Understanding Fixed Income Risk and Return (Duration, Convexity etc) Flashcards
What does duration measure?
Duration is used as a measure of sensitivity of a bond’s full price to a change in interest rates.
Duration also measures how long it takes, in years, for an investor to be repaid the bond’s price by the bond’s total cash flows.
YTM assumes
- bond is held to maturity
- no default
- coupon payments are reinvested at the same rate as coupon rate
if interest rates increase…
Bond price decreases, and reinvestment risk increases (higher reinvestment of coupon) BUT capital loss
if interest rates decrease…
bond price increases and reinvestment risk decreases (lower reinvestment of coupon) BUT capital gain
Macauley Duration (MacDur)
Macaulay duration is calculated as the weighted average of the number of years until each of the bond’s promised cash flows is to be paid, where the weights are the present values of each cash flow as a percentage of the bond’s full value.
Example Interpretation: for a single instantaneous move in interest rates of 100bps, it takes 7.0029 periods such that price risk = reinvestment risk. at 7.0029 periods, the investor realizes the original YTM of 10.4%. Now, this is not that useful b/c it’s assuming an instantaneous change of interest rate of 100 bps and no other changes beyond that - which is pretty unrealistic. Modified duration is far more useful
Modified Duration
ModDur = MacDur/(1+r)
r = interest rate per period
Important: ModDur, AnnModDur, ApproxModDur are all simply linear estimates of PV sensitivity to Δyield
AnnModDur
AnnModDur is Mod Dur that’s been annualized. You can use either use ModDur provided you know MacDur or you can use ApproxModDur (which is already annualized)
ApproxModDur
ApproxModDur = (PV- - PV+)/(2 x ΔYTMx P0)
PV- = price of bond when rates decrease
PV+ = price of bond when rates increase
ApproxModDur tells us the %age Δ in PV (the y variable) given a change in interest rate (x variable)
Let’s think back to 9th grade math. To calculate the slope of a straight line, the formula is Δy/Δx. Now, if you want to calculate the %difference in change of y given a 1 unit Δ in x, then the formula would be (Δy/Δx)/original y
Now, let’s apply in terms of ModDur. The y variable would be PV and x variable would be r (interest rates). The % difference in change in PV given a unit change in interest rates would be:
[(PV- - PV+)/(2 x ΔYTM)]/P0
We can rearrange this formula to look more familiar: (PV- - PV+)/(2 x ΔYTM x P0)
ApproxModDur is always annualized
ApproxMacDur
ApproxMacDur = ApproxModDur x (1+r)
r = market interest rates
ApproxConvexity (ApproxCon)
ApproxCon = [PV- + PV+ - (2 x P0)]/[(ΔYTM)2 x P0]
Convexity is a measure of the curvature of the price-yield relation. The more curved it is, the greater the convexity adjustment to a duration-based estimate of the change in price for a given change in YTM.
approximate percentage change in bond price
approximate percentage change in bond price = –ModDur × ΔYTM
%ΔPVfull
%ΔPVfull = (-AnnModDur x ΔYTM) + [0.5 x AnnCon x (ΔYTM)2]
You can also use ApproxModDur for AnnModDur
For AnnCon, you use ApproxConvexity
Money Duration
Money duration = -AnnModDur x PV0
Looks familiar? Of course, b/c money duration tells us the dollar amount of change in bond price given the change in interest rates
MoneyCon (Money Duration convexity)
MoneyCon = AnnCon x PV0
Dollar ΔPVfull
Dollar ΔPVfull = (-MoneyDur x ΔYTM) + [0.5 x MoneyCon x (ΔYTM)2]
bond portfolio duration
portfolio duration = W1 D1 + W2 D2 + … + WN DN
where:
Wi = full price of bond i divided by the total value of the portfolio
Di = the duration of bond i
N = the number of bonds in the portfolio
It’s really just weighted average of each duration
interest rate risk in short investment horizon
interest rate risk > reinvestment risk
interest rate risk is the uncertainty about price due to uncertainty about market YTM
interest rate risk in long term investment horizon
reinvestment risk > interest rate risk
An investor who sells a bond prior to maturity (if the YTM at sale has not changed since purchase)
will earn a rate of return = to YTM at purchase
If the market YTM for the bond, our assumed reinvestment rate, increases (decreases) after the bond is purchased but before the first coupon date, a buy-and-hold investor’s realized return will be
higher (lower) than the YTM of the bond when purchased.
If the market YTM for the bond, our assumed reinvestment rate, increases after the bond is purchased but before the first coupon date, a bond investor will earn a rate of return that is
lower than the YTM at bond purchase if the bond is held for a short period.
If the market YTM for the bond, our assumed reinvestment rate, decreases after the bond is purchased but before the first coupon date, a bond investor will earn a rate of return that is
lower than the YTM at bond purchase if the bond is held for a long period
if MacDur > investment horizon
interest rate risk > reinvestment risk
If MacDur < investment horizon
interest rate risk < reinvestment risk
