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R43: UNDERSTANDING FIXED-INCOME RISK AND RETURN, Interest rate risks -…
R43: UNDERSTANDING FIXED-INCOME RISK AND RETURN
1. Sources of returns, Durations
3 Sources of returns
Reinvestment
from coupon
Interest rate risk
Gain/loss
if sold prior to maturity
Interest rate risk
Coupon
+
Principal
Credit risk
Carrying value
vs
Selling Price of the bond
Duration
(cmt)
Modified duration
ModDur = MacDur/(1+YTM)
=> %∆PVFull ≈ –ModDur× ΔYield
=> change 1bp thì x% change in PV full (giá của bond)
(
Nhớ có dấu trừ
=> thể hiện tính ngược chiều)
Approximate Modified duration
Money duration
Macaulay Duration
đơn vị là ngày/tháng/năm/period
Effective duration
For embedded-option bonds
Bởi vì uncertain CF => dẫn đến uncertain YTM => Không dùng YTM được mà phải dùng đến benchmark
3. Interest rate risk, Money Durations
Key rate duration
Macaulay and Modified duration
is an adequate measure of bond price risk only for
parallel shifts
in the yield curve.
For
nonparallel shifts
in the yield curve, we must use Key rate duration to measure the interest rate risk of the bond.
Bond’s maturity, coupon, and yield level affect its interest rate risk
Effect of the fraction of the coupon period (t/T)
:
The Macaulay duration
declines smoothly
and then
jumps upward
after the coupon is paid.
Effect of other properties of bond duration
Effect of time-to-maturity on interest rate risk
Generally, for the same coupon rate, a
longer-term
bond has a
greater higher duration
→
more interest rate risk
than a shorter-term bond when their yields-to-maturity change by the same amount.
The exception is for the
longer-term discount bond
(interest rate risk may be less than a short-term bond) and
perpetuity bond
(interest rate risk is constant over time to maturity)
Effect of coupon rate on interest rate risk
All else being equal, a
lower-coupon bond
has a
higher duration
and
more interest rate risk
than a higher-coupon bond.
Effect of yield-to-maturity on interest rate risk
All other things remaining the same, a
lower yield-to-maturity
bond has
higher duration
and
more
interest rate risk than a higher yield-to- maturity bond.
Interest rate risk characteristics of an embedded option bond
Value of a noncallable bond = Value of a callable bond + Value of the embedded call option
Vì
call option favors the issuer
nên họ pay less
Value of a non-putable bond = Value of a putable bond - Value of the embedded put option
Vì
put option favors the investors
nên họ phải pay more
The presence of an embedded option (be it a call or a put) reduces the duration of the bond and makes it less sensitive to changes in the benchmark yield curve, assuming there is no change in credit risk
Duration of a portfolio &
Limitations of portfolio duration
Limitations of portfolio duration
This measure of duration assumes a
parallel
change in the yield curve → this is
not accurate
in practice because portfolios of bonds are composed of a variety of bonds that may have
different maturities
,
credit risks
, and
embedded options
.
Calculation
the
weighted average
of the
individual bond durations
that
comprise
the portfolio.
=> MacDur of portfolio = w1D1 + w2D2 +...+ wNDN
Money duration
MoneyDur = annual ModDur × PVFull
∆PVFull = – MoneyDur × ∆Yield
is a measure of the
price change
in
units of the currency
in which the bond is
denominated
.
Price value of a basis point (PVBP)
estimates the
change in the full price
of a bond in response to a 1 bps
change in its YTM
≈ 0.01%
PVBP = (PV− − PV+) / 2
2. Convexity & Yield Volatility
Bond convexity
%∆PVfull ≈ −AnnModDur x ∆Yield + [1 x AnnConvexity x (∆Yield)]^2 (
Nó cộng thêm 1 cái premium khác ở bên phải
)
ApproxCon= {
(PV_) + (PV+) −[2x PV0]
} / [(∆Yield)2 x PV0]
Money convexity
MoneyCon = AnnConvexity x PVfull
b. Estimate the price change using money convexity
Factors that affect convexity
Term-to-maturity ↑ ↑
Coupon rate ↓ ↑
YTM ↓ ↑
Dispersion of cash flow ↑↑
Trái phiếu
càng lồi
sẽ càng
outperform
trái phiếu ít lồi trong cả thị trường
bull hoặc bear
.
Effective convexity
Term structure of yield volatility
(Cấu trúc kỳ hạn => so sánh giữa kỳ hạn dài vs kỳ hạn ngắn)
It could be the case that a
shorter-term bond
has more
price volatility
than a
longer- term bond
with a greater duration because of the
greater volatility
of the
shorter-term yield.
=> Tức là PV của short-term bond dễ bị thay đổi hơn long-term bond => Bởi vì những cái lợi nhuận của short-term nó sensitive hơn
Bond’s
holding period return
, its
duration
, and
the investment horizon
Vì thời gian nắm giữ
ngắn
=>
Market price risk
dominated
coupon reinvestment risk
=> Bởi vì đâu có trả nhiều coupon để mà reinvestment
=> Interest rates
TĂNG
→
LOWER
total return
Nếu thời gian nắm giữ
dài
=> The
benefit
gained by
coupon reinvestment
offsets the
risk of market price
decrease.
Investors gặp risk từ lãi suất dựa vào 2 yếu tố:
Thời gian đầu tư (Investment horizon)
Duration gap (So sánh investment horizon đó với MacDur)
Duration gap = Macaulay duration - Investment horizon
Duration gap > 0 => Market price risk > coupon reinvestment risk => SỢ lãi suất TĂNG
Duration gap < 0 => Market price risk < coupon reinvestment risk => SỢ lãi suất GIẢM
Corporate bond
Components of yield
Gov benchmark yield
Expected real interest rate
Expected inflation
Spread on top of that benchmark
credit risk
liquidity risk
Empirical duration
VS
Analytical duration
Analytical
using mathematical formulas
Empirical
using historical data in statistical models
Interest rate risks
Reinvestment risk (Interest rate tăng ≈ Reinvestment
income
)
Market price risk YTM ( interest rate tăng => Capital
loss
)
Scenario
Interest rates increase => Buy & Hold
Interest rates decrease => Short-term trade
Interest rates unchange
