The post by vineviz quoted below presents several examples of cash flows, each of which he says has an investment time horizon of 15 years (under the assumption that the real discount rate is zero, so that the present value of each cash flow is equal to its future value). Each individual payment of a series has its own individual time horizon, but he calculates an average investment time horizon for a series of payments using a weighted average of all the cash flows.
vineviz wrote: Tue Feb 16, 2021 1:24 pm
Investors who have a clear understanding of their financial time horizon are able to make better decisions. Because “time horizon” is an investment term that is sometimes poorly or even incorrectly defined, I thought it might be helpful to review the topic.“Investment time horizon” is very much related to the concept of “bond duration*”, since both terms share a similar definition: the average time it takes to receive (or distribute) all the cash flows of an asset (or liability), weighted by the present value of each of the cash flows.
We generally talk about the investment horizon in long enough time frames to safely use years as the unit of measure so your investment horizon might be 3 years, 20 years, etc. I’m going to present four examples of different cash flow streams, ALL of which have a duration of 15 years (for the purposes of illustration I’m using a real discount rate of zero, so the present value of each cash flow is equal to its future value).
Note that it is NOT actually necessary to have perfect knowledge about your future cash flows to benefit from this exercise. Any financial plan will require making a certain number of estimates and assumptions, and estimating your investment horizon is just one of them. Uncertainty is easily accommodated when making such estimates, nothing about planning your financial goals prevents you from updating your estimates and assumptions when new information comes to light. Investment horizon can be used to help determine your appropriate average bond duration, your stock/bond allocation, etc.
The simplest case has just one single cash flow in the future. In this case, the investment time horizon is simply the amount of time until that cash flow. For instance, if you borrow $100 from a friend and promise to pay them back in full in 15 years, the time horizon of that promise is 15 years since 100% of the cash flows are in year 15.
Only slightly more complicated is a case involving a series of equal cash flows evenly spaced across a period of time. For instance, imagine you are planning for a series of nine annual expenses starting 11 years from now and ending 19 years from now. These nine cash flows also produce an average time, and thus investment horizon, of 15 years.
When the timing or amount of cash flows is uncertain, you can apply a probability weighting to each possible outcome. For instance, let’s say there is a household appliance or car that you will need to replace at some point in the future. You establish that there is a 40% chance you’ll need to do that in 14 years, a 30% chance you’ll need to do it in year 15, a 20% chance you’ll need to do it in year 16, and a 10% chance you’ll need to do it in year 17. This also produces an investment horizon of 15 years since 15 = (.4 x 14) + (.3 x 15) + (.2 x 16) + (.1 x 17).
These various scenarios can be combined into more robust analysis if need be. Imagine you expect to retire in five years but will delay claiming Social/Security or a pension until year 8. This produces a greater need for income in years 5,6, and 7 than in years 8 and beyond. Let’s also say that you wish to assume for planning purposes for a 100% chance of living through year 26 but a diminishing chance of living each successive year after that. Maybe this provides the margin of safety around exceeding your life expectancy that are comfortable with. For illustration let’s assume that years 5, 6, & 7 have expected cash flow needs roughly 2.4 times the needs for years 8 through 26. And let’s assume you accept an 80% chance of living to year 27, a 60% chance of living to year 28, a 40% chance of living to year 29, etc. That produces a stream of expected cash flows that looks like this, with an investment horizon (weighted average time to cash flows) of 15 years.
* There are actually several slightly different ways to estimate the duration of a bond, but for the purposes of this discussion we can safely treat them all interchangeably.
If we apply the same math to a uniform 30-year series of equal annual payments (a typical 30-year TIPS ladder), the average investment time horizon is 15.5 years. (Fifteen payments occur before year 15.5 and 15 payments occur after 15.5 years.) At time zero, the weighted average time remaining for the 30 equal payments is 15.5 years. Under the zero-discount rate approximation, the average investment horizon of an N-year uniform ladder is (N+1)/2.
Furthermore, vineviz indicates that his calculation of “average” investment time horizon is equivalent to a calculation of (Macaulay?) duration, “since both terms share a similar definition: the average time it takes to receive (or distribute) all the cash flows of an asset (or liability), weighted by the present value of each of the cash flows.” So, neglecting the discounting of present value of the payments, the average duration of an N-year uniform ladder is (N+1)/2.
I took a look at the Macaulay duration reported by tipsladder.com for uniform ladders of various lengths. I assumed that the bonds used are the earliest to mature each year and excluded interest payments in 2026. The first bond to mature is in January 2027 (approximately 14 months from now).
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Ladder Length (N yrs) (N+1)/2 Macaulay Duration
5 3 3.10
10 5.5 5.48
15 8 7.73
20 10.5 9.78
25 13 11.75
29 15 13.25 As might be expected, the duration of short ladders is very close to the undiscounted approximation (N+1)/2 but shortens noticeably for longer ladders where the actual present value of payments in the distant future is significantly reduced. I conclude from this exercise that when using duration-targeting of two bond funds to simulate the performance of a ladder, the average duration used should include an appropriate discount rate, as Kevin has pointed out.
