Question

A portfolio has a Macaulay duration of 15.8368 years and have the following assets all yielding at an annual effective rate of 5%: (a) one 10-year annuity immediate with level payment X; (b) one perpetuity immediate with level payment 0.8X; (c) one 30-year $1,000 par value bond with annual coupon rate of 5%. Calculate X.

Answer 1

Macaulay duration = 15.8368 years

rate = 5%

a)  one 10-year annuity immediate with level payment X

Present value annuity factor at 5% and 10 years = 7.7217349291848 (from PVAF table)

Time Period	1%	2%	3%	4%	5%	6%
1	0.990099	0.9803922	0.9708738	0.9615385	0.952381	0.9433962
2	1.9703951	1.9415609	1.9134697	1.8860947	1.8594104	1.8333927
3	2.9409852	2.8838833	2.8286114	2.775091	2.723248	2.6730119
4	3.9019656	3.8077287	3.7170984	3.6298952	3.5459505	3.4651056
5	4.8534312	4.7134595	4.5797072	4.4518223	4.3294767	4.2123638
6	5.7954765	5.6014309	5.4171914	5.2421369	5.0756921	4.9173243
7	6.7281945	6.4719911	6.230283	6.0020547	5.7863734	5.5823814
8	7.6516778	7.3254814	7.0196922	6.7327449	6.4632128	6.2097938
9	8.5660176	8.1622367	7.7861089	7.4353316	7.1078217	6.8016923
10	9.4713045	8.982585	8.5302028	8.1108958	7.7217349	7.3600871
11	10.367628	9.786848	9.2526241	8.7604767	8.3064142	7.8868746

We can find the PV of asset A = X * 7.7217349291848

b) one perpetuity immediate with level payment 0.8X

PV of perpetuity can be calculated as Amount / rate

Hence PV of asset B = 0.8 * X / 0.05 = 16 * X

c)  one 30-year $1,000 par value bond with annual coupon rate of 5%

We can find the PV using the financial calculator or in excel by putting in the following values,

PV of asset C = PV(rate, nper, pmt, FV) = PV(5%, 30, 50, 1000) = 1000

The formula of Maculay duration is

Macaulay duration = (PV of cash flows * time period) / Value of the asset(X)

15.8368 = X *  7.721734 + 16 * X - 1000 / X

X = 23.721734 * X - 1000 / 15.8368

15.8368 * X = 23.721734 * X - 1000

1000 = 23.721734 * X - 15.8368 * X

1000 = 7.8849 * X

X = $126.82

Hence X is $126.82

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