Bond Pricing Bond Yields Relation between YTM and Realised Return

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𝑃𝑉 = (%&')! + (%&')" + ⋯ + %&' # = ' × 1 − (%&')# + %&' #
")' Nominal yield => stated interest rates of a bond Yield To Maturity (YTM) is only a promised yield if:
Current yield (flat yield, interest yield, running yield) => Coupon/Price 1. Bond held to maturity;
Where, cpn = c/m x face value, r = r/m, n x m, par = face value Note: (sum of coupons for the year is used regardless of compounding periods)
When; Simple yield => current yield + approximate capital gain yield 2. All interest payments are reinvested at a rate = YTM (does not apply to ZCB)
Coupon rate > YTM: Premium Bond (Held to maturity => cap loss) !$$%&
6& ' 6 (%99-8)/2
Coupon rate = YTM: Par Bond =8+ , does not account for TVM Where bond is not held to maturity,
8 8
Coupon rate < YTM: Discount Bond (Held to maturity => cap gain)
Note: 1. Compute NEW price of bond
Yield To Maturity (YTM) => r that satisfies bond equation 2. Calculate capital gain/loss
- Price of disc(prem) bonds increases(decreases) as it matures Yield To Call (YTC) => r that satisfies bond equation, when bond has a call provision
- Price converges to the face value as time reaches maturity since time value = 0 • Yield to first call => based on “maturity” at first call date and call price as 3. Calculate Current Yield
This is known as the pull-to-par effect principal (bond is not callable YET) 4. Total Return (Holding Period Return) = Current Yield + Cap gain/loss
• Yield to next call => when bond is callable Note: Use Total Income/Price instead of current yield if held for more than a year
Bond quotes Yield to worst => minimum of all possible yields to call and yields to put
Price of bond = Quote/100 x Face Value Yield to sinker => yield to sinking fund payment date when the bond will be retired
Where quoted in 3nds (91-19+); Where reinvestment rate changes:
Yield to put => when the bond has a put option
First two digits reads as 91/100 = 0.91 = 91%, 2nd part shows the tick size: (91 + Yield for a Portfolio => treat entire portfolio as a bond and back out the YTM. 1. Compute NEW FV_OA of coupon reinvestments (aka Total reinvestment income)
19/32 + 1/64)/100 2. Compute NEW price of bond (if is not sold at par and/or not held to maturity)
Relationship between Various Yields => Calculate capital gain/loss
Full(Dirty) Price and Accrued Interest Premium bond => c > c/p > YTM 3. Solve for Realized Yield (YTM)
Full price (Dirty price) => Present value of future cash flows (with AI) Bonds selling at par => c = c/p = YTM 0<.), '74#=7>.:7#. 4#!<:7 & 2?@ 8'4!7 <A B<#C DE 8FE 4A G7,C .< :).+'4.5
Flat price (Clean price) => Quoted price in the market Discount bond => c < c/p < YTM P= 𝒚 (0*
(% & * )
*Accrued interest (AI) => Interest earned by seller of bond as the bond is sold
between interest payment dates YTM = r x m
Since YTM = C/P + DELTA P/P
YTM has to be greater than C/P for a discount bond and DELTA P/P has to be bigger
AI = t/T x PMT/m than 0 since pull to par where p < par value. (The reverse can be said) Reasons for the Change in the Price of a Bond
Since for a discount bond, it sells below bar and pulls to par at maturity 1. For a prem/disc bond => Pull-to-par effect, because it is moving to maturity
Where: 2. Changes in the credit quality of the issuer (aka risk) => change in yield
t = number of days from the last coupon payment to the settlement date Converting Annual Yields
T = number of days in the coupon period 3. Change in yield required by the market (change in yield on comparable bonds)
which makes other bonds more attractive and said bond less attractive.
t/T = fraction of the coupon period that has gone by since the last payment
PMT = coupon payment Note: APR = Annual Percentage Rate
M = # compounding periods per year Yield Measures for Floating-Rate Notes(FRN)
The general rule of thumb is that compounding more frequently at a lower annual => More stable price
Note: The convention used (A/A => actual days passed/actual days in period, rate is equivalent to compounding less regularly at a higher annual rate.
30/360 => 30days in a month/360 days in a year, etc) PMT must match the period! Simplified pricing model of FRN
Hence, any rate must be converted to the same for a fair comparison.
𝑃𝑀𝑇 𝑃𝑀𝑇 𝑃𝑀𝑇 + 𝐹𝑉 1(230 4 56 ×89 1(230 4 56 ×89 1(230 4 56 ×89
&*H
𝑃𝑉*+,, = %-./0 + 1-./0 + ⋯+ . 𝑃𝑉 = *
+ *
+ ⋯+ *
(1 + 𝑟) (1 + 𝑟) Dollar Return for a bond 1(230 4 :6 ! 1(230 4 :6 " 1(230 4 :6 '
1 + 𝑟 2-0 %& %& %&
Arises from: Coupon payment, Reinvestment income (from reinvesting coupon * * *

𝑃𝑀𝑇 𝑃𝑀𝑇 𝑃𝑀𝑇 + 𝐹𝑉 payments, does not apply to ZCB), Capital gain Where:
𝑃𝑉 *+,, = + + ⋯+ ×(1 + 𝑟) ./0
QM > DM => Premium bond, QM < DM => Discount bond
(1 + 𝑟)% (1 + 𝑟)1 1+𝑟 2 The quoted margin is basically the adjustment to the coupon rate
Future value of a coupon bond = FVOA(coupons) + par value
(%&'/:)(/* - % The discount(required) margin is basically the adjustment to the yield to maturity.
𝑃𝑉34'.5(*+,,) = 𝑃𝑉 ∗ (1 + 𝑟)./0 = 𝑐/𝑚 + par Index refers to the reference rate e.g. LIBOR (p.a.)
'/:
Also, 1) Where Total coupon over bond’s life = n x m x c Yield Measures for Money Market Instruments
2) Interest on interest = FV(Annuity of coupons) – Total coupons The pricing formula for money market instruments quoted on a discount rate basis
3)5> I7)' *H -8H
𝑃𝑉34'.5(*+,,) = 𝑃𝑉6,7)#(*,).) + 𝐴𝐼. 𝑃𝑉6,7)#(*,).) = 𝑃𝑉34'.5(*+,,) − 𝐴𝐼 3) Capital gain(loss) = Par or Sale price – Purchase price 𝑃𝑉 = 𝐹𝑉 × 1 − I7)' ×𝐷𝑅 , 𝐷𝑅 = (3)5>) × *H
Total dollar return = 1 + 2 + 3
Matrix Pricing The pricing formula for money market instruments quoted on an add-on rate basis.
A process to estimate the market discount rate and price of some fixed-rate bonds ' 𝑛∗𝑚 𝐹𝑉 𝑌𝑒𝑎𝑟 𝐹𝑉 − 𝑃𝑉
which are not actively traded or bonds that are not yet issued, based on the quoted Also; Total dollar return = P 1 + : −𝑃 𝑃𝑉 =
𝐷𝑎𝑦𝑠
, 𝐴𝑂𝑅 = ×
Also: Total dollar return = FVOA(coupons) + Price Sold OR Par – Purchase price (1 + 𝑌𝑒𝑎𝑟 × 𝐴𝑂𝑅) 𝐷𝑎𝑦𝑠 𝑃𝑉
or flat prices of more frequently traded comparable bonds.

Note: The discount rate can be converted to the add-on rate for a fair comparison.
Spot Rate Forward Rates Theories of Term Structure of Interest Rates
Spot rates are yields-to-maturities on zero-coupon bonds maturing at the date of The future interest rate inferred from the spot rate curve (e.g. the rate of the 1 year 1. Unbiased Expectations Hypothesis/Pure expectations theory
each cash flow. Observable market rates today (e.g. rate of the 1 year bond today) bond, 1 year from now, implied by the 2 year bond)
Calculate the bond price using spot rates that correspond to the cash flow dates. - Indicates the market consensus of what spot rates might be in the future The forward rates implied by the term structure are equal to the market's
- Can be thought of as hedgeable rates. (e.g. by buying the one-year security, the expectation of future spot rates over the same period.
Yield Curve aka "term structure of interest rates." investor can hedge the six-month rate six months from now. (Lock in the 6month t𝑓n = E(tzn)
Graphical depiction of the relationship between the yield on bonds of the same spot rate, 6 months from now by buying a 2 period security and selling a 1 period - Follows that long-term yields are geometric averages of current and expected
credit quality but different maturities ⇒ basically spot rates for bonds of the same security, paying principal at t=1 and receiving principal at t=2) short-term yields.
credit risk with different maturities. Note: Forward rates may never be realized in practice since they are just
=> Usually constructed from observations of prices and yields in the Treasury expectations of future spot rates. - Broad interpretation is that bonds of any maturity are perfect substitutes for one
L-F another (e.g. investors are indifferent between 5 1yr bonds and 1 5yr bond)
market. Because: 1 + 𝑧F F× 1 + 𝐼𝐹𝑅F,L-F = (1 + 𝑧L)L Computing forward
1. Treasury securities are free of default risk, yields not affected by credit Observation: The yield curve tends to slope up at the beginning of an expansion,
rates from spot rates and is more likely to slope down at the end of an expansion.
worthiness. 𝐼𝐹𝑅F,L-F = [(1 + 𝑧L)L/ 1 + 𝑧F F] – 1
2. largest and most active bond market => fewest problems of illiquidity or
infrequent trading. Note: 𝐼𝐹𝑅F,L-F indicates (B – A) yr bond, A years from now. Prediction of business cycles:
(Default free) Theoretical Spot Rate Curve aka zero or “strip” curve - Increase in Demand for investment during expansions.
Constructed from the yield on Treasury securities from: Relationship between t-period spot rate (𝑧_𝑡), the current six-month spot rate Results in high demand for money => higher interest rate
(𝑧_1), and the six-month forward rates (vice versa)
1. On-the-run Treasury issues => most recently auctioned issue of a given maturity. Computing spot rates
=> Estimate yield necessary to make the issue trade at par is used, - consumers smooth consumption, hence where recession
=> resulting in the on-the-run yield curve aka par coupon curve. 𝑧. = 1 + 𝑧% 1 + 𝑓% 1 + 𝑓1 … 1 + 𝑓.-% %/. − 1, from forward rates
is anticipated, save more, pushing down rates and vice versa.
Extrapolation for missing inbetween points: r + Problems:
(yield@higher maturity – yield@lower maturity)/(# semiannual periods between) where 𝑓. is the six-month forward rate beginning t six-month periods from now.
Note: Long yields are geometric average of forward rates. 1. Assumes investors maximize expected returns, with no consideration of risk.
2. on-the-run Treasury issues and selected off-the-run Treasury issues 2. No transactions costs.
3. all Treasury coupon securities, and bills 3. Securities with different maturities as perfect substitutes
4. Treasury coupon strips The curve slopes up when forward rates are
increasing, and slopes down when they are decreasing.
Bootstrapping This is because the spot yields are a weighted average 2. Liquidity Preference Theory “biased expectations theory”
Backing out the theoretical spot rate for a particular period using a par value of the forward rates. Posits that investors require a premium for investing in longer-term debt. The
coupon bond since the coupon rate is known and therefore YTM is implied. Hence, required premium is called a "liquidity premium" or “term premium”. => more
we now have a theoretical spot rate that is unstated on the market. From this, the uncertainty in longer terms.
(risk-free) interest rate can then be used to price/value other instruments. Yield Spreads
The difference between the yield of a security and the benchmark yield(usually This suggests modifying our interpretation of implied forward rates:
Given previous year’s spot (theoretical) rates: government yield curve) is known as the benchmark spread. t𝑓n = E(tzn) + tLn => explains in reality why returns are usually lower for short-term
% Par if par Where benchmark yield is: bonds vs longer term bonds
𝑐𝑝𝑛 + 𝑝𝑎𝑟 # bonds are 1. Government bond (yield curve), the yield spread is known as the G-spread.
𝑍𝑛 = −1 ∗ 𝑘 used (usually) 2. Swap rate (swap curve), yield spread is known as I-spread. Implication: Upward sloping yield curve could be the results of either higher future
𝑃𝑎𝑟 − 𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑃𝑉 𝑓𝑟𝑜𝑚 1 𝑡𝑜 𝑛 − 1 interest rate, liquidity premium, or both
Note: The rate for the fixed leg of an interest rate swap is known as the swap rate
value of the Treasury coupon security should be equal to the value of the package (paid by the fixed rate counterparty) The swap rate is derived using short-term Observation: Upward sloping yield curve that levels off
of zero-coupon Treasury securities that duplicates the coupon bond’s cash flow. lending rates rather than default-risk-free rates. Hence the swap curve is not
1000 ∗ 5.75%/2 1000 ∗ 5.75%/2 1000 ∗ 5.75%/2) + 1000 default-risk-free. 3. Preferred Habitat
1000 = 𝑃 = + + 3. LIBOR (now SOFR) and yield of matching T-bill, TED spread
1 + 0.043 1 + 0.044 1 1 + 𝑍J J Adopts the view that the term structure reflects the expectation of the future
Z = 5.76% 4. Z-spread (Zero-volatility spread)
interest rates and a risk premium. But rejects the assertion that the risk premium
Note: must rise uniformly.
2 problems with using just the on-the-run issues
1. Large gap between some of the maturities points, may result in misleading yields Investors and borrowers have preferred maturity segments(habitats).Though they
for those maturity points when estimated using the linear interpolation method. may shift out (only when risk premium is great enough).
2. Yields for the on-the-run issues themselves may be misleading since most offer
the favorable financing opportunities (since in demand, price might be inflated). 4. Market Segmentation
The true yield is greater than the quoted (observed) yield. (Recall: Inverse r/s
between P and R) Different types of investors/borrowers naturally prefer different maturities.
=> Off-the-run Treasury issues can be use to mitigate this The forces of supply and demand operate independently in these two essentially
=> Can also use all(incl semi-annual) Treasury coupon securities and bills to separate markets.
construct the theoretical spot rate curve. - whole curve does not move in tandem and may have different shape throughout
Characteristics of bonds and Price volatility Approximate Modified Duration Ways to measure bond portfolio responsiveness to non-parallel changes in
• For a given coupon rate and initial yield, longer the term to maturity, greater Alternative is to estimate the approx. modified duration DIRECTLY. interest rates
price volatility. 8H% - 8H4
𝐴𝑝𝑝𝑟𝑜𝑥. 𝑀𝑜𝑑𝐷𝑢𝑟 = 1 ∆I47,C 8H
, where +/- = PV for ∆𝑌𝑖𝑒𝑙𝑑 Yield Curve Reshaping Duration
• For a given term to maturity and initial yield, lower the coupon rate, greater $
price volatility. Focus on three maturity points on the yield curve: 2-year, 10-year, and 30-year.
• For given change in yields, lower the level of the yields, greater price volatility. 𝐴𝑝𝑝𝑟𝑜𝑥. 𝑀𝑎𝑐𝐷𝑢𝑟 = 𝐴𝑝𝑝𝑟𝑜𝑥. 𝑀𝑜𝑑𝐷𝑢𝑟× 1 + 𝑟 , Calculate the spread between the 10-year and 2-year yield and refer to this as the
since ModDur = MacDur/(1+r/k) spread for the short end of the yield curve; the spread between the 30-year and the
Macaulay duration 10-year is computed and referred to as the spread for the long end of the yield
weighted average of the time to receipt of the bond’s promised payments(interest Effective Duration (aka curve duration, the others before ”yield duration” curve.
and principal), where the weights are the shares of the full price that correspond to 𝐸𝑓𝑓𝐷𝑢𝑟 = 𝑃𝑉- − 𝑃𝑉& H>?,>%H>?,8 HA?,8%HA?,>
2× ∆𝐶𝑢𝑟𝑣𝑒 × 𝑃𝑉9 𝑆𝐸𝐷𝑈𝑅 = 𝐿𝐸𝐷𝑈𝑅 =
each of the bond’s promised future payments. 1H$(∆5) 1H$(∆5)
Estimate the percentage change in price given a parallel shift in a benchmark yield S = steepening F = flattening
1C 2C nC nPar curve. (when whole curve shifts) => used for complex bonds (e.g. callable: no
+ + ⋯+ + Value for each security in the portfolio is calculated separately for steepening and
(1 + r)% (1 + r)1 1+r M 1+r M defined YTM)
MacDur = added together to get 𝑉_(𝑆𝐸,𝑆). This is repeated for 𝑉_(𝑆𝐸,𝐹 ).
P
where Interest Rate Risk Characteristics of a Callable Bond
P = bond price, C = coupon/period r = YTM/period, n = # of coupon/period, Par 1. The price of the non-callable bond is always > callable. Key Rate Duration
OR 2. When interest rates are high, effective duration of the 2 bonds are similar.
3. When interest rates are low, the effective duration of the callable bond is lower. Most popular measure for estimating the sensitivity of a security or a portfolio
%&N/O %&N/O &P(Q-N)
MacDur = N − Q %&N/O ;<-% &N Where, This is because the callable bond price does not increase as much when benchmark Recall: Effective duration focus on level risk, key rate duration focus on shaping risk
𝑇: Time to maturity in years; 𝑟: Annual YTM; 𝑐: Annual coupon rate; yields fall.
𝑘: # compounding periods in a year Note: Callable bonds tend to get called back. A decrease in projected cashflows
Note: This eqn adjusts for compounding periods, hence do not need to Adjust the which leads to a decrease in price. This offsets the price increase, hence concave
shaped instead of convex. The prices denoted by P_ and P+ in the equation are the prices in the case of a bond
to years. and the portfolio values in the case of a bond portfolio found by holding all other
%&N/O
Perpetuity D = N , Zero-Coupon Bond D = T Spread Duration interest rates constant and changing the yield for the maturity whose key rate
Spread duration for a fixed-rate bond is interpreted as the approximate change in duration is sought.
Rules for duration the price of a fixed-rate bond for a 100-basis-point (1%) change in the spread.
1. The duration of a zero-coupon bond = time to maturity. (Note: Spread is the difference between a non-treasury security and a benchmark A 5-year key rate duration of 0.45 means that if the 5-year spot rate changes by 100 basis
2. Maturity constant, a bond’s duration is higher when the coupon rate is lower. rate (e.g. treasury yield), which reflects the credit risk) points and the spot rate for all other maturities does not change, the portfolio’s value will
∆P change by approximately 0.45%.
(more of the value tied to the future, slower to get repaid) = −DV×∆r
3. Coupon rate constant, a bond’s duration increases with time to maturity P
Spread for the floating-rate security is the margin coupon rate on top of the
4. All other factors constant, duration of a coupon bond is higher when the bond’s reference rate (e.g. treasury yield). As such the spread reflects the credit risk.
yield to maturity is lower. (bigger change at low levels)
Note: Can’t just use the duration to compare price sensitivities between bonds. Portfolio Duration
As bonds have different yield to maturity. => modified duration Measured as the value-weighted duration of the bonds in the portfolio.
Note: For a disc bond, at a certain rate, duration can actually decrease. DM(portfolioA,B) = (A/A+B)(DA) + (B/A+B)(DB)
Practical uses:
1. look at portfolio duration for sectors of the bond market.
Modified Duration 2. A spread duration for a portfolio of fixed-rate bonds can also be computed.
Duration approach to estimating price changes is only accurate for small yield Portfolio duration is divided into two durations:
changes! (Hint: Convexity) • Duration of portfolio with respect to changes in level of Treasury rates.
=
• Spread duration
S %&N/O %&<
&P Q-N ∆U Investment Horizon, Macaulay Duration, and Interest Rate Risk
DR = = , Where D = N
− = ;< U
= −D R×∆r Recall: Where IR increases, RI increases, Price decreases and vice versa
%&< Q %& -% &N
<

When:
Dollar (Money) Duration 1. investment horizon > Macaulay duration,
Dd = DM × P , where DM = Modified duration coupon reinvestment risk dominates market price risk. The investor’s risk is lower
Note: Dd the absolute value of the slope of a price curve as a function of yield. interest rates.
2. investment horizon = Macaulay duration, coupon reinvestment risk offsets
Price value of a basis point (PVBP) => Price change given 1bp change(+/-) in yield market price risk.
𝑃𝑉- − 𝑃𝑉& 3.investment horizon < Macaulay duration, market price risk dominates coupon
𝑃𝑉𝐵𝑃 = reinvestment risk. The investor’s risk is higher interest rates
2