Chapter 12

1. Chapter 12 Portfolio-Management Techniques

2. Overview • The purpose of this chapter is to examine some common portfolio-management techniques. In particular, we will look at: • Match-Funding • Immunization • Horizon Matching • Indexation • Portfolio Insurance

3. Overview • Let’s begin by discussing the economic problem, and the constraints faced by the manager of a fixed income portfolio. • First of all, what is meant by portfolio management? Basically it means tailoring the risk/return profile of the portfolio to match a set of objectives given a further set of constraints. • The objectives will largely be determined by the type of fund that is being managed. For example, a growth fund will tend to invest in riskier assets than will a conservative wealth-preservation fund. • Different types of funds will face different constraints. For example, a pension fund manager will have a very different set of constraints than will a mutual fund manager. • Some portfolio managers are “public” in the sense that they are directly owned by members of the public – such as a mutual fund. Others may be a manager within a larger entity – for example the mortgage investment manager for a bank, or the Treasurer for a corporation.

4. Overview • So who are some major fixed-income portfolio managers? • Pension Funds – Talk about the liabilities of the pension fund – the payments it will make in the future – also talk about the assets – the cash it will receive now and the investments it makes with that cash. Explain difference in defined benefits and defined contributions. Explain why defined benefits are frequently under-funded – either in Accumulated Benefit Obligations (ABO), what is owed today, or in Projected Benefit Obligations (PBO), what will be owed. • Insurance Companies – The fund must have a minimum level of assets to meet policy payouts. The goal, however, is to insure that it has more than that so as to make a profit. Assets- cash and investments, Liabilities – Long term payouts.

5. Overview • So who are some major fixed-income portfolio managers? • Mutual Fund Companies – Fixed income funds designed to make a profit for the investors in the fund. Probably the least restricted portfolios. • Banks and Financial Institutions – Liabilities – bonds and deposits, Assets – loans they make and instruments they invest in. Many portfolios within the bank – typically have both an investment portfolio and a loan portfolio. Frequently will break the investment portfolio into manageable chunks. Must be managed to protect the FDIC, and to return a profit to the shareholders.

6. Overview • The Chart 12-6 on page 375 nicely summarizes the various risks and constraints: • Liability Characteristics – What happens if they do not meet the liability? • Tax Status – Tax-exempt entities probably are not worried about investing in tax-free bonds! • Charter and Legal Restrictions • Operating Policies • Each of these can affect and even determine the type of portfolio-management techniques that you use.

7. Overview • We can now begin to examine each of the various strategies.

8. Match Funding • The idea is that if you have a stream of liabilities occurring through time, you set up a portfolio of assets at time 0 that is guaranteed to exactly match your liability cash flows. This is commonly used with the cost of failing is very high, or when the cost of actively managing the portfolio is high. • Let’s examine example 12-1. You have liabilities of $100 each in years 1, 2, and 3. You have a choice of 6 bonds to chose from, three coupon bearing bonds and three zero coupon bonds.

9. Match Funding • If the market is in equilibrium – such that you could perfectly replicate the price and yield of the coupon-bearing bonds by judiciously purchasing the zero coupon bonds, then the optimal strategy is to buy one unit each of the zero coupon bonds. • As the book demonstrates in Example 12-2, it is possible to treat this more formally as a linear programming problem. I have set up a MAPLE program to solve equation 12-2 • Formally we can set this up as:

10. Match Funding • Note the Example 12-3 they change the prices of Bonds 1, 2, and 3, as well as the cash flows of bond 1. In essence they have made the coupon bonds cheaper. This results in a slightly different linear program, which you can see in my Maple code.

11. Match Funding • Timing Mismatches • Frequently it will be the case that the best assets to use will have maturities that are sometime before when you need the cash. As a result, you need to do something with the money. What you assume will affect your reinvestment rate. To set this up as a linear program, we simply adjust the constraints:

12. Match Funding • Note in the previous equation that R is the assumed reinvestment rate, and there is nothing that says that rate must be constant across years. • Again, you can look at the MAPLE code to see an example of a linear program to solve this.

13. Immunization • Match Funding has the advantage of being relatively easy to understand – and also easy to monitor (important to a regulator or a more senior manager.) It has the downside of highly restricting what you can do with the portfolio. • If you view the portfolio management job as a “cost center”, then match funding tends to be what you do. What this means is that you are not expecting the portfolio manager to try to exploit opportunities that they may see in the market – it’s a “fire and forget” strategy.

14. Immunization • Other times a portfolio management job is more of a “profit center”, that is, the manager is supposed to manage the fund not only to meet the liabilities, but also to make a profit doing so. In such a case a common approach is to use an “immunization” strategy, sometimes called a “duration matching” strategy. • There are three basic requirements for duration matching: • The present value of the assets must match the present value of liabilities. • The duration of the assets must match the duration of the liabilities. • The assets must have a dominance patter over the liabilities for the prescribed changes in yields. In essence this means that the convexity of the assets must be greater than the convexity of liabilities.

15. Immunization • Two very important points to keep in mind: • You are not cash-flow hedged. You may have timing risk. • Since duration changes (at different rates) over time, you will most likely have to rebalance your portfolio through time – this is costly. • Following the book, we start off with simply hedging the price risk, i.e. duration hedging. We simple insist that the duration of the asset equal the duration of the liabilities. This is easy to do, since we know that the duration of both the liabilities and the asset (note that he is using Macauly duration.)

16. Immunization • Basically he simply sets up the following equality: • Putting in the appropriate numbers:

17. Immunization • So what happens if Rates instantaneously change? First, keep this in mind: we are “long” the bond we bought and we owe the liability. What is our time 0 position?Asset: 2.14*75.13 = 160.77Liability: 248.69Net -87.92 • Let’s assume that rate now fall to 9%. The value of the two components of our portfolio will change to the following: • So, what is the net change in our portfolio? Asset: 165.23Liability: 253.13 • -87.89

18. Immunization • Compared with the original position of -87.92, we can see there is very little net change in my portfolio. The $4.44 increase in my liability is almost exactly offset by the $4.46 increase in my liability value. • You could also set it a two-asset hedge so that you perfectly matched price and duration. • Why would you do either of these? One way of thinking of this is that you have a liability and you want to find an asset that will generate the cash flows. • Normally it is the other way around: you want to buy and asset and so you have to sell a liability to generate cash to purchase it. Again, this is a little bit of a backwards way to do it, but we’ll follow the book’s lead.

19. Immunization • You have issued the same liability as above. This has generated cash which you are now going to invest. You decide to invest part of the cash in Strip 1 and part in Strip 3. You will buy n1 units of strip 1 and n3 units of strip 3. • Formally, I want to meet the following two conditions: • 1. At time 0 the present value of the liabilities must equal the present value of the assets: (-1)PL = n1 P1 + n3 P3 • 2. At time 0 the duration of the assets must equal the duration of the portfolio must be zero: Dollar DurationL = n1 DD1 + n2 DD2

20. Immunization • To put this second constraint into the same terminology as was used in Chapter 4 (not 5 as the book erroneously states it), realize that DD = MD*P, so this corresponds to: MDL * PL = n1 * (P1* MD1) + n2* (P2*MD2) • So we can then divide both side through by PL to get: • Thus, we now have a system of equations:

21. Immunization • Which we can rearrange to be: • Rearranging into matrix notation:

22. Immunization • Which I can then solve as follows: • This is then easily solved either by hand or via a spreadsheet or matrix-manipulation system. There is a spreadsheet on my web-page with this equation solved.

23. Horizon Matching • This is really a combination of the match funding and immunization. Typically you will divide your portfolio into a nearer-term component and a longer-term component. • You will then use match funding for the nearer term, and immunizing the longer term. • The reason you do this is because the nearer term has more certainty regarding the cash flows, and because it is easier to find specific cash flow matching assets or liabilities (as needed).

24. Indexation • Indexation simply means that your policy is to replicate an index as closely as possible. Of course, with many fixed income assets one can do this simply by buying the actual asset. • Realize, however, that in many ways this is an operating policy, and not so much a management technique. • Note also that a lot of this section is really important if you are trying to construct a method for measuring the effectiveness of a portfolio manager that is supposed to be following an index. • Is beating the index as bad as underperforming the index?

25. Indexation • You can think of it this way, what you want to do is to minimize the variance between your portfolio and the portfolio that you are trying to mimic. Assume that you are going to purchase a portfolio with the following characteristics: • Now, the book goes through a rather long proof to arrive at the following: RP= x1 R1+ x2 R2 + … +xN RN • And then realizing that since the R values are random variables, then E[RP] = x1 E[R1] + x2 E[R2]+ … +xN E[RN] • And thatE[RP-RI] = x1 (E[R1] – E[RI]) + x2 (E[R2] – E[RI]) + … +xN(E[RN] – E[RI])

26. Indexation • Define the following: σ2j = var(Rj-RI) and σjk = cov(Rj-RI,Rk-RI) • The portfolio variance is then given by

27. Indexation • Typically then the manager is paid based on how closely they match the portfolio, and are penalized for being above or below that rate. • This can lead to an agency problem, since over-performance frequently is not criticized as harshly as underperformance.

28. Portfolio Insurance • The basic idea behind portfolio insurance is to make certain that the portfolio’s value never falls below some base amount. In the case of a pension fund they worry about making sure they meet the Accumulated Benefits Owed (ABO) and/or the Projected Benefits Owed. • Normally portfolio insurance is done through options-like contracts, such as caps, floors, swaptions. We will revisit this issue later.