This document provides an overview of key concepts related to fixed income bonds, including: - The time value of money concept of present and future value using discount factors. - Details of a sample US Treasury bond including its coupon, maturity date, cash flows. - How accrued interest is calculated when a bond is purchased between coupon dates. - How to calculate yield, forward rates, duration, convexity and other risk measures from bond prices and cash flows.

1. Fixed Income Bonds
Sapna Desai

2. Contents
• Time value of money : Present Value, Future
Value and discount factors
• Bond Basics
• Accrued Interest
• Bond Pricing
• Spot, Forward, YTM
• Risk Measures: Duration, DV01, Convexity

3. Time value of money
• What would you rather have? A dollar now or
a dollar in the future ?
• Present value = Discount factor * Future Value
• If d(.5) =.99; present value of one dollar to be
received in 6 months is 99 cents.

4. Government Bond
• Example: May 2010, the US treasury sold a
bond with coupon of 2.125%, maturity date
May 31, 2015.
• Coupon = 1.125
• Maturity Date = 5/31/2015
• Notional = 1mm.
• Purchasing 1mm face amount of these entitles
the buyer to the schedule of payments as in
table on next slide

5. Cash flows
Date Coupon Payment Principal Payment
11/30/10 $10,625
5/31/11 $10,625
11/30/11 $10,625
5/31/12 $10,625
11/30/12 $10,625
5/31/13 $10,625
11/30/13 $10,625
5/31/14 $10,625
11/30/14 $10,625
5/31/15 $10,625 $1,000,000

6. • How did we get $10,625?
Treasury promises to pay a coupon every six
months equal to half of the annual coupon
rate ie 2.125*1/2*1000000= $10,625

7. • The data on US treasury along with their coupon,
maturity and market prices is readily available.
• Almost all US treasuries settle T+1
– ie exchange of bond for cash happens one day after
the trade is done
• Price on treasury bond – mid market price
– Ie average of lower bid price where a trader can buy
and higher ask price where a trader can sell.
• Full price = flat / quoted / clean price + Accrued
Interest

8. Accrued Interest
• Say customer bought 10,000 face of UST 3.625 of
August 15, 2019 for settlement on June 1, 2010.
• Coupon will be 3.625/2*10000=181.25 on
February 15, 2010 and then August 15, 2010
• If the purchaser holds the bond to next coupon
date which is Aug 15, 2010, he is still not eligible
to collect the complete coupon from Feb to
August. He has held the bond only from June 1,
2010.

9. • Buyer should get 71/181 days of interest =
71/181*181.25=75.10
• Seller should get the rest 106/181 days of
interest= 106/181*181.25=106.15
• How do we divide the coupon?
• Market convention dictates that buyer keeps
the entire coupon but pays the seller 106.15
at the time of settlement.

10. • If the flat price of bond was 102-26 or
102+26/32; the full or invoice price would be
102.8125+1.0615=103.8740. For $10,000 face,
invoice price will be $10,387.40
Feb 15th 2010
Previous Coupon date
June 1st 2010
Settlement date
August 15th 2010
Next coupon date
106 Days 75 Days

11. US Treasury Prices as of May28,2010
Coupon Maturity Price
1.25% 11/30/2010 100.550
4.875% 5/31/2011 104.513
4.5% 11/30/2011 105.856
4.75% 5/31/2012 107.966
3.375% 11/30/2012 105.869
3.5% 5/31/2013 106.76
2% 11/30/2013 101.552
2.25% 5/31/2014 101.936
2.125% 11/30/2014 100.834

12. DFs from Bond Prices
• For the first bond in the table, we can write,
• 100.550 = (100+1.125/2)*Df(.5)
• From this we can solve for the first DF(.5)
• For the second bond, we can write
• 104.513= 4.875/2*DF(.5)+(100+4.875/2)*DF(1)
• Thus we can solve for DF(1).
• Likewise we can extract all Dfs from these bond
prices because by definition, bond price is PV of
its future cashflows and PV= FV*DF

13. Price of the Bond
• Price of a bond say 4.875% of May 31, 2011
• = 4.875/2*DF(0.5) +(100+4.875/2)*DF(1)
• We know the DF(0.5) and DF(1).

14. Spot, Forward, YTM
• Spot rate is nothing but a rate on a spot loan.
• Denote the semi annually compounded t-year
spot rate by r(t), investing 1 unit of currency from
now to t years will be :
• (1+r(t)/2)^2t
• To link spot rates and DFs:
• If $1 grows to (1+r(t)/2)^2t in t years, $1 is the PV
and we can write
• (1+r(t)/2)^2t *DF = 1
• Thus DF = 1/ (1+r(t)/2)^2t

15. Forward Rate
• Suppose two parties agree to use a 5% interest rate for a
trade to take place six months from today. This is the
forward rate.
Suppose a two-year bond is yielding 10% while a one-year
bond is yielding 8%. The return produced from the two-
year bond is the same as if an investor receives 8% for the
one-year bond and then uses a rollover to roll it over into
another one-year bond at 12.04%. So the rate 1y forward
for 1 y = 12/04%
• [(1.10)^2/ (1.08)^1] - 1 = 12.04%. This hypothetical 12.04%
is the forward rate of the investment.

16. YTM
• Single yield when used to discount the
cashflows of a security gives market price of
that security!
Example of 4.5s of November 30,2011 was
105.856
Then one can write :
105.856 = 2.25/(1+y/2) +
2.25/(1+y/2)^2+102.25/(1+y/2)^3

17. DV01, Duration,Convexity
• DV01 measures dollar change in the value of
the security for a basis point change in
interest rates
• Duration measures the %change in the value
of the security for unit change in rates

18. Convexity
• Consider a second order taylor
approximations of the price –rate function
with respect to rates after a small change in
rates
• P(y+delta y) = P(y) + dP/dy*delta y
+1/2d^2P/dy^2*delta y^2
• Solving further, we get dp/p=-Ddy+1/2Cdy^2
• Where D is the duration and C is the convexity
• C = 1/P * d^2P/dy^2

19. Convexity related to Vol
• From second order Taylor series expansion,
we see that Convexity is always with dy^2,
positive convexity will always increase returns
so long as rates move
• Like wise, with negative convexity, returns will
always go down so long as rates move.

20. Effect of rates on duration
• Increasing Yield lowers DV01
– Increasing yield lowers PV of the payments but it
lowers the PV of longer payments the most, DV01
of these longer payments goes down more
relative to the value of the whole bond!
• Duration is measured In years so higher
duration means investor will have to wait
longer to realise coupon and principal