Lecture 6

Question 1

What is a bond?

Answer 1

A debt security issued by a borrower which promises coupon payments
(C) to the lender until specified maturity date (T) and pays out the par value
(F) at the maturity date.

Question 2

What is the par value, coupon rate and maturity date?

Answer 2

Par value: debt amount, usually fixed, i.e $100 or $1000

coupon rate: a pre-determined % of par value

maturity date: “death” of the bond. This is
specified in the contract between borrower and lender

Question 3

Why is a zero-coupon payment bond priced significantly below its par value?

Answer 3

Because the holder will only receive the par value of the maturity of the bond. E.g. a zero-coupon bond with a par value of $1000 might be sold for 950, such that earnings at maturity are $50.

Question 4

What is the difference between the clean price and the dirty price of a bond?

Answer 4

The clean price: Also called the flat price, this is the price of the bond excluding accrued interest.

Dirty price: Also called the invoice price, this is the price that a buyer actually pays for the bond, which includes the clean price plus accrued interest.

Question 5

Accrued interest formula

Answer 5

Question 6

Suppose:
Par value = $1,000.
Coupon rate = 5% annually (C=$50).

Bond pays semi-annual coupons ($25 every 6 months).

90 days have passed since the last coupon payment, and the next coupon is in 90 days (180 days total).

If the clean price is $980, what is the dirty price of the bond?

Answer 6

Dirty price:
Accrued interest = 25 * (90/180) = $12.25

Thus, dirty price = 980+12.25 = $992.25

Question 7

Please explain the time value of money with relation to bonds

Answer 7

As a dollar today is worth more than a dollar tomorrow, investors estimate the future value of coupons and principal repayments in today’s dollars. Hence, the investor discounts the cash flows against their expected rate of return

Question 8

What is the price for the bond?

Answer 8

Question 9

Suppose:
Coupon payment: $50 annually
Par value: $1000
r = 5%
T = 3 years

What is the total bond price?

Answer 9

Question 10

Please describe the relationship between P, C, T and r

Answer 10

An increase in C will result in an increase in P. An increase in T will lead to an increase in P.

An increase in r will lead to a decrease in P. This is because if you are making more money from keeping it in the bank, investors will sell their bonds to do so.

Question 11

What is Yield to Maturity?

Answer 11

YTM is the IRR for a bond. It equates the bond’s current price (P0) to the present values of its future cash flows. The formula is the same as the pricing of bonds formula, here you simply solve for ‘r’.

Question 12

Consider 8% coupon bond (coupons paid semiannually) with maturity of 1 year currently trading at $1,019. Given the current price of the
bond (P0), solve for r (which is a semiannual rate).

Answer 12

Question 13

What is Current Yield? Please also provide its formula and provide the key difference from YTM

Answer 13

The CY is the annual coupon payment divided by the current bond price. It provides a quick snapshot of a bond’s return but does not account for capital gains or losses.

CY = CurrentPrice / AnnualCouponPayment

CY only considers current coupon income, not the capital gain/loss that occurs when the bond is held to maturity.

Question 14

Consider the 30-year 8% coupon bond and interest rate was set at 8% at that time. Today interest rate increases to 9% and 2 years left before maturity.

What is the effect?

Answer 14

The bond’s price falls because its fixed coupon payments become less attractive compared to the higher market rates.

Question 15

What is the holding period return? Please also provide its formula

Answer 15

HPR measures the total return earned on a bond over a specific holding period. It includes:

Coupon payments received during the period.
Any price change in the bond over the period.

Question 16

Calculate HPR for the following example:
Coupon = 80
Par value = 1000
Remaining maturity (T) = 2 years
Market interest rate = 9%

Answer 16

Question 17

Now assume that the market interest rate falls to 7% at T-1. Calculate HPR

Answer 17

Question 18

How will changing interest rates affect HPR vs YTM?

Answer 18

HPR is affected by changes in interest rates whereas YTM is not. If interest rates rise, the price of the bond will fall, causing HPR to also drop.

As YTM is calculated when the bond is purchased, all things are assumed to remain constant as such it is not ‘directly’ impacted by interest rates like YTM. As such, if interest rates remain fixed throughout the life of the bond, YTM = HPR.

Falling interest rates increase bond prices, boosting HPR above YTM.

Rising interest rates decrease bond prices, reducing HPR below YTM.

Question 19

What are the factors that determine the safety of bonds?

Answer 19

Coverage ratio (fraction of firm’s earnings relative to fixed costs).

Leverage ratio (debt-to-equity ratio)

Liquidity ratio (current assets/current liabilities)

Profitability ratios (ROE or ROA, earnings or net income divided by total assets)

Question 20

What is the default premium?

Answer 20

the difference between the promised yield on a corporate bond and the yield of another identical government bond (e.g. Treasury bond) that is riskless in terms of default. The greater the default risk, the lower the bond’s price and the higher the compensation required by investors.

Question 21

What is a yield curve and which different types exist?

Answer 21

The yield curve shows the relationship between yield to maturity and time to maturity of bonds (T).

It plots YTMs as a function of the bond’s maturity, giving insight into the cost of borrowing and investor expectations about future interest rates.

Pure yield curve
On the run yield curve

Question 22

What is a pure yield curve / on the run yield curve

Answer 22

Pure yield curve:
Constructed using zero-coupon bonds with different maturities.
Zero-coupon bonds are simpler because they provide a single cash flow at maturity, making it easier to compute yields without the effect of coupon payments.

On-the-run yield curve:
Based on recently issued coupon-paying bonds selling near par value.
Includes bonds with regular coupon payments, so the yield curve reflects current market conditions.

Question 23

What is the STRIPS program, please explain bond stripping and reconstitution and how this can be an arbitrage opportunity

Answer 23

The STRIPS program involve zero-coupon bonds created by selling each coupon or principal payment from a whole treasury bond as a seperate cash flow.

Bond stripping is when a bond is purchased and then each coupon is sold off as individual zero-coupon bonds. This can be done when the price of the bond is lower than the stripped off cash flows (arbitrage)

Bond reconstitution is when an investor buys the individual zero-coupon securities, reassemble the cash flows into a coupon bond and sell the whole bond for more than the cost of the pieces.

Question 24

Two investment options: (1) buy 2-year zero coupon bond with YTM of 6% at
$1000/1.062 = $890 and hold it until maturity or (2) buy 1-year zero coupon
bond with YTM of 5% at $890, hold it until maturity and buy another 1-year
zero coupon bond.

Assume that interest rates are known with certainty – both investments must
have the same return since there is no risk.

With that in mind, calculate r2.

Answer 24

Question 25

Describe the implications of r2<r1 and vice versa on the yield curve

Answer 25

If r2<r1, the yield curve is upward sloping and the expectation is that interest rates will rise in the future.

If r2>r1, the yield curve is downward sloping and the expectation is that interest rates will fall in the future.

Question 26

assume that r1 = 5%, but
E[r2] = 6%. Suppose that investor wants to invest only for 1 year. Two options:
(1) buy 1-year bond and get 5% return, (2) buy 2-year bond and sell it in 1 year.

Please calculate the two returns and discuss the implications of buying the two alternatives from a risk-averse investor’s perspective

Answer 26

Question 27

Please list the 6 affects of interest rate sensitivity

Answer 27

Question 28

Please distinguish between:
r2, E[r2] and y2

Answer 28

r2 is the actual interest rate between year 1 and 2. r2 only becomes known at the end of year 1.

E[r2] is the expected interest rate at T-2.

y2 is the yield to maturity for a two year bond, using the actual interest rate for r1 and the expected rate E[r2] for second year

Question 29

Please describe the difference between forward rates and expected rates. Also, describe liquidity premium and the various market assumptions

Answer 29

Forward rate is the implied interest rate calculated form spot rates (y1,y2, etc.). This is a break even rate that makes two investment strategies equivalent.

Expected rate is the market’s expectation of the future short rate (r2). This depends on macroeconomic factors and may include biases.

Forward rates may include a liquidity premium to compensate investors for holding longer-term bonds.

Question 30

Please explain the key principle of interest rate sensitivity and the asymmetry in price changes.

Answer 30

There is an inverse relationship between YTM and price of the bond. As YTM increases the price of the bond decreases and vice versa.

An increase in YTM results in a smaller price decrease than the price increase from a decrease in YTM of the same magnitude. This reflects the convexity of the bond price-yield curve (prices rise more steeply than they fall for the same yield change).

Question 31

What is the formula for duration?

Answer 31

Question 32

What are the determinants of duration?

Answer 32

1 Duration of zero-coupon bond equals its T.

2 Duration is lower when the coupon rate is higher.

3 Duration generally increases with T. When YTM is much higher than C, D falls.

4 Duration is higher when YTM is lower

Question 33

Please describe convexity and provide its formula

Answer 33

Question 34

Using immunization calculate the following example:
A company must make a payment of $100 in 1.5 years. The market interest rate is 10%. Manager thinks of investing in a zero-coupon bond with maturity of 1 year and in 3-year 8% coupon bond. What proportion of wealth should you allocate to each bond?
How many units of each bond to buy?

Answer 34

1) Determine the Dobl: 1.5 years
2) Calculate the duration of each bond: Bond 1: D0 is zero coupon payment so matures in 1 year. D0 = 1

Bond 2:
Calc. PV: 8/1.10 + 8/1.10^2 + 108/1.10^3 = 95.38
Duration D3:

Solve for weights:
w1 x D0 + (1-w1) x D3 = Dobl
SUBSTITUTE:
w1 x 1 + (1-w1) x 2.72 = 1.5

w1 = 0.709
w2 = 0.291

Question 35

Please outline the following:
Trading on forward rates
Trading on level of the terms structure
Trading on slope of the term structure

Answer 35

Trading on forward rates: if the bond’s predicted future yield is lower than forward rate, its price will have risen and return will be higher. Hence, fixed-income traders will buy a bond if their expectations of future yields are
below the forward rates implied by yield curve.

Trading on level of the term structure: if you anticipate interest rates to rise,
short any kind of bonds or bond futures. “Level” traders try to predict what
CB is doing, where economy is moving (most notably, inflation and output
growth). Example: buy bonds in countries that are likely to have falling interest rates and short bonds in countries with rising (expected) interest rates.

Trading on slope of the term structure: “curve steepener” - buy 2-year bonds while shorting 10-year bonds. It profits if the yield of 2-year bonds falls relative to the yield of 10-year bonds, steepening the yield curve.

Question 36

What is the butterfly trade?

Answer 36

sell blue cross bond and buy two black cross bonds. This strategy can
hedge both the risk of changes in level (duration matched) and slope (both short- and long-term bonds) of term structure. However, if kink widens, arbitrageur loses money