Bonds

BUSI 721: Data-Driven Finance I

Kerry Back, Rice University

Coupons and Face Value

• Pay a specified coupon at regular intervals (usually semi-annually).
• And pay face value (= par value) at maturity. Last payment is coupon plus face.
• Usually in $1,000 denominations.
• Example: a 6% bond with $1,000 face value pays 3% x 1,000 = 30 every six months.

Coupon rates

• The coupon on a bond is usually set so that it can be issued at or near face value.
• This requires setting the coupon at the market interest (for a bond of its maturity and credit quality).
• Investment banks assist companies and municipalities in setting coupons and issuing bonds.
• The U.S. Treasury runs auctions - buyers bid in rates and low bidders win. The coupon is set at the marginal rate.
• Upcoming Auctions

The bond market

• Many, many different bonds outstanding. Most do not trade in any given period.
• Trade via dealers - contact a dealer to get a quote - rather than on exchanges.
• Mostly an institutional market.
• Better to buy bonds through ETFs than buy them directly, except maybe Treasury bonds through Treasury Direct.

Chevron's debt

Coupons vs Yields

• The coupon rate of a bond is set at the time of its issue.
• However, what one anticipates earning on a bond varies with the market price.
  • Price < par $\Rightarrow$ coupon + capital gain
  • Price > par $\Rightarrow$ coupon - capital loss
• What one would earn per year on a bond if held to maturity (assuming no default) is called the bond yield.

Yield calculation example

• Bond trading at 90% of par
• Paying 5% coupon
• Next coupon in six months, matures in 2 years
• Do semi-annual discounting at the annual rate / 2
• Yield is $y=2r$ where $$ 0 = - 90 + \frac{2.50}{1+r} + \frac{2.50}{(1+r)^2} + \frac{2.50}{(1+r)^3} + \frac{102.50}{(1+r)^4}$$
• In other words, $r$ is the IRR of the cash flows from buying the bond at 90 and holding until maturity.

import numpy_financial as npf 

cash_flows = [-90, 2.5, 2.5, 2.5, 102.5]
r = npf.irr(cash_flows)
y = 2*r
print(f"The bond yield is {y:.2%}")

The bond yield is 10.69%

In this example, you are getting, roughly,

• 5% per year from the coupons
• a 10% capital gain in 2 years $\sim$ 5% per year
• so approximately 10% per year

Bond price is the PV of the cash flows

• A bond price is the PV of its cash flows when discounted at the yield.

$$\text{Price} = \frac{\text{coup}}{1+y/2} + \frac{\text{coup}}{(1+y/2)^2} + \cdots + \frac{\text{coup}+\text{face}}{(1+y/2)^{2n}}$$

where $y=$ yield and $n=$ number of years to maturity.

Example

• 5-year bond with 6% coupon rate and 8% yield
• $1,000 face value
• calculate price

years = 5
coupon = 1000 * 0.06 / 2
yld = 0.08

pv_factors = (1+yld/2)**np.arange(-1, -2*years-1, -1)
cash_flows = (coupon) * np.ones(2*years)
cash_flows[-1] += 1000
price = np.sum(PV_factors * cash_flows)

print(f"price is ${price:.2f}")

price is $918.89

# check yield

cash_flows = np.concatenate(([-price], cash_flows))
r = npf.irr(cash_flows)
print(f"yield is {2*r:.2%}")

yield is 8.00%

Long-term bonds are riskier than short-term bonds

• Let y = bond yield.
• Consider a cash flow C that is n years away. Its PV is $$\text{PV} = \frac{C}{(1+y/2)^{2n}} = C(1+y/2)^{-2n}$$
• How does this change when the yield changes? $$\frac{d}{dy} C(1+y/2)^{-2n} = - nC(1+y/2)^{-2n-1} = - n \times \frac{PV}{(1+y/2)}$$
• So the percent change in the value is $$-n(1+y/2)$$

Term structure of interest rates

• Term structure = how Treasury yields depend on maturity of bond
• Usually longer-term yields are higher
• But it varies a lot over time
• Learn Investments

Fed funds rate

• The Federal funds rate is an overnight rate that is targeted by the Federal Reserve
• The Fed borrows or lends in the market to push the equilibrium rate to the rate they want
• Long-term rates tend to move up and down with the Fed funds rate

from pandas_datareader import DataReader as pdr
rates = pdr(["FEDFUNDS", "DGS10"], "fred", start=1900).dropna()
rates.columns = ["fedfunds", "10yr"]
sns.regplot(x="fedfunds", y="10yr", data=rates, ci=None, scatter_kws={"alpha": 0.5})

<AxesSubplot: xlabel='fedfunds', ylabel='10yr'>

TIPS (Treasury Inflation Protected Securities)

• The Treasury issues bonds with payments indexed to inflation.
• 4% inflation $\Rightarrow$ all future coupons and the face value go up by 4%.
• This is cumulative. So each coupon and the face value are adjusted for all past inflation.
• Example: a $1,000 denomnation 2% TIPS issued today will pay 10 in today's dollars each 6 months and pay 1,000 in today's dollars at maturity.

Treasury yields and TIPS yields

• Get 10 year Treasury yields and TIPS yields from FRED (Federal Reserve Economic Data)
• Calculate the difference in yields
• Difference depends on inflation expectations

yields = pdr(["DGS10", "DFII10"], "fred", start=1900).dropna()
yields.columns = ["Treasuries", "TIPS"]
yields.plot()
plt.ylabel("Yield in %")
plt.show()

(yields.Treasuries - yields.TIPS).plot()
plt.ylabel("Difference in yields in %")
plt.show()

Fixed income universe

• Treasuries
• corporates
• municipals
• asset backed securities
  • mortgage backed securities
  • credit-card receivables, other receivables
  • collateralized debt obligations
• Asset backed securities enable the spreading of risks among more investors. For example, pension funds hold mortgages. Also instrumental in financial crisis.

Municipal bonds

• Municipal bonds in the U.S. are exempt from federal income tax.
• Municipal bonds are also exempt from state income taxes in the state of issue.
• So, NY investors want to hold NY municipals, California investors want to hold California municipals.
• Municipals are issued by states, cities, counties, school boards, fire districts, ...
• Tax increment financing allows limited use of municipal bonds to back private investments: sports stadiums, etc.