Group Project – Fixed Income Investment Portfolio

The goal of the project is to design a simple investment strategy using fixed-income instruments and to implement the strategy in a fixed income investment portfolio that provides best possible risk-adjusted excess returns measured against a properly designed benchmark.

Students will work collaboratively in small teams to construct the required fixed income portfolio based on each team’s investment objectives and strategies. Students are required to defend and provide a solid rationale that the portfolio will perform according to their expectations about global fixed income market going forward.

Each team is required to present their portfolio in class 7. The instructor with the rest of the class will play the role of an investment committee. Each team’s task is to deliver an effective presentation in 10 minutes and write an investment report to persuade the “committee” to pursue your investment ideas.

This project is an important component of the course. It allows you to explore investment ideas and apply the knowledge acquired in this and other relevant courses in a practical way. I recommend that you start trading a portfolio of fixed income instruments by the end of Module 2 and improve your portfolio as we progress in the course.

The overall purpose of the project is to convince an investment committee to accept your investment portfolio in the 7th class. Your group’s presentation should be short and concise, approximately ten minutes. It should follow a real-life investment committee format: each group presents its investment portfolio and submits a two-page written report explaining the investment idea, the rationale, and the investment recommendation. In order to provide you with early feedback regarding your final project, you will complete by Module 5 a brief overview of your project.

The deliverables for this project are:

1. **M5** – An investment summary introducing your investment ideas and the rational for your choice. This is an opportunity for you to receive feedback early in the development of your group project.

2. **M7** – **Submit an investment report (a 2-page write-up) and a PowerPoint presentation file, due one day prior to class 7.** Also, write down the names of all group members on submitted materials. An investment report should include the following items:

· Investment Objectives

· Benchmark and ETF selections

· Investment thesis & strategy: what it is and why it works

· Portfolio Construction & Risk Exposures

· Performance results: analyze results and make investment recommendations

3. **M7** – A live presentation that each group will use to defend the investment proposal in the 7th module of the class. This presentation is not a summary of your project. Rather, it is your opportunity to pitch your recommendation. You have one shot with a potential client, this is your opportunity to “sell” your work.

The investment universe for the purpose of this project is the global bond markets using ETFs or individual fixed income securities. You can trade any available ETF in the US including government bond ETFs, mortgage bond ETFs and so on.

The investment time-frame would be for 9 to 12 months. This is the time horizon for which to express your investment outlook. In other words, how would you expect current market conditions to change over the investment horizon?

This time-frame is also representative of the holding period for many pension funds and endowments. It is therefore a time frame that is reasonable, and it allows to keep the transaction costs low because the turnover of the investment portfolio will be low.

You can access public data available for all ETFs. Gathering and cleaning data is an important part of the project. Students are expected to search and gather data from various sources and cross check the validity of the data that they use to produce investment proposals.

For each portfolio proposal, students are required to elaborate on the risk characteristics of the proposal. Important metrics that should be considered include performance statistics and risk factor exposures.

Students are required to defend and provide a solid rationale that the portfolio will perform according to their expectations going forward. This step is probably the most important piece of the project.

In summary, students will be able to build and practically design a fixed income portfolio that institutional-quality investors would consider for their own investments. The closer one gets to a model portfolio that is forward looking and has a solid economic rationale to perform, the better are the chances of getting an investment from an institution.

__In Summary__

· The goal of the project is to construct a fixed income portfolio that consists of fixed-income ETFs or individual securities to outperform a properly designed benchmark, and to present the portfolio to the class.

· You are expected to deliver an effective investment presentation within 10 minutes to convince your peers to follow your investment recommendations.

· This project will allow you to explore investment ideas and apply the knowledge that you have learned in portfolio and fixed income theories in a practical way.

· It is important to start the project early so that you can keep track of how your portfolio would have performed as we progress in the course.

Build for what’s next TM

Fixed Income

Spring 2023

BU.232.720

1

Risk and Expected Return on Bonds

Bond Returns and Yields

Analytical measures of interest rate risk & Hedging

A Yield-based One Factor Framework

DV01, Duration, and Convexity

Effective duration & Convexity

Multi-factor Risk Metrics

Effective (Curve), Key Rate, and Empirical Durations

Today’s Agenda – Chapter 5

2

Bond Returns and Yields

3

Holding Period Returns

HPR = Holding period return

P0 = Beginning price

E(P1) = Expected Ending price

E(D1) = Expected Dividend or Coupon Payment during the time period

Holding Period Return (HPR):

The rate of return earned from holding an investment for a specified time period.

4

Potential Sources of a Bond’s HPR

An investor who purchases a bond can expect to receive a dollar return from one or more of these sources:

the periodic coupon interest payments made by the issuer

interest income generated from reinvestment of the periodic cash flows

any capital gain (or capital loss—negative dollar return) when the bond matures, is called, or is sold

Bond Immunization

5

Computing the Total Return on a Bond

The idea underlying total return computation over an investment horizon is simple.

First, compute the total future value that will result from investing in a bond assuming a particular reinvestment rate, and yield at the end of the time horizon.

Total Future Value = (coupon payments + reinvestment income) + projected sale price

The total return is then computed as the rate of return that will make the initial investment in the bond grow to the computed total future value.

Credit and Reinvestment risk

Price risk

6

Suppose that an investor has a 3-year investment horizon, and is considering purchasing a 20-year 8% coupon bond with semiannual payments selling for $828.4 (par = $1,000.)

What is the total return of the bond at the end of 3 years?

Example: Total Return of a Bond

7

Suppose that an investor has a 3-year investment horizon, and is considering purchasing a 20-year 8% coupon bond with semiannual payments selling for $828.4 (par = $1,000.)

What is the total return of the bond at the end of 3 years?

Example: Total Return of a Bond

Interest Rate Expectations:

The coupon payments will be re-invested at 5% per period

YTM at the end of three years is 10%

8

Suppose that an investor has a 3-year investment horizon, and is considering purchasing a 20-year 8% coupon bond with semiannual payments selling for $828.4 (par = $1,000.)

What is the total return of the bond at the end of 3 years?

Example: Total Return of a Bond

Interest Rate Expectations:

The coupon payments will be re-invested at 5% per period

YTM at the end of three years is 10%

Total Future value of the bond at the end of 3 years:

$40 coupon payments for 6 periods at 5% per period = FV(0.05,6,40) = 272.08

Projected sale price = 838.07 = PV(0.05,34,40,1000,0)

Total = FV(0.05,6,40) + PV(0.05,34,40,1000,0) = 1110.15

9

What is the total return of the bond at the end of 3 years?

Example: Total Return of a Bond

Interest Rate Expectations:

The coupon payments will be re-invested at 5% per period

YTM at the end of three years is 10%

Total Future value of the bond at the end of 3 years:

$40 coupon payments for 6 periods at 5% per period = FV(0.05,6,40) = 272.08

Projected sale price = 838.07 = PV(0.05,34,40,1000,0)

Total = FV(0.05,6,40) + PV(0.05,34,40,1000,0) = 1110.15

Expected (annualized) holding period return y:

(1+y)^3 = 1110.15 / 828.40

y = 10.25%

Why is y greater than 10%?

10

In general, the future value of a bond will change when yield to maturity changes.

For example, based on previous example where investment time horizon T = 3 years:

Yield = 10%: total future value = FV(0.050,6,40) + PV(0.050,34,40,1000,0) = 1110.15

Yield = 10.2%: total future value = FV(0.051,6,40) + PV(0.051,34,40,1000,0) = 1096.83

Yield = 9.8%: total future value = FV(0.049,6,40) + PV(0.049,34,40,1000,0) = 1123.83

So, the total future values are sensitive to changes in yields.

However, when T = Macaulay duration of the bond, you can verify that the total future value of the bond stays constant when yield changes by a small amount.

=> This leads to the concept of Immunization.

Sensitivity of Future Value of a Bond

11

Immunization

The accumulated value of the coupon payments and the projected sale price of the bond are perfectly balanced when investment time horizon = bond’s Macaulay duration.

In other words, price risk and reinvestment risk are precisely offsetting at the end of the time horizon T, if T = bond’s Macaulay duration.

12

Analytical measures of interest rate risk & Hedging

13

Measures of Interest Rate Risk

Money managers, arbitrageurs, and traders need to have a way to measure a bond’s price volatility to implement hedging and trading strategies.

Risk measures (yield based & one factor) that are commonly employed:

Price value of a basis point (PVBP or DV01)

Duration

Convexity

Yield value of a price change

14

PVBP or DV01

PV- and PV+ are the full prices calculated by decreasing and increasing yield-to-maturity by 1bp.

In US, it is commonly called the DV01, the Dollar Value of a 01.

Convert % to basis points: 1 (%) 100 (basis points)

Multiply by 10000

15

A Simplified Example

Consider a 20-yr 5% cpn bond issued at par.

If yield declined to 4.99%, the price of the security will increase from par to $100.12562.

In this case, the DV01 equals $0.12562, that is the price increases by $0.12562 for a 1 bp decline in yield, per 100 of par value.

2

16

Is DV01 larger/smaller: (1) par 10-yr 5% cpn? (2) par 30-yr 5% cpn?

17

DV01 is larger for the 30-yr 5% cpn par bond and smaller for the 10-yr 5% cpn par bond.

18

Exercise 1 PVBP Calculation

Consider a US T-note with 2.875% semiannual coupon payment that matures on May 15th, 2028. Its yield-to-maturity is 2.849091%, and the settlement date is 59 days into a 184-day period on 07/13/2018.

Calculate the following

PV- and PV+

PVBP or DV01

19

Hedging with DV01

20

Example: Hedging with DV01

Assume yields are 5% and a client sells a $10 million face value 20-yr bond at par to a trader

The trader wants to hedge it using a 30-yr bond

The trader should sell (or short) enough of the 30-yr bond so the changes in the value of the two bonds will offset each other

What is the amount to trade in the 30-yr bond to hedge the 20-yr long bond position?

21

Hedging with DV01, cont’d

DV01 for the two bonds are not the same:

The 20-yr bond declines in value by $0.12562 for every 1 bps increase in rates and the 30-yr bond declines by $0.15472.

The trader needs to short:

($10m)*[0.12562/0.15472] = $8.1m of the 30-yr bond

Notes:

Hedging a long position requires a short position in hedging vehicle (i.e. the 30-yr), assuming the DV01 are of same sign.

The security with higher DV01 is traded in smaller quantities than securities with lower DV01.

22

In general, hedging a position of Fp (face amount) of security p requires a position of Fh of security h with:

So that,

Hedging with DV01, cont’d

23

Empirical Hedging

Employing Least-Squares Regression

Typically yield changes of Bond A are regressed on yield changes of Bond B.

Note: stability of the regression coefficients over time could be a issue in practice

24

Hedging with Two Securities

Hedging a 20-yr bond position with a combination of 10- and 30-year bonds.

25

Duration & Convexity

26

A Yield-based One Factor Framework

– (Modified Duration) D

Convexity C

A Second-order Taylor Approximation

27

Modified duration is defined as the percentage change in price for a given change in yield as given by:

The modified duration is related to “Macaulay duration” as follows:

where r = periodic compound rate = yield / compounding frequency.

Modified Duration vs Macaulay Duration

28

Consider a 20-year 8% coupon bond with semiannual payments selling for $828.40 with par value = $1,000. What’s the Macaulay duration of the bond?

Long-term bonds tend to be ___ price sensitive than short-term bonds. (more or less?)

As maturity increases, price sensitivity ___, but at a decreasing rate. (increase or decrease?)

Interest rate risk is ___ related to the bond’s coupon rate. (inversely or directly?)

Price sensitivity is ___ related to the yield to maturity at which the bond is selling. (inversely or directly?)

Concept Check: Macaulay Duration

29

Modified Duration – Revisited

The price of a bond with $100 par value:

This modified duration is calculated in terms of number of periods. To convert the duration in years, use the following formula:

Duration in years = duration in periods / number of periods per year

Modified Duration – An Example

Consider the 25-year 6% semiannual bond selling at 70.357 to yield 9%.

Modified Duration – An Example

Consider the 25-year 6% semiannual bond selling at 70.357 to yield 9%.

Modified Duration vs DV01

Modified duration is related to DV01 per 100 of par value

It is approximately DV01*100, for par bonds

D* = = (10000 / P) DV01

if P = 100 (par bond)

= 100*DV01 for par bonds

Ex. Consider the 20-yr 5% coupon par bond when yield = 5%

DV01 says price changes by $12.562 when rates change by 100 bps which is the same as saying modified duration = 12.56

In other words, 12.56% change in price for a 1% change in yield

33

Modified Duration Can Be Used in Estimating Percentage and Dollar Price Changes

Dollar Price Change

Percentage Price Change

34

113.678/100 – 1 = 13.68%

88.443/100 – 1 = -11.56%

Notes:

D* says percentage price change should have been 12.562% and symmetric up or down when yield changes by 1%

It understates the positive change when yield goes down; and overstates the price decline when yield goes up.

Estimation Error Due to Nonlinearity

35

Convexity Correction Term

Correction for Convexity:

This convexity measure is in terms of periods squared. To convert it to an annual figure, it must be divided by m^2 (where m is the compounding frequency within a year)

36

Measuring Convexity Using Excel

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March 29, 2023

March 29, 2023

Calculation of Convexity Measure and Dollar Convexity Measure for Five-Year 9% Bond Selling to Yield 9% | |||||

Coupon rate: 9.00% Term (years): 5 Initial yield: 9.00% Price: 100 | |||||

Period, t | Cash Flow | 1/(1.045)t+2 | t(t + 1)CF | t(t + 1)CF (1.045) t+2 | |

1 | 4.50 | 0.876296 | 9 | 7.886 | |

2 | 4.50 | 0.838561 | 27 | 22.641 | |

3 | 4.50 | 0.802451 | 54 | 43.332 | |

4 | 4.50 | 0.767895 | 90 | 69.110 | |

5 | 4.50 | 0.734828 | 135 | 99.201 | |

6 | 4.50 | 0.703185 | 189 | 132.901 | |

7 | 4.50 | 0.672904 | 252 | 169.571 | |

8 | 4.50 | 0.643927 | 324 | 208.632 | |

9 | 4.50 | 0.616198 | 405 | 249.560 | |

10 | 104.50 | 0.589663 | 11,495 | 6,778.186 | |

12,980 | 7,781.020 |