Understanding Fixed Income Risk and Return (Duration, Convexity etc)

Question 1

What does duration measure?

Answer 1

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.

Question 2

YTM assumes

Answer 2

• bond is held to maturity
• no default
• coupon payments are reinvested at the same rate as coupon rate

Question 3

if interest rates increase…

Answer 3

Bond price decreases, and reinvestment risk increases (higher reinvestment of coupon) BUT capital loss

Question 4

if interest rates decrease…

Answer 4

bond price increases and reinvestment risk decreases (lower reinvestment of coupon) BUT capital gain

Question 5

Macauley Duration (MacDur)

Answer 5

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

Question 6

Modified Duration

Answer 6

ModDur = MacDur/(1+r)

r = interest rate per period

Important: ModDur, AnnModDur, ApproxModDur are all simply linear estimates of PV sensitivity to Δyield

Question 7

AnnModDur

Answer 7

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)

Question 8

ApproxModDur

Answer 8

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

Question 9

ApproxMacDur

Answer 9

ApproxMacDur = ApproxModDur x (1+r)

r = market interest rates

Question 10

ApproxConvexity (ApproxCon)

Answer 10

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.

Question 11

approximate percentage change in bond price

Answer 11

approximate percentage change in bond price = –ModDur × ΔYTM

Question 12

%ΔPV_full

Answer 12

%ΔPV_full​ = (-AnnModDur x ΔYTM) + [0.5 x AnnCon x (ΔYTM)²]

You can also use ApproxModDur for AnnModDur

For AnnCon, you use ApproxConvexity

Question 13

Money Duration

Answer 13

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

Question 14

MoneyCon (Money Duration convexity)

Answer 14

MoneyCon = AnnCon x PV₀

Question 15

Dollar ΔPVfull

Answer 15

Dollar ΔPV_full = (-MoneyDur x ΔYTM) + [0.5 x MoneyCon x (ΔYTM)²]

Question 16

bond portfolio duration

Answer 16

portfolio duration = W₁ D₁ + W₂ D₂ + … + W_N D_N

where:

W_i = full price of bond i divided by the total value of the portfolio

D_i = the duration of bond i

N = the number of bonds in the portfolio

It’s really just weighted average of each duration

Question 17

interest rate risk in short investment horizon

Answer 17

interest rate risk > reinvestment risk

interest rate risk is the uncertainty about price due to uncertainty about market YTM

Question 18

interest rate risk in long term investment horizon

Answer 18

reinvestment risk > interest rate risk

Question 19

An investor who sells a bond prior to maturity (if the YTM at sale has not changed since purchase)

Answer 19

will earn a rate of return = to YTM at purchase

Question 20

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

Answer 20

higher (lower) than the YTM of the bond when purchased.

Question 21

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

Answer 21

lower than the YTM at bond purchase if the bond is held for a short period.

Question 22

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

Answer 22

lower than the YTM at bond purchase if the bond is held for a long period

Question 23

if MacDur > investment horizon

Answer 23

interest rate risk > reinvestment risk

Question 24

If MacDur < investment horizon

Answer 24

interest rate risk < reinvestment risk