Author Topic: Deep theoretical shit: CAPM with a mortgage as the risk-free rate  (Read 1754 times)

ChpBstrd

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Investment pricing models like the CAPM require you to plug in a "risk free" rate of return. This is usually understood to mean the yield on 10-year US Treasury bonds, which are currently <2.4%. However, for an individual investor with a mortgage (or any debt really), the risk-free rate of return is in fact their mortgage rate - e.g. my mortgage rate is about 50% higher than the 10y treasury at 3.625% (and I take the standard deduction, so no adjustment needed). So does a person with debt have a different risk-free rate than the institutional investors they're competing against? Do they have a different required return on an asset?

If I plug my mortgage rate into the CAPM model as my risk free rate, it makes no difference if I invest in the entire market with a beta of 1. At beta=1, the expected return is the historical average return.

Ra - expected return on asset, i.e. the expected ROI you'd require to buy it.
Rf - risk free rate of return
B - beta, a measure of the volatility of the asset's price compared to the market.
Rm - historical return of the market

Ra = Rf + B(Rm - Rf)

E.g.

10% = 2.4% + 1(10%-2.4%)
10% = 4% + 1(10%-4%)

However, at different betas, the numbers start to diverge.

For higher-volatility assets, with beta >1, the expected return on the asset is lower for a person with a mortgage than it is for someone using the 10y treasury as their risk free rate.

17.6% = 2.4% + 2(10% - 2.4%)
16% = 4% + 2(10% -4%)

For low-volatility assets, with beta <1, the expected return on the asset is higher for a person with a mortgage than it is for an investor using the 10 year treasury.

6.2% = 2.4% + 0.5(10% - 2.4%)
7% = 4% + 0.5(10% - 4%)

So the 2 investors do their respective math, and the debt-free investor finds beta = 2 assets to be not worthwhile unless they can expect at least a 17.6% return. The indebted investor, on the other hand, will bite if the return is expected to be at least 16%.

For low-beta assets, it's the opposite. At beta = 0.5, the debt-free investor will buy assets yielding as low as 6.2%, but the indebted investor won't buy unless they expect 7%.

My question is, does this observation mean that investors with a mortgage or other debt should be more inclined to buy risky assets and less inclined to buy low-risk assets? Do they have different security market lines? 

At some level, this makes sense. Why buy a 4% corporate bond with some risk when you could just pay down your 4% mortgage at zero risk? Maintaining a big savings account / CD balance while paying a mortgage is also an obvious bad move. However, both investments might make sense for a millionaire with zero debt. Even if both investors have an X% portfolio allocation for risk-free assets, they will make different decisions because the indebted investor has the advantage of a higher-returning risk-free investment. But CAPM implies to me that they should also consider different investments throughout their portfolios.

It's the high-risk observation that throws me off. Should indebted investors be more willing to take higher risks at lower expected returns than debt-free investors? Usually, debt makes people more cautious. Does the math prove that behavioral tendancy a fallacy, or does this mean we should throw out the model?

If this was 1982 and your mortgage rate or treasuries were 15% the model would spit out some illogical outputs, such as buying a 5% yielding investment at beta = 2!

Interesting side note: the CAPM has historically done a poor job predicting actual future returns based on beta. Returns of low-beta stocks have tended to be underestimated. Maybe the lesson is that all this is moot and the model is irrelevant.

https://www.investopedia.com/terms/c/capm.asp

YoungInvestor

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #1 on: November 22, 2017, 10:42:02 AM »
First, the risk free rate is generally the rate on 3-month t-bills.

Generally, you need to use the rate available to all market participants, which is why a mortgage would not be ok.

The expected return on an investment is the same for anyone buying it, obviously. The real question you may ask is who would benefit most from a risky investment, which is obviously the person with the lower guaranteed rate.

Intuitively, this should make sense to you: if you have a risk free investment available that will give you a higher rate of return, you would need a lawyer expected rate of return to buy a different investment. For an extreme case, look at someone with credit card debt at 20%+, who would need an incredible expected return to do something else than pay their debt.

bh2115

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #2 on: November 22, 2017, 02:20:14 PM »
First, the risk free rate is generally the rate on 3-month t-bills.

Generally, you need to use the rate available to all market participants, which is why a mortgage would not be ok.

The expected return on an investment is the same for anyone buying it, obviously. The real question you may ask is who would benefit most from a risky investment, which is obviously the person with the lower guaranteed rate.

Intuitively, this should make sense to you: if you have a risk free investment available that will give you a higher rate of return, you would need a lawyer expected rate of return to buy a different investment. For an extreme case, look at someone with credit card debt at 20%+, who would need an incredible expected return to do something else than pay their debt.

First, just a couple comments on the reply above...1) 10-Year T-Bill is widely more accepted as the risk-free rate than the 3-month. I guess if you were to use the 3-month you need to at least annualize the number for it to make any sense. 2) No you don't use the market rate available to all participants. If that logic were true everyone would use the same WACC and would clearly be flawed...

So back to OP's original questions of does someone with a loan at a higher rate use that as the risk-free rate and does this lead someone into riskier assets since their expected rate of return is higher?

Before I try to answer either question I think it's very important to define what the CAPM model represents. The (%) output is the required rate of return needed to move forward with the investment (not the expected return). Therefore, I do believe it's appropriate for someone with a loan that represents a higher rate to plug that in and therefore require a higher rate than someone with no debt. Then to your second question, I don't believe it makes people with debt move into riskier assets but rather raises the bar on the expected rate of return required to proceed with that investment. Therefore, it's the opposite...someone with no debt is likely to accept a risky (high beta) deal for a return of say 15% where someone with a mortgage or other debt would require something higher (let's say 20%) because you have a better alternative to deploy your capital. Hope that makes sense!

Financial.Velociraptor

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #3 on: November 22, 2017, 04:00:02 PM »
Of interest on what proxy to use for risk free rate (academic paper): https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1876117

ChpBstrd

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #4 on: November 24, 2017, 08:46:30 PM »
Yes, it's confusing how the academic descriptions of CAPM describe the "expected" return, because they're describing what people actually buying the investment must be expecting/requiring, not what you as a potential buyer should expect. This switch of perspective is even more confusing because often it is lumped in with models like DCF or IRR, which actually tell you the rate of return as a prospective investor. Expected = required!

I agree that the 10y treasury is the most common proxy for a risk-free rate and that this is relevant when you sell securities, because you're probably selling to an institution for which the 10y treasury is the best risk-free rate they can get. However, that doesn't change the fact that I personally have better risk-free opportunities, and these opportunities logically change my behavior (i.e. I would never buy 2.4% treasuries while my 4% mortgage exists).

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I don't believe it makes people with debt move into riskier assets but rather raises the bar on the expected rate of return required to proceed with that investment. Therefore, it's the opposite...someone with no debt is likely to accept a risky (high beta) deal for a return of say 15% where someone with a mortgage or other debt would require something higher (let's say 20%) because you have a better alternative to deploy your capital.

But the math shows the opposite - that a higher risk-free rate lowers the required return for higher-beta investments and raises the required return for low-beta investments. E.g.

Debt-free investor for whom 2.4% treasuries are highest-yielding risk-free investment and beta =2 :
17.6% = 2.4% + 2(10% - 2.4%)

Indebted investor for whom 4% mortgage is highest-yielding risk-free investment and beta = 2 :
16% = 4% + 2(10% -4%)

This is certainly counterintuitive because we would think an indebted person would be more reluctant to gamble. They must make their payments, after all. In real life, this is the advice we'd give; don't gamble on startups while you have debts.

Granted, I understand you don't buy into this concept of "everyone gets a personal risk-free rate" but consider the possibility that 10 year treasuries will be raised by the 1.6% difference we're talking about within within the next year and a half. Would this change alter your required return for volatile vs. non-volatile investments just because it applies to everyone?

 

marty998

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #5 on: November 26, 2017, 01:20:12 PM »
It's quite difficult to actually calculate a Beta for starters. Knowing the covariance between a security and the market (which will change over time) is a tricky thing to figure out.

Interesting thread all things considered... I've forgotten most of my finance knowledge from university so will refrain from adding misinformation here.

Systems101

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #6 on: November 27, 2017, 08:33:31 PM »
My question is, does this observation mean that investors with a mortgage or other debt should be more inclined to buy risky assets and less inclined to buy low-risk assets? Do they have different security market lines? 

You could assert they are different lines, but in reality they are one, non-linear system.  What you have actually isn't one risk free rate.  It's one risk free rate IF YOU ARE A LENDER (treasury), and one risk free rate IF YOU ARE A BORROWER (mortgage).  It should be calculated as such.

Look at problem 3 in this dataset:
http://privatewww.essex.ac.uk/~rbailey/book/ex/ch05es00.pdf

This shows you graphically that to stay on the efficient frontier, the higher risk free rate system is a flatter line and will thus shift to riskier investments as you have described in the math.  It also means there is now a valid range on the efficient frontier, not just a single point.


JetBlast

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #7 on: November 28, 2017, 09:33:11 AM »
I suppose one consideration is whether you consider pre-paying a mortgage to be ďrisk free.Ē

Whether they are six month t-bills or 10 year treasuries, there should be a liquid market for the securities should you need to sell to raise cash. The transaction costs are minimal and the market is vast.

Without an existing HELOC your home is rather illiquid. Costs to sell are high, the time to sell is much longer than government bonds, and that assumes you can find a buyer at all for the home. Since treasuries arenít subject to the vagaries of individual housing markets you shouldnít have any problem finding an eager buyer at market prices.

Thereís also the difference in cash flow. If the mortgage isnít paid off entirely, refinanced after your payment, or perhaps no longer required to pay PMI because of the prepayment your cash flow hasnít improved at all. With bonds you are getting the coupon payments that add to your cash flow while you wait for maturity.

To put it bluntly I think a mortgage is a very low risk investment but still riskier than the government bonds usually used to determine risk free rates.

JetBlast

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #8 on: November 28, 2017, 09:50:09 AM »
A couple more thoughts. If the mortgage is on a rental and not your residence Iíd find it to not be close to risk free, as the value of the asset being bought is a function of the rental income that can be generated, which is subject to the local housing marketís conditions.

Also, I donít have to pay maintenance costs on bonds. The asset purchased with a mortgage has continuing and variable costs after the home is owned free and clear.

ChpBstrd

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Re: Deep theoretical shit: CAPM with a mortgage as the risk-free rate
« Reply #9 on: November 28, 2017, 01:41:08 PM »
My question is, does this observation mean that investors with a mortgage or other debt should be more inclined to buy risky assets and less inclined to buy low-risk assets? Do they have different security market lines? 

You could assert they are different lines, but in reality they are one, non-linear system.  What you have actually isn't one risk free rate.  It's one risk free rate IF YOU ARE A LENDER (treasury), and one risk free rate IF YOU ARE A BORROWER (mortgage).  It should be calculated as such.

Look at problem 3 in this dataset:
http://privatewww.essex.ac.uk/~rbailey/book/ex/ch05es00.pdf

This shows you graphically that to stay on the efficient frontier, the higher risk free rate system is a flatter line and will thus shift to riskier investments as you have described in the math.  It also means there is now a valid range on the efficient frontier, not just a single point.

Great reference!

By extension, investors who expect an increase in the risk free rate might begin shifting their portfolios toward greater risk (and perhaps borrowing to do so if they expect borrowing rates to rise). Thus we have a mathematical explanation for why leverage and risk-taking should be expected to increase near the top of economic cycles, when rate hikes become expected.

The paradox is that those rate hikes both justify risky investments and stifle the economic activity required to justify risky investments. The boom-bust cycle seems inevitable. For example, investor money flows to highly indebted (risky) firms just as those same firms begin to face higher interest charges.

More importantly, it seems like indebted retail investors are incentivized to pay more for high-risk investments than their institutional or debt-free counterparts are willing to pay. To the extent CAPM is a behavioral model, this means being in debt probably persuades an investor to move off the efficient frontier.

Maybe their 30 year mortgage is why our parents gambled on penny stocks instead of building a blue chip portfolio. :)