Redirecting Taxable investments to Roth 401(k) - An Improved Analysis

This is a follow-up to Redirecting Taxable investments to Roth 401(k). The analysis is conducted in terms of capital appreciation and dividend yield rates. It also incorporates the additional basis created in a taxable account by dividend reinvestment. 

Refer to the aforementioned post for the definitions of "Default strategy" and "Roth strategy" used below.

Term	Description
$P_0$	Amount withdrawn from the taxable brokerage in the Roth strategy
$B_0$	Cost basis of $P_0$
$B_N$	Normalized cost basis; $B_N = B_0/P_0$ $(> 0)$
$r_c$	Annual capital appreciation rate
$r_d$	Annual dividend yield
$T_c$	Annual capital gains tax rate; $0 < T_c \le 1$
$T_d$	Annual dividend tax rate; $0 < T_d \le 1$

Assumptions

1. $r_c$ and $r_d$ are annual returns measured relative to the beginning-of-year portfolio value
2. Dividends reinvested are net dividend taxes
3. For passive long-term investing it is reasonable to assume that both $r_c, r_d > 0$ and $0 < B_N \le 1$

All other assumptions from Redirecting Taxable investments to Roth 401(k) continue to apply.

Redirecting Taxable investments to Roth 401(k)

Preface

This article is for those in the USA who

• have access to after-tax 401(k) contributions with immediate in-plan conversion to Roth 401(k), aka, Mega Back Door Roth (MBDR), and
• have a taxable brokerage account with sufficient assets that can be leveraged to facilitate the MBDR.

Introduction

Usually, people focussed on saving and investing set their retirement contributions so that their net pay covers living expenses. Assuming one is already maxing out contributions to tax-advantaged retirement accounts and has accumulated a nontrivial amount of investments in a taxable brokerage, the question naturally arises:

Does it make financial sense to take advantage of the MBDR even if it means that one has to draw down on the taxable brokerage to fund living expenses?

Comparing mortgages for buying or refinancing a house

Questions that come up in scenarios related to buying or refinancing a home:

1. Is mortgage A better than mortgage B?
• Each mortgage may be amortized over a different duration (e.g., $15$ vs. $30$ years)
2. Given that I'm part-way through mortgage C, should I refinance to mortgage D?

Setting aside the emotional aspects, there is almost always a clear answer based on the Net Present Value ($NPV$) calculation of the cashflows related to the principal, interest, and loan fees. For a fixed-rate mortgage with no taxes or unusual features:$$NPV = F + P_0 + \sum_{k=1}^N \frac{PMT_k}{(1+d)^k}$$where

• $F$ is the amount in fees paid to acquire the loan
• $P_0$ is the initial cash outflow ("downpayment"), if any, outside of fees
• $N$ is the total number of payment periods
• For a $30$ year mortgage with monthly payments, $N = 30 \times 12  = 360$
• $PMT_k$ is the payment due in the $k^\text{th}$ period
• Usually the $PMT$ is the same across the entire loan duration for a fixed-rate mortgage
• $d$ is the chosen discount rate per period

I've made available a Mortgage Comparison Template.ods file that implements the calculations referred to in this post.

Sidebar

For question $1$, the Loan Estimate would include an APR value that can be the basis for a comparison if the loan durations being compared are identical. It's not immediately obvious how to use the APR to compare loans of different durations. Also, the APR cannot be used to answer question $2$ related to a refinance.

Sometimes the total (undiscounted) cashflow of interest payments over the span of the loan is used for the comparison, but as you'll see below that may give you the wrong answer.

Discount rate

For the $NPV$ calculation you'll need an appropriate discount rate (aka, opportunity cost). This would be the expected risk-adjusted after-tax return on an investment that you'd redirect your cash into if you didn't have to pay the mortgage.

One option is to use the "risk-free" rate of return (yield on T-bills/T-notes of the appropriate duration).

I use a conservative after-tax expected portfolio return of $7.1$ %/year (nominal rate which includes inflation). Note that the risk characteristics of a portfolio that includes stocks is completely unlike a fixed-rate mortgage which is a risk-free liability. Ergo, the choice of $7.1$ %/year for the discount rate is a personal one; it is not a universal discount rate.

Caveat

While the discount rate is specified in terms of %/year, most mortgages require monthly payments. If so, payments must be discounted on a monthly basis using $1/12^\text{th}$ the assumed discount rate. For the $7.1$ %/year choice above, it would be $7.1/12$ $=$ $0.5917$ %/month.

Buying a house

Inputs needed:

• Initial downpayment
• Starting principal
• Interest rate (%/year)
• Amoritization duration (years)
• Amortization period (usually monthly; to wit, $12$ periods/year)
• Loan fees (origination fees, title/insurance/deed fees, recording fees, etc.)
• Do not include any escrow amounts for home insurance and/or property taxes as they are not pertinent to the cost-benefit analysis

Set-up your amortization table for each mortgage and then calculate:

Net Present Value of the loan = 

Downpayment +

$PV$ of cashflows for principal +

$PV$ of cashflows for interest +

Loan fees

Choose the loan that has the lowest present value. That's it.