Fixed Income terminology Present Value/Future Value Principal and Interest Periodic cash-flows (usually semi-annual) Compounding rate (usually semi-annual) We care about: –Contracted return or the bond “Coupon” –“prevailing rate” / “market” / the interest rate environment –The length of the investment : “maturity” / “term” –What would it be worth if rates change? –It’s sensitivities –Default probability and other “credit” risk measures On to our pricing formula...

Yield To Maturity Formula

Yield to Maturity Equation - Closed Form (started with a geometric series…) Calculate price of bond with par value of $1,000 to be paid in 10 years, a coupon of 10% and YTM of 12%. Assume coupons are paid semi-annually to bond holders: 1)Determine number of coupon payments (2 per year for 10 years = 20) 2)Determine value of each coupon payment (divide coupon in half since semi- annual). Each payment will be $50 ($1000 * 0.05). 3)Determine the semi-annual yield: Like the coupon rate, the YTM of 12% must be divided by 2.

YTM - Frequency Parameterized Accounting for different payment frequencies: Most bonds pay semi-annually but to make our formula more general we extend formula with parameter “F” below. if a bond was paying annual coupons F = 1, quarterly = 4

While more programmer friendly, what do we assume away with this formula? Assumes all intermediate flows are discounted at same rate (the YTM) A more general implementation is the “cash-flow series” A collection of individual cash-flows which can: –Support assigning different interest rates to individual cash-flows –Represent changing principal amounts (e.g., for amortizing instruments) –Support independent processing of individual cash-flows (e.g., date adjustments)

Break then we’ll examine our spreadsheet living spec... Yield Goes up/price goes down Price goes up/yield goes down Standard sensitivity calculation

A Fancy Bond Prospectus

Debt Issuance/Secondary Market Class Example #1 $10 loaned amount - called the “Face” 2 year term – return money at “maturity” date Interest to be paid annually 10% interest rate – called the “coupon” After 1 year the loaner of $10 receives interest of $1 After 2 years receives $11 which is the second coupon + the loaned amount. Total received after 2 years is $12

Debt Issuance/Secondary Market Class Example #2 $10 “Face” 2 year term Interest paid semi-annually 20% coupon After 6 months loaner receives interest of $1 After another 6 months receives second coupon of $1, etc. Total received after 2 years is $14 What if after year 1 the loaner wants money back? Could terminate the deal OR sell the bond on the secondary market. But at what price? The “model” price is different than the actual price which is affected by supply/demand, perception of credit quality, etc.

Deliverables for next week: -Download the “living spec” prototype spreadsheet. -Build calculator class around PV function -Implement the closed form equation -Calculate price from yield and price sensitivity to yield change (first partial derivative) : “DV01” (dollar value of 1/100 of a percent) -Download sample code to load bond data file -File access utility class in “SBB_util.*” -Run your main() loading data.txt and call your executable from run.sh -You will need to calculate number of periods… (download the SBB_date.* code) Output 3 rows of results 2 values per row to stdout: –“Price” “dv01” in each row separated by a space –Example: (in each row)