential
liquidations of the issuer in case of bankruptcy. While most investors will stick with “investment
grade” (that is, higher quality) bonds, not all bonds are investment grade and not all bonds that
started as investment grade keep that standing – these latter bonds are called “fallen angels”.
Debt seniority is broken out as:
• senior secured (first at liquidation and holding assets as collateral)
• senior unsecured (second to senior secured at liquidation but have no specific
collateral)
• senior subordinated (comes in after the secured and unsecured senior debt holders)
• subordinated (last in the queue)
Note that not all unsecured debt will have no claim on assets. Some bonds are issued
without explicit collateral but with the right to priority in the event of liquidation of the firm. The
bond markets as such are driven by these layers of credit quality, security, collateralization,
callability, and sinking fund requirements.
One of the ways to determine what are the probabilities of a bond of a given credit standing
either keeping that credit position, improving that position, or worsening that position is to look at
historical values for bond ratings over time using the S&P ratings scale.
The table below shows this:
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Rating
at Start
AAA
AA
A
BBB
Rating at End
AAA
AA
A
BBB
BB
B
C/D
Total
91%
1%
0%
0%
8%
91%
3%
0%
1%
7%
91%
6%
0%
1%
5%
88%
0%
0%
1%
5%
0%
0%
0%
1%
0%
0%
0%
0%
100%
100%
100%
100%
Note: within each category, S&P may add a + or -. So, for AA and for A, there are three
effective tiers: AA+, AA, and AA-, for example. BBB is lowest rating that is investment grade;
after that, there is a BB rating that is considered non-investment grade.
As we can see, the ratings don’t usually change for corporations. In effect, nine out of ten
times the rating will not change. When they change, though, they are usually for the worse.
For the purposes of most investment funds, though, the key number to keep in mind is the
BBB to BB change – 5% if one goes by the table above. Many funds are not allowed to invest in
non-investment grade securities. So, if any securities fall from grace, their fall is further
accelerated because investment funds must sell them to keep with the fund’s restrictions.
For 1Q20, as an example, we list the largest bond issuances in the U.S. in the table below.
We use 1Q20 as it was the first quarter before the pandemic hit the macroeconomy with full force):
Issuer
AT&T
Exxon/Mobil
HCA Healthcare
Verizon
Intel
AT&T
United Technologies
Gazprom
Energy Transfer
Exxon/Mobil
Nvidia
Amount (in
billions)
3.0
2.8
2.7
2.4
2.3
2.2
2.0
2.0
2.0
2.0
2.0
Coupon
(%)
4.0
4.3
3.5
3.6
4.8
2.9
2.2
3.3
5.0
3.5
3.5
Maturity
2049
2050
2030
2060
2050
Perpetual/Callable
2025
2030
2050
2030
2050
Credit Default
Spread (bps)
246.6
83.3
313.8
137.5
62.7
246.6
42.2
271.2
186.8
83.3
100.0
Worldwide, the total non-financial bond issuances were:
1Q19
1Q20
Source: Bloomberg
Issuances
(trillion $)
0.9
1.2
Average
Coupon (%)
2.77
2.59
Average
Maturity (yrs.)
5.7
4.8
The table above shows credit default spreads in basis points. These spreads are an indicator
of the credit risk associated with the bond – the higher the credit spread, the higher the risk.
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High in this example, I’m going to walk you through how I would calculate bond yields, bond prices and bond durations, I will start by looking first at a seminar and coupon paying bond. I will then go to the coupon paying bond. OK, so let’s get started and calculate a yield. For this example, we’re going to use a four percent coupon. OK, so the coupon basically it’s the annual amount of face value that gets paid us interest. It’s the face value of called par, and it’s almost always a thousand dollars so far, this course just assume it’s a thousand. Because this bond is paying semiannually, that is every six months, you have to divide the coupon by two. OK. So that’s what we’re going to do in order to calculate the payments that we’re going to receive now for this exercise. Let’s assume that we paid nine hundred dollars for this bond because it was a payment that we made as a cash outflow. So it’s minus nine hundred dollars. OK, you can see it right there. I’m going to market us right now that it’s an input. And then we know that we’re going to be receiving. Half of the coupon every six months and half is two percent times the bond par value. So let’s do that. So. The par value and we’re going to look at using the F key, we can copy it down and it won’t change times the coupon again, we’re going to lock that using the F or key. And then we’re going to divide that coupon by, OK, but we’re going to get 20 dollars. Interests, as it were, every six months, now we’re going to get that interest until year 10. That’s the interest bit of the bond. And maturity, we get the coupon payment that we’re due plus the thousand dollars back. OK, that’s Veloz that. Remember, we paid nine hundred to get a stream of payments and to get that thousand dollars and maturity. But we get a thousand twenty what is a yield, the yield is easily calculated by using the IRR function from Excel. So I’m going to put in all the values every single cash flow gets put in. OK, then I’m going to close the parenthesis. And because it is semiannual, remember, I have to multiply the IRR result by two. OK, so I can make it. And that is my calculated yield on this four percent coupon bond. OK, so why is the yield higher than the coupon? The two are unrelated. The yield is what the the market will pay for this kind of bond at this point in time. The reason why this is because the bond is paying four percent. Which is below the market rate, so you have to make up that by giving the investor an additional gain. That is why the investor paid nine hundred dollars for something that is, on its face, worth a thousand. So you’re going to get returns added together to get your total yield. What is going to be the coupon payments that you get over time? And one is going to be the appreciation of the bond from nine hundred dollars that you pay for it today to the thousand dollars that you get in 10 years. OK, so that is the thing. Normally what happens is a company, when they issue a bond, they issue the coupon equal to the market rate of return. So almost always, but not always. Almost always, the the price that is paid for the bond is a thousand dollars. Why? Because the coupon is equal to the market return. There is no capital gain. There is no appreciation. It’s all the interest you’re being paid is the interest. So that’s where it’s at. Now, we done the deal and we know that for this case, it’s five point three zero one three. We’ll go to four significant digits. Now, let’s take a look at a bond that pays a six percent coupon when the market says that it should be paying only five. OK, so this bond will be selling at a premium. It will sell above a thousand. So what are the payments, the payments, again, are the bond per volume, OK? Times the coupon that’s divided by two because again, we are Minyanville. OK, so you get thirty dollars every six months. OK, so now we cuppy that down. Kaposi shift down arrow. We can’t and now we go to year 10 and we take ad. The. Thousand dollar face value right here.
Now, what is the price, the price is going to be the net present value. Of those bombs. Payment. OK, so. We’re going to take the market return, which is what it is, it’s a reference, it’s our discount rate, and because the payments are semiannual, we have to make the market return semiannual. The market return that’s shown here, it’s annual, it’s five percent per year, so what I do is I think the five percent in the budget by. Now on semi-annual. And then what I do is I. Highlight all the cash flows from the bond payments, close out the differences, and that’s the price of the bond, OK, that’s where it’s at. So I said to you that you would be paying more for the bond than a thousand dollars because they’re paying you six percent in a market that says, no, no, no, you shouldn’t be paying six, you should be paying five, OK? And because you’re paying more than what the market wants, you as a buyer have to pay a premium for it. OK, you’re getting a return that is not due to you, so you pay extra for it. OK, and that’s how we calculate the price again. For calculating prices on semi-annual coupons, we use the MPV formula and we divide the market rate by two. OK, not to calculate yields. What we do is we use the IRR function. OK, showing all the cash flows, make sure that your initial purchase price is negative because you paid for it, cash on the door and if it’s imminent, you have to multiply the IRR by two. OK, so keep that in mind.
We now go to the annual coupon pain bonds, and these are a bit easier to do because they’re annual. So for us, it’s easier to just visualize, not let’s just say that the purchase price is still the same. So we’ll make it nine hundred dollars out the door, plus minus nine hundred. I’m going to make it red again so we know that it’s a mental input. OK, and then what I’m going to do is I’m going to calculate the yield, but the yield is going to be again. And so what’s going to be the coupon payment? The coupon payment is going to be the par value again a thousand times the annual coupon. OK, because it’s and I don’t have to divide it by two. OK, I’m good now. I just leave it as it is. It’s just the face value of a thousand times to four percent. It’s an animal bond once every 12 months. Forty dollars, not just a 20 year bond. But what I’m going to do is I’m just going to copy this down, OK, and it’s going to be paying 40 dollars per year and then at maturity, it’s going to pay me. My thousand dollars back. OK, I can put it in a thousand, but I’d rather just put it in the palm of Will, OK? Now we go and we calculate the. Calculated based on these cash flows, again, we use a higher function of these other values for Halit, all the values, and then I don’t have to multiply by two a.m. So I’m good on these filter and hence the IRR, Samuel, and that’s my yield. OK, so for this one, I get that my yield is four point seven eight eight one.
Right now, why is this a better if this yield worse than this yield, which is basically well, for one thing, it’s. You’re not getting payments six months earlier here, you’re getting payments over 10 years, you’re getting payments over 20 years. There’s there’s a steep drop off in value to these payments are made after year 10 and that’s where you’re going to notice.
OK, now that we’ve completed our yield calculation for the coupon, that’s a price calculation for the coupon payment. Now, again, what we’re going to do here is we’re going to use the NPV function to calculate price. So what’s going to be the payments? The payments are going to be, again, the bond par value, OK, times the annual coupon. OK, OK, because if the coupon is six percent, it’s going to pay six percent of a thousand sixty dollars every year. Now we’re going to bring this this down.
Ten year bond, and then we’re going to add the thousand dollar coupon. OK.
Back in here and away we go, so we get 60 dollars every 12 months, and then at maturity 20 years, we get 60 dollar coupon plus a thousand dollar face value. What’s the price? Well, it’s going to be MPB function. OK, we’re going to use the market returns, which is five percent. That’s the discount rate. OK, we don’t have to turn it into semiformal because this is an animal bond. So I don’t divide by two. I just leave it at it and then I highlight all the cash flows. OK, plus parentheses. And that’s a price. OK, so what does this tell me tells me that the market says I should be getting paid five percent instead of getting paid six on a 20 year bond, which means I’m getting paid extra means I have to pay extra for it because I’m getting a benefit that it’s not due to me. Not the market says I should be getting paid less. So in order to buy the bond, I have to pay one thousand one hundred twenty four dollars and sixty two cents.
Every year I’ll get sixty dollars and then in 20 years I get thousands. So again, to recap, with an online coupon paying bond, I use the TV formula fame, but I do not divide by two because it’s a.. So my discount rate is going to be the market return. OK, and then the arguments are going to be all the casualties, which are from times. The payments made from year one through 20. When I look at calculating yield, I use the IRR function in here, the argument is I have to put in the negative cash flow as a purchase price. I paid for the bond and then all the coupon payments that I receive and the final phase of out of the bond paid at maturity. So I don’t have to divide multiply by two because it’s already in. So I’m good.
In this example, I’m going to show you how to calculate not just a price, but also to duration, and then we’ll see what the role of duration is and bond trading. So for this example, I’m going to use that we have a five percent coupon and expect that market return of six percent. So the bond is paying less than the market. So it should be selling at a discount Parvati sealife. So what we’re going to do is we’re going to first calculate what the coupon payments are. Then we’ll calculate the price, then we’ll cut duration. Then we’ll see what duration means in the grand scheme of things. So the coupon is going to be the par value. Times the coupon rate. OK, so the coupon rate is going to be five percent and I like that because a semiannual we have to go to buy to remember Sam. I am. If I buy. And we got twenty five dollars, so we know we’re going to be receiving twenty five dollars every six months, so we strike that up to 10 years and then in the final year, we had a thousand dollar face value from PA. And so now we’re going to price it. OK, we have our stream of payments. They’re all here.
So how much are we going to pay for it? We’re going to pay for it, the MPB. OK, we’re going to use the market yield. Divided by two, because the semi-annual we have to turn it into a semiannual rate for this example, and then we’re going to put in all the cash flows in the MPB and we’re going to get that. We’re going to pay nine hundred and twenty five dollars and sixty one cents. OK, that’s that’s where we are.
OK, so that’s the price, so let’s calculate now the duration, the duration from Hezbollah’s. I’m going to take.
The payment times the coupon and I put it in the numerator. OK. I’m not going to use the 14th. Everyone is different and I’m going to present value using one plus. Expect expected market yield. OK, I’m going to lock that, I copy it down to one number. OK. Close parentheses 56, which is a correct that’s exponent shaded to the time of payment. OK, so what we’re doing is we’re really pressin valuing.
That amount of money that we’re going to get at just for the time of payment.
There would well do it right, so I got this formula. I mean, just going to copy it down, OK? And I get.
All these values. OK, so you can see that most of the value is going to be in the year 10, so migration is going to be the sum of all of these numbers. OK.
Close parentheses divided by the price.
And that’s my direction. So what does it tell me that this 10 year bond has preloaded to the back end so it has a long duration? And then what happens if interest rates rise from ten point zero zero to ten point zero five? That’s an increase of five basis points or five bips, as they would say in the industry. So it’s basically a basis point is one one hundredth of a percent. OK, so from ten point zero zero to ten point zero five, the change is. Five basis points. OK, so the formula for the change in the bond price is the duration times, the change in interest rates times two divided by one plus the semi-annual coupon is equal to the change in bond price or by the original bond price. Now, I know this part. I know the right side of the formula. I also know the original bond price because it’s nine. Twenty five. Sixty one. So what I do is I solve the equation for you so that the change in bond price divided by the original bond price, which is zero zero seven eight, there is a formula right there.
Then with point zero zero seven eight, I multiply it by the original bond price to get the change in bond price and the original bond price being nine. Twenty five. Sixty one. So you multiply the two and I get that the change in price because of a five basis point increase in rates is that the bond one from nine twenty five sixty one point I’m twenty five. Sixty one. Yeah. It dropped by seven dollars and 19 cents. OK, the bond value will reduce by seven dollars and 19 cents, so a little bit less than one percent. But here’s the issue. If you had a one hundred billion dollar bond portfolio with this duration of seven point nine six, the five basis point rise would reduce your portfolio’s value by about seven hundred seventy six thousand dollars. OK, so that’s a big hit you’re going to take. So in bond pricing, duration matters. It’s key because it tells you both you are to interest rate changes. If interest rates rise, your bond portfolio falls. If interest rates fall, bond portfolio rises. So most bond fund managers where they’re looking for is estimating what they think is going to happen to interest rates. If they think that if they think that rates will rise, which means that that bond portfolio will fall, they want to shorten the duration so that they don’t take as big a hit as they otherwise would. OK, duration can magnify losses or magnify gains. Now, if you think that interest rates are going to fall, which means that your portfolio value is going to rise, you want to go to longer duration so that you can magnify the gains. OK, so bottom line, duration is a metric that bond fund managers use to play the interest rate game of trying to maximize value and minimize losses.