Bonds Exam Cheat Sheet

Quotation of Bonds: Bond Prices are in 1/32 nds; i.e.: 101:20 = 101 + 20/32 Coupons are given like 7,25s (s: semiannual payment) Maturity is stated like 08/15/01 (mm/dd/yy) Canada and Europe use decimal system and dd/mm/yy Treasury Strips = Zero Coupon Bonds (=ZCB) Bonds that make no payment till they mature. Strips are created by stripping coupon (C-Strips) and principle payment (P-Strips) from regular Treasury. Strips that mature on same date should have same price Advantages of ZCB: only one payment, no reinvest. risk The Discount Factor for a particular time period gives the value today or the PV of $1 to be received at the end of that time, i.e.: d(t) = discount factor for t years Because of the time value of money the discount factors fall with maturity. Discount factor can be obtained for each maturity date from respective Strips (if enough strips are in the market)  Discount Factor =Strip Price / 100 To compute future values: 1/d(t), i.e., $1 invested for 6 months grows to 1/d(.5) = $1/.9825 =$1.02 in 6 months  Future Values (t) = 1/d(t) Forward loan is an agreement made today to lend money at some future date at an agreed upon rate called a forward rate. The Expectations Hypothesis states that the forward rate is the market consensus expectation of the future interest rate. r(t) = semi-annual compounded rate earned on a six-month loan (t - 0.5 yrs) forward: By definition:

r(.5)  rˆ(.5)

 rˆ(t)  1  2 

2T

 r(.5)   1   ... 2  

 r(t)  1  2 

Trading rich: The discounted CFs of a Bond are worth less than quoted market price of bond Trading cheap: The discounted CFs of a Bond are worth more than quoted market price of bond Replicating Portfolio: Needed for arbitrage - Long cheap Bond and short Rep. Portfolio or - Short expensive Bond and long Rep. Portfolio To set up replication portfolio, each of CFs of the Bond have to be matched by a fair priced bond with exactly that maturity (mimic CFs!!) Start from the last date, then the second last... and calculate the FVs of different Bonds with the different coupons and maturities.Example:

 6.25    %  105.37  F1  0  F2  0  F3  0  F4  100  2     FV4 =102.182 …

now the next one…

6.375     6.25    F1  0  F2  0  F3  100  2 %  F4   2 %  5.375

Spot (Interest) Rates (= Return on ZCB) The spot rate is the rate on a spot loan: Only 2 exchanges of cash  (borrow(+) and repay principal and interest (-))

 100 120    - 1  5.385%   58.779    

 r $ X 1    2

2T

with T  years

Example: 5% annual on a semi-annual bond with a $100 face value corresponds to: 2

 0.05  $100 x 1    $105.0625 2  

YTM of 2-year bond will be below 2-year spot rate - In a downward-sloping curve, any blend of the four rates will be above the two-year spot rate  Coupon Effect on YTM, different YTM but no one is a better bond (same credit risk) Example:

Term structure of Interest Rates (Spots): Relationship between spot rate & maturity Example: Monetary policy can influence short term, but long term is controlled by market.

rˆ(t)  rˆ(10)  2 

A complete specification of return needs to detail: (1) the annual rate, and (2) the number of times the rate will be compounded during the year. Most bonds are semi-annual: the annual rate divided by two is compounded twice.

rˆ2  rˆ1.5  rˆ1  rˆ.5

Bootstrapping: Obtaining discount factors from bonds. Example: 1.) d(0,5): Quoted Price = [CF Coupon + CF FV] * d(0,5)  d(0,5) 2.) d(1): Quoted Price = CF Coupon * d(0,5) + [CF Coupon + CF FV] * d(1)  d(1) 3.) d(1,5): Quoted Price = CF Coupon * d(0,5) + CF Coupon * d(1) + [CF Coupon + CF FV] * d(1,5)  d(1,5)

 ř(t)= semi-annually compounded return from investing in STRIPS (ZCBs) that mature t years from now Example: STRIP due 2/15/11 = $58.779. Today is 2/15/01, therefore there are 20 six-month periods:

When the forward rate is above the spot rate, the spot rate curve upward sloping (rising), since the spot rate [ř(t)=ř(t - 0,5) * r(t)] is a blend of the previous spot rate and the respective forward rate. When the forward rate is below the spot rate, the spot rate curve is falling or downward sloping. Example: Upward sloping Spot curve and correspon. forward rates.

The YTM is in fact a blend of spot rates, and for a bond that pays most value at maturity, the YTM of the bond will be close to the spot rate for the same maturity - In a flat term structure, all spot rates are the same and equal to the YTM - In an upward-sloping curve,

The law of one price: same CF with same risk must sell for same price. Arbitrage (violation of law of one price): Make money with no risk, and no equity (= 0 = square position). Reverse Repurchase Agreement: Selling Short a Bond and lending the Cash received to the original owner of the Bond and receiving the Bond as collateral which is the bond sold in the first place.

 250  130  HPR  2   - 1  6.2% 100   

Yield-To-Maturity (YTM) = rate so that the discounted value of a bond’s CFs at that rate equal bond’s price. Example: T 6 1/4 due 2/15/03 with p = $102:18-1/8

3.125 1 y

2



3.125 (1  y ) 2 2



3.125 (1  y )3 2



103.125 (1  y ) 4 2

 102  18.125 / 32

  c F  1     1 -  y   1  y/2    1  y 2    2T

P(T)  

t 1

c/2



or

F

1  y 2t 1  y 22T

3 ways to compute Bond prices: Example: As of 2/15/01, price of $100 face bond of T’s 14 1/4 due 2/15/02 1.) Returns of ZCB (discount factors):

108  31.5 / 32  7.125 x d(.5)  107.125 x d(1)

2.) Spot Interest Rates:

 YTM is not the same as realized return nor a good measure of relative value. The uncertainty about the reinvestment rates for coupons is named reinvestment risk There is no reason to assume that they will be reinvested at initial YTM. Clearly, the realized return will depend on the future rates. If rates are greater than YTM, the realized return will be higher than the YTM, and vice-versa YTM Formulas: 2T 2T  

P(T) 

 Different rates for different CF maturities Holding Period Return (=Realized Return) Semiannually compounded return from investing $X for T years and having $W at the end:      W 12T  1    d(t)   r  2   - 1   rˆ(t)  2T   1  2    X      and Example: initial investment $X = $100; at end of 15 years $W = $250. semi-annual HPR?

with

C=Coupon payment y=YTM T=Years P(T)= Price of Bond F= Face Value of Bond Useful YTM Formula:

108  31.5 / 32 

7.125 107.125  rˆ(.5)  rˆ(1)  2 1 2 1  2 

3.) Spot and Forward Interest Rates

108  31.5 / 32 

7.125 107.125  r(.5)  r(.5)   r(1)  1  1  2 2   2 

1

 Price increases whenever c>r over period of maturity extension  Price decreases whenever c
P  A[1  (c/2 - y/2)  a(y/2, 2T)]

Yields across fairly priced securities of the same maturity vary with the cash flow structure of the securities 1.) ZCB: YTM = spot rate for that maturity 2.) Par bond: P = FV which implies that c = y 3.) Non-prepayable Mortgage: CFs are all the same one each date  In an upward-sloping term structure, YTMZCB > YTMPAR since the YTMPAR is blend of spot rates  Mortgages have lower YTMs because more of their total value discounted at lower spot rates. In downward-sloping term structure, the inverse will hold.

A= Face amount of the bond, a(y/2, 2T)= annuity factor

The interest due to the seller is called Accrued Interest. The convention is that, at the time of purchase, the buyer pays the seller accrued interest and keeps the coupon when received later. Accrued Interest is used when sale of bond  coupon payment day. The system in bond markets is different to the system in equity markets. The quoted price in bond markets is always net of accrued interest. P = PV(CFs) - AI P = Quoted Price or Flat Price PV(CFs) = Invoice Price or Full Price AI = Accrued Interest With constant yield, the quoted price of a bond does not fall as a result of a coupon payment – in contrast to what happens with stocks

When cash flows do not follow a semi-annual cycle, semi-annual compounding is clearly not suitable Money market convention is the actual/360 convention, or actual/365 convention Example: Say that from 2/15/01 to 8/15/01 there are 181 days. So the final date is 181/365 or .4959 years away. If the discount factor for d(.4959) is .97561, then the market interest payment is

a( y / 2,2T ) 

1 1  y 2 y 2 (1  y 2) 2T

- c = y, P = 1, bond sells at face value Par Bond - c < y, p < 1, bond sells at discount  Disc. Bond - c > y, p > 1, bond sells at premium  Prem. Bond If T  , P = c/y = price of perpetuity

1 1 1   1  2.50% d .4959 .97561

If YTM remains unchanged over a 6-month period, the YTM equals the annual return. Volatility of Bond reduces towards maturity The market interest payment of 2.50% is unique. Depending on the compounding conventions, we will obtain different rates: 6

 rmonthly  1  12   1  2.50%  rmonthly  4.9487%  

rsemi annual  2.50%  rsemi annual  5% 2 181rsimple 360

 2.50%  rsimple  4.9724%

181

 rdaily  1   365   The convention is to discount the next coupon payment at a semi-annual comp. rate even though the payment does not occur in six-month intervals: 1 2.75 2.75 P    .... 1  y / 2166 /181 … to.. full (1  y 2)166/ 181 (1  y 2)166/ 1811

Curve fitting: The set of traded bond prices from which the discount factors are extracted are limited. Therefore, it is necessary to construct an entire, continuous discount function to price cash flows at any interval in the future. A common but unsatisfactory technique is linear yield interpolation.

 1  2.50%  rdaily  4.9798%

Bad Days refers to the fact that some of the days in which coupon payments are due are not business days. Market convention uses a distinction between conventional yields and true yields, which adjusts for the discrepancy in dates

Formula for Discounting (when coupon payment date ≠ sales date):  = fraction of semi-annual period before next coupon, N = number of semi-annual periods until maturity after next coupon C= Coupon and y = YTM

P

N1  c  1 1     (1  y/2)   t  N 1  y 2 2  t 1  (1  y/2)  (1  y 2)

1

with annuity formula: N 1    1 (1  y ) n  1   A (1  y / 2)    t  y  (1  y) n t 1  (1  y / 2)  

DV01: Dollar Value of a Basis Point (0,01%) change in interest rate yield:

1 P 1 dP( y ) DV01      10.000 y 10.000 d ( y)

It involves par bonds and it entails connecting the chosen points with straight lines. Knowing the par yield curve, it is possible to extract the discount factors. Problems: Unrealistic kinks in the spot curve because slope is different before and after a point. Convex regions are often an artifact of the technique used, beyond a certain point, curves should be concave. The forward curve is extremely irregular and has major jumps. It is therefore better not to force a curve to price bonds exactly and attempt to fit a smooth curve to the data A callable bond is a bond that the issuer may repurchase or call at some fixed set of prices on some fixed set of dates. A callable bond is negatively convex (concave) in all but the highest rates Hedging with DV01: DV01 is expressed in a fixed face amount FA = Face Amount of A to be hedged FB = Face Amount of B to be used for hedge

Derivative of the price-rate function at that point  slope of the line connecting the two points used. Usage: Determine the $ amount to hedge

Duration (= risk measure  similar to  for equities): It measures the percentage change in value of a security for a unit change in rates (or 10.000 basis point). When an explicit formula for the price-rate function is available, the derivative may be used.

1 P 1 dP D-  -  P y P dy

D approx

P  P  2Po y

or If Duration is given, a Δ change yield will lead to % price change!

- D  y  P% 

P P

Usage: Sensitivity analysis and risk assessment From Duration to DV01 for Δ change in yield

- D  y 

P  P%  P  DV01 P

Paradigms: parallel yield shifts & fixed CFs Yield based DV01 (yield = interest factor, FV=100): 1 1  2T t c/2

 DV 01   T   10,000 1  y/2 t 1 2 1  y 2t 1  y 22 T 

DV01 

c  1 1  1 10,000  y 2  (1  y / 2) 2T

100

   T   100  c    2T 1  y  (1  y / 2)  

DV01 is the sum of the time-weighted PVs of a bond’s CFs divided by 10,000 multiplied by one plus half yield The meaning is as follows: a one-basis point decline in the bond’s yield changes its price by DV01 Modified Duration (P=Invoice Price)

 1 1  2T t c/2 100 DMod  T   t 2 T P 1  y/2 t 1 2 1  y 2 1  y 2 

1c  1 DMod   2 1 P  y  (1  y / 2) 2T

   T   100  c    2T 1  y  (1  y / 2)  

1% change in yield leads to DMod% change in P of Bond Macaulay duration is a transformation of modified duration

 y DMac  1   DMod  2

1  y/2  c  1 1 P  y 2  (1  y / 2) 2T

   T   100  c     y  (1  y / 2) 2T 1  

Zero Coupon Bonds and a Reinterpretation of Duration: A convenient property of Macaulay duration is that the Macaulay duration of a T-year zero coupon bond equals T;

DMacc 0  T

hence the Macaulay duration of a 6-month zero is 0,5 while that of a 10year zero is 10. Longer-maturity zeros have larger durations and thus greater price sensitivity. Interpretation: its price sensitivity is that of a zero coupon bond with the same duration equal to its maturity. Furthermore, since a coupon bond can be seen as a portfolio of ZCBs, the D MAC of a coupon bond equals duration of its repl. portfolio of zeros. The PV of CFs in the calculation of the DMAC is weighted by its years to receipt bcs. years to receipt are the duration of the corresponding zero in the repl. portfolio.

DModc 0 

T (1  y / 2)

DV01c 0 

T 100(1  y / 2) 2T 1

Zero Coupon Bonds and their Yield-Based Convexity

Cc 0 

T(T  .5)

(1  y/2)

 FA  DV 01A DV 01B

The security with the higher DV01 is traded in smaller quantity than security with the lower DV01. DV01 hedging is due to differences in convexity local; as a result, rehedging the position is necessary. Convexity: measures how interest rate sensitivity (slope  line  duration) changes with rates.

C

1 d2P P dy 2

C approx 

P  P  2 P0

2

Clearly, longer-term maturity ZCBs have greater convexity, as convexity increases with the square of maturity. Hence, the price-yield function of a longer-term ZCB will be more curved than that of a shorter-maturity ZCB. Moreover, since a coupon bond is a portfolio of ZCBs, longer-maturity coupon bonds will tend to have greater convexity than shorter-maturity coupon bonds. Clearly, the convexity formula may be viewed as the convexity of the portfolio of zeros making up the coupon bond The Barbell versus the Bullet: Barbelling refers to the use of a portfolio of short- and long-maturity bonds rather than intermediate-maturity bonds. Need of a liability with duration equals to 9. Two Options: Buy several bonds with durations approximately equal to 9 (“bullet portfolio”) Buy, say, 2- and 30-year securities with combine duration equal to 9 (“barbell portfolio”) Assume the yield curve is flat at 5%. 9-year ZCB will have a duration of 9 and a convexity given by: 2  2.5 30  30.5 9 (9.5) .75  .25  221.30  81.38 ( 1  0.05/2)2 ( 1  0.05/2)2 (1  0.05/2)2 In contrast, the duration of a portfolio constructed with 75% of 2-year ZCBs and 25% of 30-years ZCBs is .75x2+.25x30=9, and its convexity is:

Estimating Price Changes and Returns with DV01, Duration, and Convexity A second-order Taylor approximation for the price of a security after a small change in rate:

P 1  - D y  C (y)2 P 2

In words, the percentage change in the price of a security (i.e., its return) is approx. equal to minus the duration multiplied by the change in the rate plus half the convexity multiplied by the change in rate squared. In general, the duration term is much larger than the convexity one, thus the duration effect dominates. Hence a first-order approximation for the change in price:

P  - D y P

2

DV01 of a portfolio:

 i DV 01   DV 01i

Duration of a portfolio:

P D   i Di P Convexity of a portfolio:

P C   i Ci P

Par Bonds and Perpetuities: The DV01s and durations of par bonds and perpetuities are given by:

DMacP 100 

d (t )  1  at  bt 2  ct 3 where a, b, and c are constant to be estimated. A more advanced technique is piecewise cubic splines, which knits several cubic polynomials so that the spot curve is continuous.

P (y ) 2

o or where d P/dy is the 2nd derivative of the price-rate function. Positive convexity is a basic feature of most fixed-income securities; There are securities that, at certain rates, need not be convex and exhibit negative convexity.Example:Callable bonds, mortgage-backed secur.  2T t t  1 c / 2  1 100 C  T(T  .5)   P(1  y/2)2 t 1 2 2 1  y 2t 1  y 22 T  yield based Short Convexity: The hedged pos. loses whether rates rise or fall  sell volatility Long Convexity: The hedged position wins whether rates rise or fall  buy volatility Convexity in Investmt & Asset-Liability Context Duration of very convex security change dramatically as rates change  exposure of a portfolio to interest rates may change quite sudden. Exposure to convexity is an exposure to volatility. Since Δy2 is always positive, positive convexity increases return so long as interest rate move: the bigger the move, the bigger the gain; vice-versa for negative convexity. Greater protection against interest rate changes by setting duration and convexity of assets equal to those of liabilities  less need for rehedging Value of a portfolio: P  P 2

 1 y / 2  1 1  y  (1  y/2)2T 

 1 1  DModP 100  1 y  (1  y/2)2T 

 1  2T t c/2 100 DMac    T 2T  P t 1 2 1  y 2t 1  y 2  DMac 

FB 

Piecewise cubics The first step is to assume a functional form for the discount function., for instance a polynomial:

DV01P 100 

1  1 1 100 y  (1  y / 2) 2T

The graph of the approximation shows a good improvement reached with the second-order approximation. For less convex securities, both approximations work quite well Par Bonds and Perpetuities: The DV01s, and durations of par bonds and perpetuities are given by:

 1 1  DModP 100  1 y  (1  y/2)2T   1 y / 2  1 1  DMacP 100  y  (1  y/2)2T  DV01P 100 

Duration, DV01, Maturity, and Coupon: A Graphical Analysis: The following figure assumes that all yields are fixed at 5%. Duration: At this yield, the duration of a perpetuity is a horizontal line. This line is a benchmark for the duration of any coupon bond with a sufficiently long maturity. The Macaulay duration of a ZCB is a 450 line since its duration equals its maturity. The premium curve is constructed assuming a coupon of 9%; the discount curve with a coupon of 1%. Duration falls as coupon increases Results: Higher-coupon bonds have a higher fraction of their value paid earlier, hence the higher the weights on the duration terms of earlier years relative to those of later years: they are like “shorterterm bonds” For very deep discount bonds, duration rises above that of a perpetuity and then falls. The major difference between DV01 and duration is that the former is an absolute change in price whereas the latter measures a % change

P  DMod P  DMac DV01   10,000 10,000(1  y / 2)

n For a given duration, bonds with higher prices tend to have higher absolute price sensitivities. While duration almost always increases with maturity, the behavior of DV01s depends on how price changes with maturity. The duration effect tends to increase DV01 with maturity while the price effect can either increase or decrease DV01 with maturity. As yields increase, the derivative and the DV01 fall; duration also falls: increasing yields lowers the present value of all payments, but lowers the present value of the longer payments most, those with higher duration. Vice-versa, decreasing yields, increases DV01 and duration The two have the same duration but the different convexities, since duration increases linearly and convexity increases with the square of maturity; the convexity of the 30-year ZCB compensates for the lower convexity of the two-year zero. The graph on the right shows the price-yield curve of the bullet and barbell portfolios Note that the values of the two portfolios are equal at a yield of 5%. As rates rise and fall, the barbell portfolio with greater convexity will outperform the bullet portfolio Again, the barbell portfolio does not dominate the bullet portfolio; the latter outperforms if rates move by a small amount up or down, while the barbell outperforms if rates move by a large amount Finally, spreading out the cash flows of a portfolio (without changing its duration) raises its convexity

   

The DV01 and duration expressions of a perpetuity provide a limiting case for any coupon bond: if the maturity of a coupon bond is extended long enough, its DV01 and duration will approximately equal the DV01 and duration of a perpetuity with the same coupon

DV01T      

1  1 1 100 y  (1  y / 2) 2T

PT  

c y

c 10,000 y 2

DMod T   DMac T  

1 y

1 y / 2 y

DV01 Results/Description (no graph): The DV01 of par bond always increases with maturity; since price is always 100, the price effect does not come into play. In general, extending the maturity of a prem. bond increases its price; price and duration effects combine to make the DV01 of premium bonds increase with maturity faster than the DV01 of a par bond. The opposite holds for a disc. bond: the duration effect dominates first and DV01 increases then the price effect catches up and the DV01 declines with maturity. The DV01 of a ZCB behaves like that of disc. bond and finally falls to zero, as its PV does with longer and longer maturity. Unlike duration, the DV01 rises with coupon: higher coupons  higher $ prices and absolute price sensitivity.