R43: UNDERSTANDING FIXED-INCOME RISK AND RETURN
1. Sources of returns, Durations
3. Interest rate risk, Money Durations
2. Convexity & Yield Volatility
3 Sources of returns
Reinvestment from coupon
Gain/loss if sold prior to maturity
Coupon + Principal
Credit risk
Interest rate risk
Interest rate risk
Carrying value vs
Selling Price of the bond
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
Duration
(cmt)
Modified duration
Money duration
Macaulay Duration
Effective duration
đơn vị là ngày/tháng/năm/period
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
For embedded-option bonds
Bởi vì uncertain CF => dẫn đến uncertain YTM => Không dùng YTM được mà phải dùng đến benchmark
Key rate duration
Bond’s maturity, coupon, and yield level affect its interest rate risk
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.
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
Effect of coupon rate 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)
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
Calculation
the weighted average of the individual bond durations that comprise the portfolio.
=> MacDur of portfolio = w1D1 + w2D2 +...+ wNDN
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.
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
Bond convexity
Money convexity
MoneyCon = AnnConvexity x PVfull
b. Estimate the price change using money convexity
Factors that affect 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]
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
Spread on top of that benchmark
Expected real interest rate
Expected inflation
credit risk
liquidity risk
Empirical duration VS Analytical duration
Analytical
Empirical
using mathematical formulas
using historical data in statistical models