Glossary

Bond principal - Lump sum at the end (Other name: Pay value)

Bond coupon - Payment to holder periodically

Bond

Interest Rates

Divy the annual compounding by the number of periods, and then compound repeatedly. (10% semianually means 1.05 * 1.05 = 1.1025)

and if

Then you'll get the

Continuous compounding case

Treasury Rates

Treasury rates are the rates an investor earns on Treasury bills and Treasury bonds.

These are the instruments used by a government to borrow in its own currency. Treasury rates are therefore totally risk-free rates in the sense that an investor who buys a Treasury bill or Treasury bond is certain that interest and principal payments will be made as promised.

Imagine Bank A has spare cash for the next 3 months, while Bank B needs cash for the next 3 months.

LIBOR/LIBID - London Interbank Offered Rate

Bank A can effectively say:

“Give me your money for 3 months and I’ll pay you interest.”

Economically, that is just Bank A borrowing money from Bank B.

Historically, 3-month LIBOR was roughly:

the interest rate a reasonably creditworthy bank would have to pay to borrow money from another bank for 3 months.

Repo Rates

Discrepancy between price you sell something to someone, and price you buy it back at.

Risk Free Rate/Overnight Index Swapped

(Used to use LIBOR but that grew out of style after 2007)

Coupons

Now discount each one back to today Because the rates use continuous compounding, the present-value formula is

The negative exponent is just the reverse of the compounding formula you learned:

so:

Bond Yields

A bond's yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price.

i.e. its the y where the following equation is true

(that y is the bond yield, and it turns out its value is 6.76%)

Par Yield

If you force the market value to the principal value, and make the coupon the unknown, that value of the coupon which would make market value = principal value is actually the par yield.

Determining Treasury Zero Rate

Boostrap method

Reverse engineer the zero rate from data and assume continuous compounidng effect. Use

Sub in for A = onberved value, n = timespan (as a fraction of a year), F = 100. Then reverse engineer R out.

I think it's called bootstrap, because you use the simple cases of zero-coupon instances to decouple away the coupon parts of the Bond's market value. R acts only upon the principal value.

Forward Rates

Calculated from a determined Treasury Zero rate.

You can get the forward rate of the second year by dividig the the forward rate of 2 years by the forward rate of 1 year.

Proof:

Then if you make T2 come very close to T1, you can derive the

Instantaneous forward rate

Forward Rate Agreeement

Here's the intuition I think you're reaching for.
Imagine the difference between your fixed rate and the market rate means you deserve:

\$100 \quad \text{at } T_2

But the FRA says, “We're going to settle with you now, at T_1.”
You shouldn't receive the full $100 now, because if I gave you $100 now, you could invest it at the current market rate R_M, and by T_2 you'd have more than $100.
So I instead ask:
“How much can I give you now such that, if you invest it at R_M, you end up with exactly $100 at T_2?”

Suppose the market growth factor is:

1+R_M(T_2-T_1)=1.05

Then I give you:

\frac{100}{1.05}=95.24

because:

95.24(1.05)=100

And that is exactly why the denominator exists.
So your intuition can be rewritten as:
The denominator tells me how much today's FRA payment needs to be reduced, given that I could invest that payment at the market rate R_M until T_2.

Duration

(y = bond yield rate)

A bond with cash flows c_i paid at contractual times t_i has price:

B=\sum_{i=1}^{n} c_i e^{-yt_i}

This decomposes into:

B=c_1e^{-yt_1}+c_2e^{-yt_2}+\cdots+c_ne^{-yt_n}

where y is the continuously compounded bond yield. Each term is the present value of one future cash flow, so the bond price is simply the sum of the present values of all future cash flows. The bond's duration is:

D=\frac{\sum_{i=1}^{n} t_i c_i e^{-yt_i}}{B}

This decomposes into:

D= \frac{ t_1c_1e^{-yt_1} +t_2c_2e^{-yt_2} +\cdots +t_nc_ne^{-yt_n} }{B}

Equivalently:

D=\sum_{i=1}^{n} t_i \left[ \frac{c_i e^{-yt_i}}{B} \right]

Duration is therefore the present-value-weighted average time at which the bond's cash flows are received. The weight assigned to cash flow i is:

\frac{c_i e^{-yt_i}}{B}

For a zero-coupon bond, there is only one cash flow and it occurs at maturity. Therefore t_1 = n and:

B=c_1e^{-yn}

Hence:

D= \frac{nc_1e^{-yn}}{c_1e^{-yn}} =n

So the duration of a zero-coupon bond is equal to its maturity.

Also, if you differentiate B by y, you will get an equation describing change in bond. Theres a negative relationship between B and y.

(note theres a whole proof, but my intuition tells me that the only reason Duration reappears in this equation is because its defined in a convenient enough way to appear derivable in the equation)

Then if we switch from the continuous compounding case to an n-nual compounding case, we get...

and the D/denominator is known as the modified duration.

Convexity

For a bond,

where B is the bond price, D is duration, C is convexity, and \Delta y is the change in yield.

The important intuition is:

• Duration is a straight-line approximation. It says, roughly, "if yields move by this much, bond price moves by this much in the opposite direction."
• This works well for small yield changes.
• But the true bond price/yield relationship is curved, not straight.
• Convexity measures that curvature, so it becomes important for larger changes in yield.

That is what the graph is showing. Portfolios X and Y have the same duration, so at the current yield they have the same slope:

For a tiny \Delta y, they therefore behave almost identically. But X has more convexity, so as you move farther away from today's yield, the two portfolios increasingly behave differently.

(For why, see liquidity preference theory)

Banks earn money from the difference between the interest received on loans and the interest paid on deposits.

Deposits are liabilities because the bank owes depositors money. Mortgages are assets because borrowers owe the bank money.

Customers generally prefer short-term deposits because they provide flexibility, but long-term mortgages because they lock in borrowing costs. This creates a maturity mismatch: the bank may fund a 5-year fixed-rate mortgage using deposits that must be renewed every year.

If interest rates rise, the bank must pay higher rates on new deposits while still receiving the old fixed mortgage rate, reducing its net interest income.

Banks therefore try to match the maturities of their assets and liabilities. They can encourage customers to choose longer-term deposits and shorter-term mortgages by offering higher long-term rates.

Because investors usually need extra compensation for locking up money for longer periods, long-term interest rates tend to be higher than the average expected future short-term rates. This is the liquidity preference theory.