R41: INTRODUCTION TO FIXED-INCOME VALUATION

Bond Valuation & Yield to Maturity

Spot rates & Accrued Interest

Yield Measure
Most important

Yield Curve & Yield Spread

Price = PV of ( ∑PMT)


=> YTM ≈ discount rate of PV ≈ Ls nđt nhận khi nắm giữ đến đáo hạn

YTM incre => PV decre => Value decre
(Vice versa)

NOTE: Semiannual:

  • Coupon / 2
  • YTM / 2
  • N (new) = N (old) * 2

Relationship

Bond price vs Market discount rate (YTM)

Bond price vs Coupon rate

Inverse effect
Market discount rate increases (decreases) → the price of the bond decreases (increases).

Convexity effect
Screen Shot 2022-03-06 at 13.42.51

Bond price vs Maturity

Maturity effect
LONG maturity => ∆ YTM ≈1% => HIGH ∆ Price
SHORT maturity => ∆ YTM ≈1% => LOW ∆ Price
=> Thời gian càng dài => càng RỦI RO => giá biến động càng mạnh

a LOWER coupon bond is MORE sensitive to changes in the market discount rate than a higher coupon bond


HIGH coupon => ∆ YTM => LOW ∆ Price
LOW coupon => ∆ YTM => HIGH ∆ Price
=> coupon bé thì giá biến động mạnh

The constant-yield price trajectory
Screen Shot 2022-03-06 at 14.07.49


Bond value sẽ hội tụ lại tại Par value của bond có YTM = cp rate

Spot rate


Thay thế cho lãi suất ko đổi (YTM, discount rate) bằng lãi suất thay đổi


Spot rate bond ≈ no-arbitrage value

Flat price Accrued interest Full price

AI = PMT(≈cpon) × t/T

Full price = PV( tại lúc t ngay trước) × (1+r)t/T

Flat price = Full price - AI

Matrix pricing

3-year YTM is estimated as the 2-year YTM + one-third of the difference between the 5-year and 2-year YTM


3-year YTM = 2-year YTM + [ (3-2) / (5-2) ] * (5-year YTM - 2-year YTM)

EAR = (1+ (APR / n) )^(n-1)

Fixed-rate Bond

Floating-rate Bonds

Option-free

Embedded option

Government equivalent yield

Current yield = Annual cash cp pmt / Bond price

Street convention yield and True yield
(Scheduled cp pmt OR Actual cp pmt)

Simple yield = ( Annual cash cp pmt + gain/loss (∗) ) / Bond price


Gain/loss = the straight-line amortization of a discount/premium

Yield-to-call and yield-to-worst


Tính ra từng cái YTM (FV khác nhau, N khác nhau)
YT-1st call: FV = giá bán tại t=1, N = t1 - t0
YT-worst: Tính thêm YTM, YT-2nd call rồi chọn cái thấp nhất

Option-adjusted price (Value of noncallable bond) = Flat price of callable bond + Value of embedded call option

Quoted margin (QM): Coupon rate = MRR + quote margin (QM)

Required margin/Discount margin (DM): Discount rate = MRR + discount margin (DM)

QM để tính coupon
DM để tính discount

Money market instruments

Discount rate basis

Add-on rate basis

Yield curve

Spot rate YC
refers to YTMs on a series of zero-coupon government bonds at various maturities.

YC for coupon bonds

Par curve
YTMs for each bond trades at par.

Forward curve
the series of forward rates for bonds or money
market securities for the same maturities for annual periods in the future.


(1+Si)i = (1 + S1)(1 + 1y1y)(1 + 2y1y) ... (1 + (i-1)y1y)

Yield Spread

Benchmark Spread

G-spread (Government)

I-Spread (Interpolated)

Z-Spread (Zero-Volatility Spread)

OA-Spread (Option-adjusted Spread) = Z-spread − Option value