Interest Rates and Fixed Income Instruments

In this lesson, we looked at fixed income instruments, such as annuities that pay out a certain amount every period, based on a return on investment of r (as well as a borrowing rate of r).

Definitions:

Simple interest: amount A invested for n periods with return r will return A(1+n*r) at maturity.
Compound interest: amount A invested for n periods with return r will return A(1+r)^n at maturity.
Continuously compounding interest: amount A invested for n periods with return r will return A*e^(rn) at maturity.

We then applied no-arbitrage from the previous lesson to find a price for a portfolio of continuing cash flows:

Buy a portfolio of price p with return c = (c0,c1,c2,...cT) where the c's are returns for each period.

This gives us a lower bound of:

Sell a portfolio of price p with return c = (c0,c1,c2,...cT) where the c's are returns for each period.

This gives us an upper bound of:





The final price was actually both of these prices, since the upper bound and the lower bound are the same.



Then we looked at if there was a different borrowing/lending rate than the return on investment rate.  This is typically true, as the return on investment is often lower than the rate for borrowing to invest.

Say we borrow at rate rb to buy the same contract.  By using the same formulae from up above, we get an lower bound for the price:

Then say we sell the same contract and accept a rate rl for lending out the money.  By using the same formulae, we get an upper bound for the price:

Putting these both together we find the final bound for the price:

The actual price is set by supply and demand.  If there are more suppliers of contracts, the price will shift more to the lending rate, and if there are more borrowers, the price will shift more to the borrower's rate.



We then took a look at fixed income securities and their risks: fixed income instruments are securities that guarantee a fixed cash flow.  Some risks are:

• Default risk - the entity that guarantees the fixed cash flow goes bankrupt 
• Inflation risk - less power in the fixed income because there inflation brings down the value of a money (TIPS hedge against this)
• Market risk - securities might become less and more valuable over time and if you wanted to sell these securities in the market, you would be open to these price fluctuations 

Let's try to price certain fixed income securities:

Perpetuity - fixed income instrument that pays amount A for all times in the future, and assuming a borrow or lend rate, the price will be:





Annuity - fixed income instrument that pays amount A for periods 1 to n (can be thought of as two perpetuities: one perpetuity that pays A  and another perpetuity that pays -A starting in period (n+1))
The price for an annuity is:




where (A/r) is the price of the first perpetuity and (A/r)(1/(1+r)^n) is the price of the second perpetuity discounted back n periods, where (1/(1+r)^n) is the discount factor.

Bond - characterized by 5 parameters
Face value - F - amount paid to holder at maturity
Coupon rate - alpha - pays (alpha)(F)/2 every 6 months
Maturity - T - date of payment of both the last coupon and the face value
Price - P - cost to buy the bond
Quality rating - rating on likelihood to default by different rating agencies

Yield to maturity - lambda - annual interest rate at which the price P is equal to the current value of the coupon payments plus the face value - this is a way to compare bonds:

Sum all the coupons (c), discount them at the rate (lambda/2) at time k, which is when each coupon is paid + the final face value discounted at the rate (lambda/2) at time T.

This equation gives a single number to start comparing bonds by summing the face value, coupon, maturity, and quality.  If a bond is lower quality, we expect a lower price, and higher yield since there is more of a likelihood of default.