Bonds (Finance)

Motivation. When a company needs money to invest in a new project, Loans are the simplest way. But banks are wary because once a loan is made they have to keep it to maturity and can’t offload it. We want a loan that is tradable. def. Bonds are loans that are tradable, often of low principle amount, that pays coupon payments (instead of interest, like in loans.) thr. Buying a bond is a way of betting on the current/future interest rates, because $\text{Price}\propto 1/\text{Yield}$, and price of a bond perfectly matches the market yield (linked to federal Fed Funds Rate).

• Tradable Inflation-Protected Securities (TIPS) are inflation protected bonds.

Cash Flow Structure of a Bond §

• Lender lends money to the government (usually US Treasury)
• Coupon payments pay regularly:
  • Semi-annually for US Treasury
  • Annually for other countries
• Principal (=par, base) amount is paid at the maturity;
• Time To Maturity (TTM) is time from now to when the principal pays
• Yield
  • Current Yield
  • Yield to Maturity: Expectation Hypothesis defines yield to maturity as the expected rate of return if held til maturity.

Governments are the borrowers of money in this transaction.

• Government “issues” bonds regularly, which is
  • Primary Market: Treasury sells bonds (millions of $ worth) mostly to private Investment Banks
  • Secondary Market: Investment banks sell smaller chunks to private investors / bond holders sell to each other
• Governments pay coupon and principal through
  1. Tax revenue
  2. Issuing more bonds

Bonds are an important part of a portfolio because…

• Are near-riskless investments as they generate stable cash flows. The only risks are: inflation risk—when the government doesn’t have the money, they’ll just print more money. This causes inflation which devalues the bonds.
• Are very liquid, since many people are willing to buy bonds in most scenarios, even in recessions.

Bond Pricing §

thm. The price of a bond is the following:

$$B(n)=\frac{1-{\left(1+r_y/2\right)}^{-n}}{r_y/2}C+\frac{M}{(1+\frac{r_y}{2}{)}^n}$$

where…

• $B(n)$ is the current market price of the bond
• $n=2\tau$ is the periods left to maturity (where $\tau$ is the years left to maturity)
  • not periods passed! periods left
• $M$ is the Principal amount
  • Normally use $100or$1000
• $C$ is the per-period (=semi-annual) coupon payment
  • normally, coupon is quoted at annual coupon rate $r_C$ (in percent) so $C=\frac{r_C}{2}M$
• $r_y$ (or $y$) is the yield to maturity ($\approx$discount rate)
  • $r$ is the current yield. $r=\frac{2C}{B(n)}$
  • $n$ is the number of coupon payments (semi-annual for Treasury bonds)

The Three Rates of Bonds

1. $r_y$: Yield to Maturity—“What’s the return rate for the coupons if I hold until maturity?”
2. $r$: Current Yield—“Coupon rate, but at the current bond price”
3. $r_C$: Coupon Rate—“Coupon rate (at principal price).”

• Price and rates:
  • Trades at premium $B(n)>M$ ⇔ $r_y<r<r_C$
  • Trades at discount $B(n)<M$ ⇔ $r_y>r>r_C$

Observe…

• $B(n)\propto \frac{1}{\text{yield}}$.

• ${\lim }_{n\to 0}B(n)=M$

• Causality: Exogenous factors → $\Delta$yield → $\Delta$price.

• In a portfolio of bonds…

$$\text{Market Value}=\frac{\text{Price of par}}{100}\times \text{\# of pars}$$

The causality of the variables of a bond. The market’s expected rate of return of a bond is the yield. With this yield we can calculate its price.

Price—Yield Curve (mathematical) §

Price and Yield are inversly correlated → Price-yield curve looks like this:

Duration is the gradient of this price-yield function:

$$\text{Duration}:=\frac{dP}{dy}$$

Dollar Duration (DV01) is the change in price due to 1% change in yield [=$1 change in yield per par]

$$\text{DV01}:=\frac{\Delta P}{\pm 1\%\cdot y}$$

• Dollar Duration can be only used to approximate price changes for small changes in yield (~50bp)
• Duration is used to estimate risk (Price sensitivity against yield = DV01 risk
• Dollar Duration is quoted as an absolute value. (We all know that it’s mathematically negative)
• DV01 Risk:= $\frac{\Delta \text{Market Value}}{\Delta 1\%\text{yield}}$ = “$\Delta$ market value for 1% yield change” = $\frac{\#\text{par}}{100}\cdot \text{DV01}$
• DV01 has no relationship with volatility

Example calculation

Time to Maturity	DV01	Yield Vol.	e.g. change in yield	change in price
2 year	$1.8	10%p	20bp	$0.36
10 year	$7.2	5%p	10bp	$0.72

Yield(–Maturity) Curve (empirical) §

Determinors of the shape of the yield curve:

1. Yield $\propto$ Maturity. The farther away the cash flows are, the more risky it is.
2. Interest Rate changes. Yield $\propto$ 1/Price, and yield is linked to Fed Funds Rate. Thus buy at high yield, sell at low yield. Consider
   • Bond investors think a recession is looming in 1yr → Treasury will increase rates in 1yr → sell 1yr mat bonds
   • Left intercept is the fed rates & short-mat bonds → high movement right side is long-mat bonds → low movemen
3. Liquidity Preference Theory. Shorter maturity bonds are more liquid simply because more of them exist in the world. Higher liquidity means less risk.

Risk & Volatility §

def. Price[rate of return] volatility = $\mathbb{SD}[\text{Price}]$ (in $)

def. Yield volatility = $\mathbb{SD}[\text{Yield}]$ (in %p)

def. DV01 Risk = Change in MV per change in 1% yield

• Yield Vol > RoR Vol ← mathematical relationship (price formula)
• Maturity $\propto$ Price Vol.