Nominal spread, Static spread, Zero-volatility spread, Option-adjusted spread

Nominal spread – simpliest

Nominal spread = (YTM of the bond – YTM of a Treasury security of similar maturity)

Limitations of nominal spread:
· Does not account for term structure of interest rates (no arbitrage pricing).
· Ignores affect of embedded options.

Static spread (Z-spread)
· Spread not over the treasury’s YTM but over each of the spot rates in a given treasury term structure, ie. same spread is added to all risk-free spot rates.
· more accurate than nominal spread as it is based upon the arbitrage free spot rates.
· Equal to nominal spread if yield curve is perfectly flat.

PV = CF1/(1+Z1 +SS) +CF2/(1+Z2 +SS)³+ CF3/(1+Z3+SS)³ , by trial & error to find SS (Z-spread)

Zero-volatility spread (Z-spread):
· Spread over the entire Treasury spot rate curve.
· Accounts for the term structure of interest rates, but ignores the affect of embedded options.
· Difference between Z-spread and nominal spread is greatest for amortizing bonds, for longterm high-coupon bonds, and for steep yield curves.

Option-adjusted spread (OAS):
· Spread over entire Treasury spot rate curve after accounting for embedded options.
· OAS is highly dependent on the model used to calculate the spread and the assumption for interest rate volatility.
· Higher interest rate volatility assumed » Lower OAS.

Option cost = Z-spread - OAS.

For option-free bonds OAS = Z-spread.
For option value greater than 0( call), OAS smaller than Z-spread;
For option value smaller than 0 (put), OAS greater than Z-spreadNominal spread may be misleading in the case of securities with embedded options. Buy securities with largest OAS for a given duration.

Forward Rate

A forward rate is the borrowing / lending rate in the future. The logic is that the return from investing in a 2-year bond is the same as that from investing in a 1-year bond and then rolling the proceeds into a second 1-year bond one year from now.

Forward rate from period 1 to period 2 = (1 +Z2)² / (1 + Z1) – 1

Where:
Z1 is the spot rate for 1-year
Z2 is the spot rate for 2-year.

Given the forward rates, calculate spot rate
(1+Z3)³ = (1+ f1)(1+f2)(1+f3)

Calculate N-period spot rate (SN)
ZN)3 = [(1+ f1)(1+f2)…..(1+fN)]^1/(N+1) –1

Bootstrapping Spot Rate

Bootstrapping is the process of calculating spot rates from coupon bond yields.

Example:
Given a 2-year bond price (P) with annual coupon (C) and 1-year spot rate Z1,
You can calculate 2-year spot rate in the following formula:

P = C/(1+Z1) + (C+PAR)/ (1+Z2)²

Given Z1,Z2 and the price of a 3-year bond, you can calculate Z3 in the same manner.

Annual pay, semi-annual pay, Reinvestment income

Annual pay versus semi-annual pay:
Many non-US governments and corporate bonds that pay annual coupons.

Annual pay yield = (1 + BEY/2)² - 1.

BEY = [(1 + Annual-pay yield)^1/2 - 1] x 2.

Total Dollar Return = coupons + principal + reinvestment income, i.e. to calculate the FV of the reinvestment coupon + principal at maturity

Reinvestment income
Reinvestment income can make up a large portion of the return for a bond due to its compounding effect.

Factors That Affect Reinvestment Risk
There are two characteristics that affect reinvestment risk:
· The longer the maturity, the more the total dollar return depends on reinvestment income to realize the yield to maturity at the purchase time. Longer maturity = greater reinvestment risk.
· The higher the coupon rate, the more the total dollar return depends on the reinvestment of the coupon payments. Higher coupon rate = greater reinvestment risk

Yield Measurements

Current yield
= Annual coupon / Price

Problem: do NOT consider capital gain/ loss or reinvestment income

Yield to maturity (YTM)
IRR of an investment in the bond that equates the PV of all expected cash flows from bond to its full price.

At par, nominal yield(coupon) = current yield = YTM
At discount, nominal yield(coupon) smaller than current yield smaller than YTM
At premium, nominal yield(coupon) greater than current yield greater than YTM

Note:
· for semiannual bond, IRR is not YTM which is equal to BEY

Assumptions and limitations of YTM:
· Security will be held to maturity
· All intermediate cash flows will be reinvested at the same rate as yield. Higher term to maturity leads to higher reinvestment risk. Higher coupon rate implies higher reinvestment risk.
· All coupon payment are received promptly & timely fashion, ie no credit risk. So YTM is also referred to as promise yield.
· A flat yield curve (rarely exist)

Bond-equivalent yield (BEY)
· Discount rate calculated using six-monthly periods and annualized by multiplying by two.

Realized return (annualized basis) = YTM =BEY

Equivalent annual yield (EAY) = (1+YTM/2) ² for semiannually paid bond

Yield to call
Value bond on the basis that it will be called on first call date. Market conversion is to use the lower /more convervative measure of yield (YTM or YTC) as the appropariate indicator of value.

For premium bond, YTC smaller than YTM
For discount bond, YTM smaller than YTC


Yield to put

Yield to worst(YTW)
The lowest expected return among YTC, YTP, YTM etc.

Cash flow yield (CFY)
Based on the cash flows that include an assumption regarding prepayment rate. Useful for MBS that have monthly payments. The estimated rate of principal repayment may be different from the actual rate of prepayment.

EAY= (1 + Monthly CFY)¹² - 1

Sources of return for a bond

Sources of return includes:
· Coupon
· Capital gain or loss
· Income from reinvestment of interim cash flows

Total return = Coupon interest payments + Reinvestment income + Capital gain/loss.