Fixed Income Analysis
1. Bond Price
2. Zero Coupon Bond
Mt
P 
1  r  n
3. Price of bond whose next coupon payment is due in less than six
months
n
C
P 
t = 1 1  r 
v
1  r 
+
t 1
M
1  r v 1  r t 1
4. Duration
1C
Macaulay duration 
1 +
1  y 
2C
1  y 
modified duration 
+ . . .+
nC
1  y 
n +
nM
1  y 
n
P
modified duration 
d u ra tio n in yea rs 
2
Macau lay du ration
1 y
d u ra tio n in m p erio d s p er yea r
m
C 
1
 n 1 0 0  C / y 
1

n 
n1
y 2 
1  y  
1  y 
P
2
d P
dollar convexity measure 
2
dy
approximate duration 
P _  P
2  P0    y 
duration of an inverse floater 
1  L   d u r a t i o n o f c o l l a t e r a l  
collateral prices
inverse prices
5. Approximate percentage change in price for a given Yield=
dP 1
  modified duration
dy P
1
dP
 ( m o d i f i ed d u r a t i o n )  d y 
P
6. Approximate Dollar Price Change=
dP = (dollar duration)(dy)
7. Approximate change in price due to convexity is:
dP   dollar convexity measure  dy 
2
8. Convexity
2
d P 1
convexity measure 
2
dy P
c o n v e x ity m e a su re in y e a r 
c o n v e x ity m e a su re in m p e rio d p e r y e a r
m
2
9. Percentage price change due to convexity is:
dP 1
2
  convexity measure  dy 
P 2
approximate convexity measure 
P  P  2 P0
P0   y 
2
10. Realised coupon Yield to Maturity (RCYTM):
RCYTM = n
Terminal Value
-1
Initial Value
11. Single Period Immunisation:
F = I(1 + r)𝐻
12. A measure of risk for immunized portfolios, denoted by M2:
M
2

CF1 1  H 
1  y 
2

CF2  2  H 
1  y 
2
2
 . . .
CFn  n  H 
1  y 
2
n
13. Target accumulated value
 (1 + y ) n  1 
 annual coupon rate 
=
(P)


+ P

2

y


2
 accumulated value 1/ n 
Total return = 2 
  1
 policyholder payment 

Future value of an annuity= = 𝐴0 [
(1+𝑟)𝑡 −1
𝑟
]
14. The monthly mortgage payment
 i 1 + i  n 
MP  MB0 

n
1
+
i

1






15. To calculate the remaining mortgage balance at the end of any month:
 1  i  n  1  i  t 

MBt  MB0 
n


1

i

1




16. To calculate the portion of the monthly mortgage payment that is the
scheduled principal payment for a month:
 i 1 + i  t 1 

SPt  MB0 
n
 1 + i 


1


17. A formula can be used to determine the SMM for a given CPR:
SMM = 1 – (1 – CPR)1/12
18. Average Life of MBS
T
average life  
t 1
t  principal received at time t 
12 total principal 
19. MOD and percentage change in price: (dP/dy)*(1/P) = -MOD
20. PV annuity=𝑉𝑡 = 𝐴0 [
1−(1+𝑟)−𝑡
𝑟
]
21. Floater Coupon Rate= 1- monthly LIBOR + x
22. Inverse Floater Coupon Rate= K – L (1 – month LIBOR)
23. Price Value of a Basis Point
=
PV(−) – PV(+)
2
24. Forward rate for multiple periods
(1+𝑌𝑇𝑀𝑛 )𝑛
= [(1+𝑌𝑇𝑀
𝑛−1 )
3
𝑛−1
− 1]
25. Conversion Value= Underlaying share price x Conversion ratio
26. Market Conversion Premium Per Share= Market Conversion Price Underlaying Share Price
27. Market Conversion Price=
𝐶𝑜𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒
𝐶𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜
28. Market conversion premium ratio=
𝑀𝑎𝑟𝑘𝑒𝑡 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒
𝑈𝑛𝑑𝑒𝑟𝑙𝑎𝑦𝑖𝑛𝑔 𝑆ℎ𝑎𝑟𝑒 𝑝𝑟𝑖𝑐𝑒
29. Effective Duration
30. Effective Convexity
4