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Reading 44: Introduction To Fixed-Income Valuation - Coggle Diagram
Reading 44: Introduction To Fixed-Income Valuation
Bond Valuation
Spot Rate
A Spot Rate is a market discount rate for a single payment (either coupon or principal payment) to be received in the future.
Yields on zero-coupon bonds = spot rates, also called zero rates, denoted \(Z_1 , Z_2 , Z_3 \) (for year 1, year 2, year 3 respectively)
simply speaking: spot rate is the discount rate for each specific year
Use PV of cash flow with coupon every year and principal at final year
Pay attention to compound period: annual, semiannual, etc.
Yield to Maturity (YTM)
YTM assumes that:
Held-to-maturity
All payments made
Coupon payments reinvested at YTM
= Calculate IRR
Relationship
bond price vs. YTM
Yield down : Price up
Yield up : Price Down
Coupon rate vs. YTM
Coupon Rate < YTM → Price < Par (Discount)
Coupon rate > YTM → Price > Par (Premium)
Convexity
Price increase (from decrease in yield) > Price decrease (from increase in yield) (look at graph for easier understanding)
Maturity Effect
Values of bonds with longer maturities are more sensitive to a change in YTM
Coupon Effect
Values of bonds with lower coupons are more sensitive to a change in YTM
Flat vs. Full Prices
Flat Price (clean price or quoted price by bond dealer): Does not include accrued interest
Full Price (invoice price or dirty price) = flat price + accrued interest
Accrued interest
= \(\frac{t}{T} \times PMT\)
t: number of days from the last coupon payment to the settlement date
T: number of days in the coupon period
PMT: Coupon payment per period
Day-basis
Corporate bonds: use 30 / 360 method
Government bonds: use actual / actual number of days
Matrix Pricing
Use matrix pricing when bonds trade infrequently so that market YTM is unavailable
Matrix Pricing uses YTMs of traded bonds of same credit quality to estimate bond YTM
Use average of YTMs for same maturity
Use linear interpolation to adjust for differences in maturity (example: Video Module 44-2 16:29)
Yield Valuation
YTM for annual-pay bond
\( Price = \sum_{I}^{i}\frac{Coupon_i}{(1+YTM)^{i}} + \frac{Par}{(1+YTM)^{n}}\)
Be careful for multiple-compounding per year (periodicity)
Adjusting for periodicity
if periodicity = M
Solve r for: \( PV= \frac{FV}{(1+r)^{N \times M}}\)
Then, effective yield = r x M
Equality:
The annual rate and the periodic rate (after adjustment) must be equal
Annual rate = \( (1+\frac{YTM_m}{m})^m=(1+\frac{YTM_n}{n})^n \)
m: monthly
n: any given periodicity
Yield Conventions
Street Convention: assumes payments are made on scheduled dates (don't care about calendar)
True yield: uses actual payment dates, taking account of holidays and weekends (based on calendar)
Government Equivalent yield: Corporate bond yield restated based on actual/actual; used for calculating spread to benchmark government bond yield
Current yield
\( Current.Yield = \frac{Annual.Coupon.Payment}{Flat.Price}\)
Properties
Current yield ignores movement toward par value
for discount bond: Current Yield < YTM
for premium bond: Current yield > YTM
Simple Yield:
Assumes that premium or discount will be amortized on a straight-line basis
\( Simply.Yield = \frac{Annual.Coupon.Payment \pm Annual.Amortization}{Flat.Price}\)
Yield To Call (YTC)
Yield To First Call: Substitute the call price at the first call date for par, and number of periods to the first call date for N (example: Module 44-3 14:20)
YTC for each of a bond's call dates and prices
Yield To Worst: is the lowest of YTM and the YTCs for all the call dates and prices
Option Adjusted Yield
More precise yield measure for callable bonds
Value the call option using a pricing models and expected yield volatility
Add the call option value to the bond price
Calculate the option-adjusted yield based on the option-adjusted price
For putable bond, substract the value of the put option to get option-adjusted price
Floating-Rate Notes
Coupon at next reset date = reference rate at previous reset \( \pm\)
Quote margin
Required margin (discount margin) is the margin that would cause the note's value to return to par at reset date
Required margin vs. Quote margin
If required margin > quoted margin, price < par
If required margin < quoted margin, price > par
\(PV=\sum_{i=1}^{n}\frac{\frac{(Index + QM)\times FV}{m}}{(1+\frac{Index + DM}{m})^{i}}\)
Index: Usually LIBOR
QM: Quoted margin
DM: Discounted Margin or Required Margin
m: Number of compounding period per year
Money Market Instrument
(Maturity < 1 year)
Yield
Yield quoted as simple annual interest
Maybe discount or add-on yields
May use 360-day or 365-day year
Compare money market instruments based on bond equivalent yield (add-on yield based on 365-day year)
Bond equivalent yield
= \( \frac{Year}{Days} \times \frac{FV-PV}{PV}\)
Treat periodicity of yield as \( \frac{365}{Days.to.maturity}\)
Example: Module 44-3 28:41
Banker Acceptance
\( PV_{Banker.Acceptance}=FV \times (1-\frac{Days}{Year} \times DR)\)
DR: Discount Rate
Yield Curve
Yield curve (or term structure) shows yields for bonds at different maturities.
Bonds have similar:
Currency denomination and tax treatment
Credit risk
Liquidity
Coupon rate / reinvestment risk
Periodicity and yield calculation method
Spot (zero) yield curve
is yield curve constructed from a sequence of yield-to-maturities on zero-coupon bonds
Same currency, credit risk, liquidity, tax treatment, and coupon rate/ reinvestment risk
However, not actively traded across all maturities, so yield curves are constructed using coupon bond yields
Coupon Bond Yield Curve
Semi-annual bonds issued for specific maturities (e.g., for 1,3,5,7,10 years)
Newly issued bonds are close to par (similar tax effects), actively traded (similar liquidity)
Other maturities based on linear interpolation
Yields for 1-, 3-, 6-, 12-month maturities often converted to semi-annual bond equivalent yields
Par Curve
Calculate the coupon payment for each bond maturity that would value the bond at par, using spot rates
Find PMT for \( 100 = \sum_{i=1}^{n} \frac{PMT}{(1+Z_{i})^{i}}+\frac{100}{(1+Z_{i})^{n}} \)
All bonds used to derive the par curve are assumed to have same credit risk, periodicity, currency, liquidity, tax status, and annual yield
Forward Yield Curve
Forward Rate are rates for loans to be made in the future.
[1y2y] : The rate for 2-year loan, to be made 1 year from now
[\(S_n \)] (a.k.a [\(Z_n \)] ): is the spot rate for N-year loan
Calculating forward rates
(Example: Module 44-4 6:02)
Cost of borrowing
$1 for 3 years =
Borrowing 1 year at S1, then 1 year at 1y1y, and then 1 year at 2y1y
\( (1+S_3)^3=(1+S_1)(1+1y1y)(1+2y1y)\)
Borrowing for 1 year at [\(S_1 \)], then for 2 years at 1y2y
\((1+S_3)^3=(1+S_1)(1+1y2y)^2\)
Borrowing for 2 years at [\(S_2 \)] and for 1 year at 2y1y
\( (1+S_3)^3=(1+S_2)^2(1+2y1y) \)
Approximation method
Faster to calculate but only approximate result
Forward rate can be interpreted to be incremental or marginal return for extending the time-to-maturity of an investment for an additional time period
Yield Spreads
Yield spread (to risk-free security) = Yield - Risk-free rate
Risk-free rate (RFR) is driven by macroeconomic factors
Spreads driven by microeconomic factors
Spreads are also considered based on other factors
Government Spread (G-Spread)
Spread to benchmark government bond yield for same maturity (US, UK, Japan)
Yield on governments can be on-the-run bond yields or interpolated yields
Calculate G-Spread
1) Calculate YTM of both government bonds.
2) Convert YTM into basis point (bps). Ex: r= 0.0456 →bps = 456
3) G-spread = Higher bps - lower bps
Interpolated Spreads (I-Spreads)
For eurozone bonds, spreads are relative to swap rate, the fixed rate of a fixed for floating swap
For example, a 5-year corporate euro coupon bond might be quoted as "midswap+80bp (base point)"
This spread represents the credit risk difference between interbank lending (LIBOR) and the bond
Static Spreads (Z-Spreads)
Are the amount added to each spot rate (government or swap) to get bond price
Find Z for:
PV (Bond Price) = \( \sum_{i=1}^{n}\frac{Coupon_i}{(S_{i}+Z)^i} + \frac{PAR}{(S_{i}+Z)^i} \)
\(S_i\) is the spot rate at year i
Option-Adjusted Spreads
Option-adjusted spreads = Z-spread - option value
Option value is the cost to issuer in basis points per year
Example: Z-spread = 180 bp; call option increase yield by 60 bp → OAS = 180-60=120 bp