Jim Tuck is the trustee of a family trust company, the Tuck Family Trust. The trust was formed by his grandfather Thomas Tuck with the equity of the Tuck magazine company, which he founded, and managed most of his life. The trustees sold the magazine company in 1980 for 70 million dollars, and have been in charge of managing the money since then. The assets have since grown to 100 million, and are held in what is termed a "sprinkling trust" -- it is designed to benefit future as well as present generations. The trust is allowed to pay out only interest income to the beneficiaries, until the last child of Thomas Tuck dies, at which time, the trust is dissolved and it pays out the entirety of its assets -- tax free -- to the beneficiaries.

There are currently 48 beneficiaries who are receiving quarterly checks from the Tuck Family trust, and the 53 children of these beneficiaries are themselves future potential beneficiaries. Jim's fiduciary duties require him to address their needs as well as the needs of the current recipients. There are four living children of Thomas Tuck, and the youngest is 75 years old.

The current allocation of the trust is 40% stocks and 60% bonds. The equities are invested in an S&P indexed account with a dividend yield of 1.8%. The bonds are managed by Weller and Beame Investments. Recent pricing for the securities is provided in the table below. All the bonds are government bonds. About 2/3 of the portfolio is invested in intermediate- term government. The purchase of these intermediate-term bonds was one of Weller and Beame's real coupes. They moved the portfolio into 10-12 year maturities in the summer of 1991 and the prices of these securities shot up by about 10% when interest rates dropped in 1992. The remaining third is invested in treasury bills, some long bonds and a FNMA issue. If the trust sells the intermediate term bonds to lock in profits, it will pay capital gains taxes on them -- at the highest marginal rates.

Jim is faced with difficult family problems. About half of his relatives want the trust to pay out a substantial income, and about half do not. Those who do not need the income from the trust would prefer that the assets be allocated to growth securities that will be transferred to them tax-free when the trust is dissolved. Those who want the current income are feeling constrained by the cash needs of putting children through college, buying homes and other major expenses. Needless to say the two factions rarely agree. Both sides pay taxes on the interest income, and those that have higher incomes are in the higher tax brackets.

Jim is considering the recommendation from Patricia Dow at Weller and Beame that he sell half of the intermediate term bonds now and recognize a handsome profit. She suggests that they take the money and purchase newly issued five year government bonds. He trusts Weller and Beame's advice -- after all they called the interest rate movements pretty well, but he also realizes that they are motivated to sell in order to show a good profit on this year's performance statement.

Note: Assume T-bill face values are the same as T-bonds, i.e. $1,000 each.

Quantity | Security | Rate | Maturity | Bid | Ask | Ask Yield |
---|---|---|---|---|---|---|

500 | T-Bill | NA | 1/4/1996 | 5.39 | 5.79 | 5.58 |

10,000 | T-Bond | 4 | 1/96 | 99:05 | 99:07 | 5.66 |

30,000 | T-Bond | 8 | 1/97 | 102:28 | 102:30 | 5.84 |

5,000 | FNMA | 7.60 | 1/97 | 102:01 | 102:05 | 5.99 |

5,000 | T-Bond | 5 | 1/99 | 96:12 | 96:14 | 6.16 |

5,000 | T-Bond | 7 1/2 | 2/05 | 106:20 | 106:22 | 6.55 |

5,000 | T-Bond | 8 1/2 | 2/20 | 117:08 | 117:10 | 7.01 |

5,000 | T-Bond | 7 5/8 | 2/25 | 108:23 | 108:25 | 6.92 |

The trustee's meeting is next week, and Jim must decide upon a reasonable course of action to recommend to his fellow fiduciaries by then.

**Computing Yields**

Goverment securities have certain conventions about pricing and valuation. Treasury bonds have interest compuonded semi-annually. the price of a coupon bond with N coupons left to pay is:

P = C/2 * 1/(1+y/2) + C/2 * 1/(1+y/2)^2 + C/2 * 1/(1+y/2)^3 + ...+ [C/2 + 100] * 1/(1+y/2)^T

In order to calculate equivalent yields and prices, various conventions must be used.
The *invoice price* of a bond or bill is the price paid for it. The invoice price for a T-bill, using the bid discount yield is:

P = 100 * [1-(n*d)/360]

Where n is the number of days remaining from the settlement date until the maturity of the bill. d is the discount yield at the bid. P is the percentage of the par amount of the T-bill face value.

The invoice price for a T-bond is made more complicated by the need to calculate acrued interest. First, determine the last coupon date: [LCD] it occurs in the same month of the year that the bond matures. Next, calculate the next coupon date [NCD]. This is generally six months later, [i.e. 182 days] since bonds pay semi-annnual coupons. the number of days from the last coupon until the settlement date has interest that accrues at the yield. For a 6% yielding bond with say, 91 days since the last coupon, the accrued interest would be:

91/182 * 1/2 * 6 = 1.5

Thus, if the quoted bid price is say, 101.5, then the invoice price would be 101.5 + 1.5 = 103, which is expressed as a percentage of the pricipal amount.

For T-bills with maturities less than six months, the 360 day year is a poor approximation. A **Bond Equivalent Yield** can be calculated to make its quoted yield comparable to T-Bond yields.

BEY = (365*d) / (360 - d*n)

The calculation is more complex when the T-bill has a maturity greater than six months, since T-bonds have coupon payments and are thus not equivalent instruments.

T-Bonds with less than six months left to maturity are called **short governments**. Take a short government with a 6% coupon and 64 days since last coupon, with a bid price of 100.19 or 100 19/32. The invoice price is 64/182*1/2*6 = 1.68, so the invoice price is 100 19/32 + 1.68 = 101.2746. The yield maturity for short governments is calculated as:

y = [(100+c/2)/P - 1]*2x/z

where x is the days from last coupon to next coupon (182 in the example), and z is the number of days until next coupon (182-64 = 118 in the example). thus, the short goverment would have yield to maturity of:

[(100 + 6/2)/101.2746 -1] *2*182/118 = 5.2554

**Assignment:**

In preparation for the Tuck analysis, address the following:

1) Calculate the portfolio weights for each security

2) Estimate the duration of the portfolio, assuming that the T-Bonds are non-callable, and that the FNMA issue has a similar payment structure as a T-Bond. Are these assumptions reasonable? If not, how would you adjust the duration calculation?

3) Why are the T-Bill and T-Bond maturing in January yielding different rates?

4) Why do the 1/97 FNMA and 1/97 T-Bond have different yields?

5) Use the yields on the securities to plot the yield curve.

Prepare a report for the Tuck trustee meeting. It should address the following questions:

1) What are the tax consequences of the sell decision? Is Jim favoring one faction over the other?

2) Are there liquidity differences that might be a concern upon the sale of the bonds?

3) Is the bond portfolio sufficiently diversified? How safe is the income flow to the beneficiaries? How great is the re-investment risk they face when the intermediate term bonds are due?

4) Suppose that the most risk averse of the beneficiaries cannot afford to have a drop of 20% in his income flow. When would such a drop probably occur, and how likely is it that it would be of that magnitude? How can you characterize the probability distribution of a 20% drop in income from the portfolio?

5) Assume that Jim decides to sell. Given the current shape of the yield curve, what do you recommend he do with the proceeds? You are constrained to purchasing only government bonds with the money.

6) How can you work the liabilities of the beneficiaries into your portfolio management problem. Is the duration of the liabilities of material concern?

7) Can you suggest some kind of side agreements among the parties that might make both faction better off? Here you can use any kind of contracting and/or derivative securities that might help.