VALUATION OF FIXED INCOME SECURITIES Presented By Sade Odunaiya Partner, Risk Management Alliance Consulting
OUTLINE Introduction n Valuation Principles n Day Count Conventions n Duration n Covexity n Exercises n
INTRODUCTION n
Valuation ¨ determination
of fair value i.e. Price ¨ Yield and Yield spread is relative measure of value
TERMINOLOGIES n
Fixed Income Security ¨A
financial obligation that promises to pay specified sums at specified future dates Debt obligation – borrower/lender with a promise to pay interest and principal n Preferred stock – ownership interest with fixed dividend payment from after tax profit n
Terminologies (Contd) n
Indenture & Covenants ¨ Rights & obligations managed by a ¨ Affirmative & Negative covenants
n
trustee
Term to Maturity ¨ No of years issuer promises to pay ¨ Same as tenor only on issue date
n
Par Value ¨ Amount
payable at maturity ¨ Principal, face value, redemption value and maturity value ¨ Assumed 100
Terminologies (Contd) n
Trading Value ¨ Below Par Value – Discount ¨ Above Par Value – Premium
n
Coupon Rate ¨ Interest rate issuer pays ¨ Called nominal rate
n n
each year
Coupon is the actual amount paid/payable annually No coupon instruments are Zero Coupon instruments
BASIC PRINCIPLES OF VALUATION n
Value or Price ¨ Present
Value (PV) of expected cash flows ¨ 3 basic steps Step 1: Estimate expected cash flows (coupon & principal) n Step 2: Estimate appropriate discount rate using comparable bonds n Step 3: Calculate the PV of cash flows in Step 1 using discount rate in Step 2 n
STEP 1: ESTIMATE EXP CASH FLOW n n n
Cash expected to be received in the future Principal & interest Difficult to estimate for some instruments where¨ Option
to change contractual date e.g. callable/putable bonds, mortg backed sec. ¨ Coupon reset based on reference rates, prices / exch rate ¨ Investor has choice to convert or exchange security
STEP 2: DETERMINING APPR RATE n n n n
Minimum rate should be credit-risk free rate with same maturity i.e. Treasury rate Premium should reflect additional risk embedded in the instrument & issuer Comparative risk or alternative investment return Differing rate for each of the expected cash flows
STEP 3: DISCOUNTING THE CASH FLOWS In general, the price of a bond is given by P = C + C + 1+ r (1+ r)2 or: P
where:
C + ……+ C + M (1+ r)3 (1+ r)n (1+ r)n
=
Σ C + (1+ r)t
C P
= =
coupon payments usually semiannual price of the investment
n r M t
= = = =
number of periods periodic interest rate maturity value time period when the payment is received
M (1+ r)n
STEP 3: DISCOUNTING THE CASH FLOWS n
Annuity Concept - valuation with the same discount rate The present value of an annuity is equal to – 1Annuity payment X
1 (1 + r)n r
STEP 3: DISCOUNTING THE CASH FLOWS n
Present Value Properties
A bond that matures in 4 years time has a coupon rate of 10% with an annual interest payment frequency. a.
Assuming applicable discount rate for similar security is 8%, what is the price of the bond today?
b.
If the discount rate is changed to 12%, what will the price be?
STEP 3: DISCOUNTING THE CASH FLOWS NOTE: · TIMING: The longer the term to maturity, the lower the value of the cash flow today · DISCOUNT RATE: The higher the discount rate, the lower the security value · RELATIONSHIP OF COUPON RATE, DISCOUNT RATE & PRICE RELATIVE TO PAR: Coupon rate = yield required; price = par value Coupon rate < yield required; price < par value Coupon rate > yield required; price > par value · CHANGE IN VALUE AS TIME MOVES CLOSE TO MATURITY: A bond’s valueo Decreases over time if the bond is selling at a premium o Increases over time when the bond is selling at a discount o Is unchanged if the bond is selling at par value o At maturity is equal to its Par Value
VALUATION USING MULTIPLE RATES Proper way to value cash flows n Illustration Suppose we have a 4-year 10% coupon bond with annual interest payment and the appropriate discount rates are as follows: Year 1 Year 2 Year 3
6.8% 7.2% 7.6%
Year 4
8.0%
VALUATION USING MULTIPLE RATES Practice: What is the value of a 5-year 7% coupon bond assuming the payments are received annually and the discount rates for each year are as follow: Year 1 Year 2
3.5% 3.9%
Year 3 Year 4 Year 5
4.2% 4.5% 5.0%
VALUING SEMI-ANNUAL CASH FLOWS Same Computation n Each period is now 6 months n No of periods will double n Exercise 5 n
¨ assuming
semi-annual cash flows
VALUING ZERO COUPON BONDS n
One cash flow at maturity Maturity value
(1 + i)n x 2 Where i is the semi-annual discount rate n is the number of years to maturity Note: The number of periods is n X 2 for consistency with the pricing of coupon bearing bonds in the market.
VALUATION COMPLICATIONS n
Next Coupon less than 6 months away Interest earned by seller Last coupon payment date
Interest earned by buyer
Settlement date
Next coupon payment date
Interest (AI) – interest earned by seller ¨ Full or Dirty Price – based on discounted value ¨ Clean Price = Full price – Accrued Interest ¨ Accrued
VALUATION COMPLICATIONS n
Full or Dirty Price Present valuet
=
expected cash flow (1 + I )t -1+w
Where: w period
=
days btw settlement and next coupon date no of days in coupon period
DAY COUNT CONVENTIONS n
Different day-year conventions in different markets Eurodollar (LIBOR) market
Actual/360
Eurobond market (AIBD)
360/360
US Treasury/ Ghanaian Money Market Actual/365 Nigeria Money Market (NMM)
Actual/Actual
The numerator depicts the number of days interest is to be earned while the denominator depicts the number of days in a year in the relevant market.
DAY COUNT CONVENTIONS A coupon bearing US Treasury security whose previous coupon payment was March 1 and the next is September 1. Suppose this Treasury security is purchased with a th
settlement date of July 17 .
CASH FLOWS ARE UNKNOWN Mainly in respect of options embedded securities n Valuation models n
¨ Binomial ¨ Monte
Carlo Simulation
Fixed Instrument Price Volatility Measures n
Bond price moves in an inverse relationship to yield changes
Price
Yield
Bond price volatility n n n
Measured in terms of percentage price change in price Influenced by more than yield behavior alone Factors such as: ¨ Par
value ¨ Coupon rate ¨ Term to maturity ¨ Prevailing interest rate
Relationship between Yield/Price n n
n n
n
Price moves inversely to yield Price volatility is directly related to term to maturity i.e. longer maturity bonds post larger price changes Price volatility increases at a diminishing rate as term to maturity increases Price movements from equal absolute increases or decreases in yield are not symmetrical Higher coupon issues show smaller percentage fluctuations for a given change
Illustration 1: Effect of Maturity on Bond Price Volatility ---------------- Present Value of an 8% Bond ($1,000 Par Value) -----1 Year
10 Years
20 Years
30 Years
Yield to Maturity
7%
10%
7%
10%
7%
10%
7%
10%
Present value: - interest - principal Total value % change
75
73
569
498
858
686
1,005
757
934
907
505
377
257
142
132
54
1,009
980
1,074
875 1,115
828
1,137
811
-2.9%
-18.5
-25.7%
-28.7%
Illustration 2 Effect of Coupon on Bond Price Volatility -----------Present Value of an 20 Year Bond ($1,000 Par Value) -----0 Coupon
3% Coupon
8% Coupon
12% Coupon
Yield to Maturity
7%
10%
7%
10%
7%
10%
7%
10%
Present value: - interest 0
0
322
257
858
686
1,287
1,030
- principal
257
142
257
142
257
142
257
142
Total value
257
579
399 1,115
828
1,544
1,172
% change
142 -44.7%
-31.1
-25.7%
-24.1%
Trading Strategies n
What type of portfolio of bonds will you seek to have if you expect a major decline in interest rate going forward?
n
Zero coupon ( or very low coupon) with long term to maturity in order to post maximum gains
Measures of bond price volatility Measure of interest rate sensitivity is Duration n Three key measures: n
¨ Macaulay
duration ¨ Modified duration ¨ Effective duration
Macaulay Duration n n n
Measure of time (xteristics) flow from a bond Weighted average number of years over which total cash flows occur Weightings used are the market value of cash flows D
=
SCt (t) (1+i)t S Ct (1+i)t
Price of the bond
Example: Consider two bonds with the following features: Face value
$1,000
$1,000
Maturity
10 years
10 years
Coupon
4%
8%
Calculate the Macaulay duration for each of the bonds assuming an 8%market yield.
Example: Bond A
Year 1
Cash flow
PV @ 8%
PV of flow
PV as % of
PV as % of Price time
Price
weighted
.9259
37.04
.0506
.0506
2
40 40
.8573
34.29
.0469
.0938
3
40
.7938
31.75
.0434
.1302
4
40
.7350
29.40
.0402
.1608
5
40
.6806
27.22
.0372
.1860
6
40
.6302
25.21
.0345
.2070
7
40
.5835
23.34
.0319
.2233
8
40
.5403
21.61
.0295
.2360
9
40
.5002
20.01
.0274
.2466
10
1,040
.4632
481.73
.6585
6.5850
731.58
1.0000
8.1193
Total Duration = 8.12 years
Example: Bond B
Year
Cash flow
1
80
.9259
74.07
.0741
.0741
2
80
.8573
68.59
.0686
.1372
3
80
.7938
63.50
.0635
.1906
4
80
.7350
58.80
.0588
.1906
5
80
.6806
54.44
.0544
.2720
6
80
.6302
50.42
.0504
.3024
7
80
.5835
46.68
.0467
.3269
8
80
.5403
43.22
.0432
.3456
9
80
.5002
40.02
.0400
.3600
10
1,080
.4632
500.26
.5003
5.0030
1,000.00
1.0000
7.2470
Total Duration = 7.25 years
PV @ 8%
PV of flow
PV as % of Price
PV as % of Price tim e weighted
Macaulay Duration n
Characteristics ¨ Less
than term to maturity due to interim cash
flows ¨ Inverse relationship between coupon and duration I.e. higher coupon lower duration ¨ Positive relationship between term to maturity & Macaulay duration but duration increases at a decreasing rate with maturity ¨ Sinking fund and call provisions have dramatic impact on bond’s duration
Modified Duration n n n
Measures the price volatility of a non-callable bond Modified Duration = Macaulay duration 1 + YTM/n Greater modified duration, the greater the price volatility for small price changes Therefore dP 1 = dY P
- modified duration
Modified Duration Example: n Consider a bond with Macaulay Duration of 8 years, yield of 10%. If you expect YTM to decline by 75 basis point (from say 10% to 9.25%) n What is the expected percentage price change?
Modified Duration If you expect a decline in interest rate, you should increase the average modified duration of your portfolio n Note that duration changes in a nonlinear fashion with yield changes – a concept called convexity n Necessary to recalculate and rebalance the portfolio as rate changes n
Effective Duration Direct measure of interest rate sensitivity of any asset (bond inclusive) n Based on observed market prices surrounding interest rate changes n
n
%age change in Price - D*
=
=
- Modified Duration X change in Yield
%age change in Price Change in yield
The D* obtained this way is the effective duration.
Effective Duration n
Note:
Effective duration greater than maturity is possible with CMO n Negative effective duration as in the case of bonds with embedded options n
THANK YOU ANY QUESTIONS??
