*Photo by Standard Travel Fotos*

A reader writes:

The Vanguard Total Bond Market Index fund has a SEC yield of 1.99% and an average maturity of 7.8 years and average duration of 5.7 years . I can get a 5-year CD with a rate of 2.09% and APY 2.11%. Which is the better choice?

Here we have two kinds of investments: a bond fund and a CD. They have similar yields and slightly different durations. But the behavior of each is fundamentally different.

**Type 0 and Type 1 Investments**

There are basically two kinds of investments. Let’s call them Type 0 and Type 1. The difference between the two is important and very basic, but it is often glossed over in Investing 101 and in pop investing books.

Type 0 investments are guaranteed to pay a certain amount on a certain date. When they are held to that date, there is only one outcome. In Modern Portfolio Theory (MPT) terms, the standard deviation of return is zero, i.e. it is a risk-free investment.

Type 1 investments do not guarantee the amount of money paid on any date. No matter what the holding period, the outcome is uncertain. Instead, there is a distribution of possible outcomes. In MPT terms, these are risky investments.

Examples of Type 0 investments are default-free government bonds like U.S. Treasury bills, notes and bonds, Treasury STRIPS, Treasury Inflation-Protected Securities (TIPS), certain U.S. government agency bonds, and FDIC and NCUA insured CDs.

Everything else is a Type 1 investment: investment-grade corporate bonds, municipal bonds, junk bonds, stocks, gold, stock funds, and bond funds. None of these investments guarantee a certain amount on a certain date. The return of stocks is obviously unpredictable. But how about investment-grade corporate bonds?

AAA Corporate bond will usually pay exactly what you were expecting on the date of maturity. But not always. They can default on payments and investors may only get back 50 cents on the dollar. There is a similar risk with municipal bonds. Since they can default, the payments are not guaranteed, and not Type 0.

A Treasury bond fund, which holds only Type 0 investments, can not guarantee a certain amount on a certain because the bond funds never mature. To maintain a constant maturity funds will sell bonds before maturity at market prices. So Treasury bond funds are Type 1 investments.

What about something like an FDIC-insured savings account? FDIC saving accounts are certainly a safe place to keep your money. But when you deposit $1,000 into a savings account, which is also called a demand account by the way, there is no way to know exactly how much the account will hold in 5 years. It depends on the daily interest rate, which fluctuates. Savings accounts are very safe, but they are not Type 0 investments.

**Bond Fund vs. CD**

Going back to the original question, The choice is between Vanguard Total Bond Market Index Fund (VBMFX), a Type 1 investment, and a 5-year FDIC CD, a Type 0 investment.

A CD will return a certain amount on a certain date. $10,000 invested in a CD for 5-years at APY = 2.11:

11,110.47 $

By constrast, over 5 years, VBMFX has a distribution of outcome. Thus, it makes sense to treat the return of VBMFX over some holding period as a random variable with some mean and standard deviation, because the actual return that will be realized is unknown beforehand. Economists like to say that the return is unknown *ex ante*, but fully known *ex post.*

Now we could model the return of VBMFX as a probability density function (p.d.f), run 1,000 Monte Carlo simulations and see what the distribution of outcomes looks like. A complication is that, ex ante, we don’t know the p.d.f or it’s parameters mean and variance. Let’s make some assumptions about the p.d.f. and create a computer model to run Monte Carlo simulations.

**Monte Carlo Simulation Using Past 5-Year Return**

Suppose we assume that the distribution is a log normal distribution and mean return and standard deviation will be the same as the past five years of VBMFX. We can look up those numbers in morningstar.com at Vanguard Total Bond Market Index Inv VBMFX Performance and Returns. The 5-year mean annual return was 4.00% p.a. and the standard deviation was 2.98%. In R, I coded up a Monte Carlo simulation. ^{1} Here is one outcome from the simulation:

` Year Bal.Start ROR Bal.End`

1 2014 10000 4.639 10370

2 2015 10370 4.981 10886

3 2016 10886 3.310 11247

4 2017 11247 -1.571 11070

5 2018 11070 2.342 11329

For this case, after 5 years $10,000 invested in VBMFX result in $11,329. Compared with the 5-Year CD which gave $11,110, VBMFX came out ahead. But that was only one possible result. Let’s try 10 Monte Carlo runs.

` Bal.Start Bal.End Gain TR TR.annualized`

[1,] 10000 11541 1.154 0.1541 2.909

[2,] 10000 11984 1.198 0.1984 3.686

[3,] 10000 11826 1.183 0.1826 3.412

[4,] 10000 11983 1.198 0.1983 3.684

[5,] 10000 13548 1.355 0.3548 6.261

[6,] 10000 11274 1.127 0.1274 2.427

[7,] 10000 11487 1.149 0.1487 2.812

[8,] 10000 11465 1.147 0.1465 2.773

[9,] 10000 13009 1.301 0.3009 5.403

[10,] 10000 13143 1.314 0.3143 5.618

There were ten cases with various outcomes. They all came out better than the CD. This is not surprising, since the average historical return was 4.00% and CD has APY = 2.11%. Let’s run 1,000 Monte Carlo simulations and summarize the results.

` Bal.Start Bal.End Gain TR TR.annualized`

Min. : 10000 Min. : 10158 Min. : 1.02 Min. : 0.0158 Min. : 0.314

1st Q: 10000 1st Q: 11648 1st Q: 1.16 1st Q: 0.1648 1st Q: 3.098

Medi : 10000 Medi : 12103 Medi : 1.21 Medi : 0.2103 Medi : 3.891

Mean : 10000 Mean : 12141 Mean : 1.21 Mean : 0.2141 Mean : 3.929

3rd Q: 10000 3rd Q: 12570 3rd Q: 1.26 3rd Q: 0.2570 3rd Q: 4.681

Max. : 10000 Max. : 14823 Max. : 1.48 Max. : 0.4823 Max. : 8.190

Both the mean and median annualized return are fairly close 4%, as expected. But there’s a range of outcomes and the worst case return is less than the CD. We can use the quantile function to see how many were worse than the CD’s $11,110.

` 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%`

10158 10695 10848 10913 10988 11063 11098 11156 11202 11250 11275

About 6-7 percent of the time, the bond fund had worse outcome than the CD. With 90+% chance of the bond fund winning, all but the most risk adverse would pick the bond fund over the CD.

**Monte Carlo Simulation Using SEC Yield**

Now someone might argue that you can’t expect to get past returns with a bond fund. Interest rates are lower. You should use something like the Yield-to-Maturity (YTM) as the mean return. There’s a figure-of-merit for bond funds called 30-day SEC Yield which is something like YTM. SEC Yield for VBMFX is currently 1.99%. Let’s use SEC Yield as the mean return in the model, run 1,000 Monte Carlo simulation and see the results.

For standard deviation, I kept the ratio of mean / sd the same as the past 5 years.^{2} The ratio was 4.00/2.91 = 1.375, so sd = 1.99 * (2.91/4.00) = 1.45. Here are the results:

` Bal.Start Bal.End Gain TR TR.annualized`

Min. :10000 Min. : 10041 Min. : 1.00 Min. : 0.0041 Min. : 0.082

1st Q:10000 1st Q: 10799 1st Q: 1.08 1st Q: 0.0799 1st Q: 1.549

Medi :10000 Medi : 11024 Medi : 1.10 Medi : 0.1024 Medi : 1.968

Mean :10000 Mean : 11036 Mean : 1.10 Mean : 0.1036 Mean : 1.983

3rd Q:10000 3rd Q: 11271 3rd Q: 1.13 3rd Q: 0.1271 3rd Q: 2.421

Max. :10000 Max. : 12238 Max. : 1.22 Max. : 0.2238 Max. : 4.121

The median and mean annualized return are not far from the 2% mean. But there is a range of outcomes. No cases showed a net loss over five years. What percentage beat the CD?

`0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%`

10041 10462 10583 10677 10747 10799 10853 10896 10944 10980 11024

`55% 60% 65% 70% 75% 80% 85% 90% 95% 100%`

11071 11115 11161 11216 11271 11342 11419 11500 11631 12238

Looking at the quantiles, About 60% percent of the time, the outcome was worse than the 5 year CD’s $11,110. On the other end, the max case gave us over $12,000. It’s the luck of the draw.

If, ex ante, you expect to do worse than the CD 60% of the time, the risk averse would opt for the guaranteed return of a CD. Risk seekers might take a chance with the bond fund. But they need to go in with their eyes open and realize that they may get unlucky and do worse than the safe investment.

**Summary & Conclusion**

- The outcome of a Type 0 investment is known ex-ante. The outcome does not depend on any future state of the world, such as future interest rates.
- The outcome of a Type 1 investment is _unknown_ ex ante. The outcome depends on future states of the world, such as a future yield curve and the path it takes to get there.
- Type 1 investments have a distribution of outcomes. Some outcomes are better, and some worse, than a similar Type 0 investment.
- The p.d.f of a Type 1 investment is generally not known.

___________________________

^{1} Download the Monte Carlo simulator R script and run it yourself. Type1MonteCarlo.R

^{2} This seemed logical: as mean goes down, the sd goes down. But it is really just an arbitrary choice made for the presentation.