FIE432E Flashcards
Net tax on all assets –>
Net tax on specific asset –>
Gross tax on all assets –>
Gross tax on a specific asset –>
Net tax on all assets –> higher portfolio weight risky assets
Net tax on specific asset –> higher weight on that specific asset (and less on risk-free asset)
Gross tax on all assets –> likely more risk taking
Gross tax on a specific asset –> likely but less of the specific asset
Two period model
Plain tax subsidy
Income effect
- Subsidy makes you richer (red moves out)
- -> Implying: Increase in both c1 and c2 –> smaller s(=y-c1)
- Need utility function to determine exact points
Substitution effect
- Subsidy makes consumption during worklife less attractive compared to retirement consumption (y is stille the same)
- -> Implying: Increase in c2, and decrease in c1 –> bigger s(=y-c1)
Combined
- c1 ??, s ??
- c2 will definently not be smaller
- -> Need utility function to be more precise
Return is divided into thee components
Which?
- (+) Yield, net benefit of owning asset as a percentage of the value of the asset.
- (+) Capital gain, increase (or decrease) in the price of the asset as a percentage of the value,
- (-) Inflation
How much to save –> two main risks
- Risk in returns
- Risk in time of death
Both are important, but the second risk is likely bigger (wider range of outcomes plausible. exDie at 70 (3-y retirement), or at 90 (23-y retirement). Both plausible)
Risk in time of death add a lot of uncertainty to retirement plans, and insuring youself is costly! Rest of course will mainly look at without risk in time of death (assuming full insurance)
Capital Income Tax
Gross: Return + Volatility
Gross and net taxation reduce the expected return on investment provided the expected excess return is positive, the effect of gross is stronger.
Gross and net both reduce portfolio variance, the effect is equally strong.
Graph
Capital income tax effect, gross and net.
X-axis - Return
Y-axis - Percentage of times
Red before taxation (higher expected return, borader distribution (risk))
Black after net taxation (compared to blue: same distribution, higher expected return)
Blue after gross taxation (lower expected return, narrower distribution (risk))
Partial vs Generel Equilibrium
Beta increase
- In partial equilibrium prices do not change
o Asset simply offers a lower rate of return to its Beta
o Meaning few people will buy it - In general equilibrium the asset moves back to the security-market line
Formula for perfect offset
Capital Income; net tax
How much you should invest in X1 after taxation depends on:
- The excess return prior to tax reform (and therefore your intital X1)
- The tax rate
- The excess return you can earn on the risky asset
TAKE AWAY: With proper respone net taxation changes nothing (offset by more risk) –> Net taxation leads to more risk taking
Bonds vs Housing
Housing return is less volatile than the bond rate, BUT:
Bonds fix the nominal rate of return over (say) 10 years
–> Then only inflation (and bankrupcy) risk –> you are safer from volatility for 10 years
Volatility in the price of housing on the other hand has a direct effect on your wealth, and you are still subject to inflation risk as well
==> Over the duration of the bond, bonds are safer than housing.
Two period model
BSU Ex. First 10’ saved exempt from tax
(Picture $$ not same, but same principal nevertheless)
For somebody who was only planning to save only 5’ NOK the reform has both an income and a substitution effect.
Outcome: ???
For somebody who was planning to save 15’ NOK the reform only has an income effect. His last NOK is taxed exactly the same rate as before, no incentive to save more, but there is an income effect (reversed income effect)!
Outcome: Save less!
==> (Skattelette) that do not affect the last NOK saved only have an income effect, so you save less.
Graph: Wealth Tax
Ideal setting (very similar in reality probably)
Red - Before tax
Blue - After tax
Substitution effect
Tax on wealth makes consumption at retirement age more expensive
Consumption now is more attractive than consumption at retirement
–> we want to consume more, save less
Asset-pricing model
Standard
We have these because we want to study in genereal equilibrium, and need to take the price effect of the policy into account as well.
Formula prvides the rate of return on an asset. Important: Relationship between return and price.
Ex. Asset becomes more attractive:
Demand goes up –> price goes up –> return of asset goes down
- Current owners see capital gain
However: In asset-pricing models, the rate of return is always the rate for future owners of asset. –> decrease in return is good for current owners, bad for future
–> Asset-pricing models show relationship between attractiveness of asset (Beta, tax rate) and the rate of return for future investore (prices and effect for current investors are often left implicit)
When to use?
Partial vs General Equilibrium
Partial equilibrium is relevant when:
- The tax cut applies only to a few market participants
- The tax cut is local but the market is global
General equilibrium is relevant when:
- The tax cut is local and the market is local
- The tax cut is global and the market is global
Tax variants
- Capital-Income Tax (focus in course)
a. Gross
b. Net - Wealth Tax (focus in course)
- Labor-Income Tax
- Corporate Tax
Capital Income Tax
Net: Return + Volatility
Gross and net taxation reduce the expected return on investment provided the expected excess return is positive, the effect of gross is stronger.
Gross and net both reduce portfolio variance, the effect is equally strong.
Two period model
Plain tax increase, explain effect
Income effect
- Increase in tax makes you poorer (red gives less opportunity than blue)
- -> Implying: Reduction in both c1 and c2 –> bigger s(=y-c1)
- Need utility function to determine exact points
Substitution effect
- Increase in tax makes consumption during worklife more attractive compared to retirement consumption (y is still the same)
- -> Implying: Reduction in c2, and increase in c1 –> smaller s(=y-c1)
Combined
- c1 ??, s ??
- c2 will definently not be bigger
- -> Need utility function to be more precise
Fixed investment model
Collateral
Same incentive-compatibility constraint, because the borrowr will not be able to recoup any of the collateral!
Zero profit condition has to be updated. to include collateral if failure. (see picture)
Combining we will now get (see picture)
Two period model
Explain
Period one: Working period (earn y, consume c1 and save s=(y-c1))
Period two: Retirement period (live of savings c2 = (1+Rp*)s
Leas to intertemporal budget constraint: (Picture)
S is in the model implicity, same for taxation (slopw is determined by Rp*, the higher tax, the lower is Rp*)
Danish pension study
Main takeaways
Most workers can be best described as passive savers
–> They do not respond to incentives, have noe clear retirement goals, and simply do as they are told.
Not really suprising!
Housing vs Equity
Very similar rates of return, but housing has much lower variability.
There may be two reasons as to why housing are doing so well!
- Equity can be easily diversified, housing cannot
- -> Implication: Risk of housing is larger than the numbers in the study reveal (paper has macro perspective, and assumes diversification) - Combination of leverage, taxation and asset pricing (may wear off)
Leverage in the housing market has increased substantially in later years. Due to different in tax treatment (see formulas) this have led to capital gains for housing equity has not had. But this does not benefit future home owners!
(Current home owners benefit from tax rebate, and capital gain.
Future home owners benefit from tax rebate, but suffer for increase in price –> 0 in perfect market)
Seperation theorem, Stage 1:
I want to retire after x years. How much should I save each year?
Split in two:
- How much wealth do I need to accumulate by age y in order to pay myself P NOK for the next (z-y) years?
- How much should I save during (y-x) years to acquire this wealth?
To calculate:
- Use the PV formula to calculate the PV of your pension at time of retirement (x)
- Use the FV-formula to calculate the annual payments you require to end up with x at retirement.
TAPM
Ex. Gov provides a large tax cut on housing
Adjustment mechanism: Suppose the gov provides a large tax cut on housing –> tax rate goes down –> asset becomes more attractive –> General equilibrium: The demand for the asset increases –> the price of the asset increases –> return of the assets decreases
Current owners: Win twice
- Tax break
- Capital gain
Future owners: Neutral effect
- Tax break
- Lower rate of return
(we are not accounting for the fiscal cost of the tax break)
(PS! This example show how the many tax breaks have helped current hous owners, but may not affect future home owners)
Two important irrationalities
For personal finance?
- Unaware of changes in incentives
- Aware, but do not understand how to change their actions
Wealth Tax
In practice
Not the same for all activities, real estate and unlisted firms are severely undervalued.
–> Wealth tax in practive favors investments in real estate and in small unlisted firms
(certainly not bad for entrepreneurial activity)
Issue: Becoming a listed firms is very costly
TAPM
Explained
Almost same as CAPM, but we use tax rate (t) instead of Beta. (see formula (Ri is taxed asset, Ru is untaxed asset))
A1: Assets are the same, except for the tax rate t.
- You could think of TCAPM where assets differ in both Beta and tax rate.
- Model becomes needlessly complex
- ->Price/return can only vary by t
Assuming tax rate is a wealth tax (very easy mathematically)
- A wealth tax only reduces the expected return
A2: There exists at least one untaxed asset with return Ru
- This assumption is really just for mathematical convenience
Taxation & Real World Policies
Many real world policies only have an income effect
Some policies only have a substitution effect
Ex. Many countries allow you to save an amount x without paying taxes on it (or even subsidize savings) (Norway examples: BSU, wealth tax)
Purpose of these policies is often to increase savings, but unfortunately they are often counter-productive. –> Because they have an (reverse) income effect, but no subsitution effect
Many useful financial products bannot be bougt, and households are far from diversified!
Why?
- Consumers are irrational
- Regulation: Reasons why fully rational consumers do not hold diversified portfolios
o Most important regulators in this context: People that provide credit
Households most strongly affected:
- Home owners
- Entreprenerus
Crediots require you to overinvest in your own firm or your own home! Because of incentives (to workd hard or maintain)
Important assumption for Capital Income counteracting?
The most important assumption Sandmo makes is that losses can be perfectly offset.
Loss offset: A loss leads to a reduction in tax payments.
R* = (1-t)R even if R<0
Real world: Loss carry-forward
- Loss offset opportunities are pretty good for household
- Losses can often be carried forward to next year
o Great if you have positive income in the next year(s)
o But you do lose interest
Real world: Diversification
- Helps oppset losses
o Losses in one asset can cancel out against gains in other assets
o Negative capital income cancels out against positive labor income in many countries
- Ensures that resulting income is rarely negative
–> Loss offsets are close to perfect for most household
Takeaway
Pretty good assumption for households
Potentially bigger issue for companies
Portfolio formulas
Portfolio return, portfolio variance, min risk (p=0 and p=/0) and sharpe
Danish pension system
Similar to most developed countries
Wealth Tax
In ideal setting
Wealth tax reduces expected return (both portfolio and asset), but has no effect on risk!! (see formulas)
Wealth tax has no direct effect on portfolio compsition (may have an indirect effect, but this is likely very weak: Investors are worse off, may imply less willingness to take risk)
Why do we have SIM?
And what are the assumptions?
Shortcut, magic formulas work, but are impossible to do for 1000s of assets.
Assumption: Risk on assets consist of two components
- Idiosyncratic (specific to asset)
- Systematic (moving with the “market”)
Asset prices move toghether, because all of them move with the markets (this is the only reason for correlation between assets.
Asummptions work reasonably well for stocks.
Gross tax, offset?
Two effects
Reduces the expected return more than net tax, and same effect on volatility.
Implication: Impossible to fully counteract –> Cannot get as good risk/return
Therefore effect is not clear:
- Direct effect: Investor would like to make up for loss in expected return –> More risk
- Indriect effect: If you are worse off you may be less willing to take risk
Sandmo shows that it is very likely that the direct effect outweighs the indirect. (First effect is orders of magnitude larger than the second according to conventional empirical estimates)
–> Gross taxation will most likely lead to more risk taking (weaker effect than for net taxation)
Three types of return
Three Asset Model
Gross Capital Income Tax
Two risky assets, one risk-free.
Gross taxation of one asset makes you like to buy LESS of that asset.
–> Even more likely with more than 3 assets.
Intuition: Gross tax makes asset less attractive (can’t fully counteract). You can no longer attain previous risk/return for portfolio. Asset is less attractve –> You want less of it.
Exception: The asset is unique in its risk/return payoff (which is less likely with more assets avaliable)
Mandates vs. Tax Policy
Advantages / disadvantages
Mandate:
- Advantages
–>If people do not counteract, gov chooses how much you save
–>Increase in mandate likely increases overall savings (retirement account + other savings)
–>Do not have a direct effect on tax revenue
- Disadvantages
–>Little effect on rational people
–>One size may not fit all
- *Tax incentives:
- **Advantages
- ->Increase retirement income for everybody
- ->May increase savings for rational individuals (substitution effect)
- Disadvantages
- ->May reduce savings for rational individuals (income effect)
- ->Does not affect retirement savings for irrational individuals
- ->Goes at the expense of tax Revenue
(Most countries use a combo of both, which is better depends on rationality)
CAPM
Explained
SIM model, but in equilibrium! –> Meaning asset prices/returns change with demand.
Investors will only get payed for systematic risk, as idio risk can be diversified –> Asset with highet Beta are more riske, and should therefore give a higer rate of return. (this is security market line (se formula))
Beta: Relationship between sotck return and stock index.
So remember: Beta up = return up (for future owners)
A1: Assets are equally attractive, except for their Beta
- Prices/returns can therefore only vary by Beta
A2: There exists a risk-free asset with Beta = 0
- This assumptions i really just for mathematical convenience
INCOME VS SUBSTITUTION EFFECT
The effects work in opposite direction.
(If tax goes up you will)
- Save less through substitution effect
- Save more through income effect
–> There is noe way to determine which effect dominates for general wealth-/gross-income tax
(But for some policies the effect is more clear)
Bonds vs Bills
The difference is the investment premium: Term premium = r(bond) - r(bill).
Which is not neccesarily postiive –> risk for bond: unexpected inflation (and banckrupcy). risk for bills: changes in rate of return.
–> This leads to different asset choice for different expectations (the asset that offers the most attractive features from the market’s perspective yields the lowest rate of return)
Bills offer lower return if markets expects
- Inflation risk
- Future increase i the interest rate
Bonds offer a lower return if market expects
- Future decrease in the interest rate
CAPM
Adjustment mechanism and outcome (Beta up)
The Beta suddenly increases –> Investors will sell the asset because it’s now unnatractive (at same rate of return) –> Price reduces –> This increases the rate of return of the asset –> moving it back to the security market line
Current owners: Loose twice
- Increase in risk
- Negative capital gain
Future owners: Neutral effect
- Increase in risk
- Increase in return
Three assets Model
Net Capital Income Tax
Two risky assets, and one risk-free.
Introduction of net tac that only affects one of the risky assets.
Investor should buy more in the taxed asset to fully counteraxt the tax!
Investor buy more of risky taxed asset, and buy less of risk-free asset.
(The ratio of the taxed asset will increase compared to the other risky asset, but since it is bought insted of risk-free, the other risky asset will remain constant as % of totalt portfolio)
==> Portfolio E(Rp) and st.dev will remain unchanged.
Fixed Investment Model
Diversification
Formal analysis beyond a point becomes quite complex.
From analysis need to know: Borrower only gets paid if both projects succeed! Projects are uncorrelated)
Main outcome: Asset requirement is less than twice as large when you invest in two projects.
–> Diversification is good for the borrower, despite him only beeing rewarded when both succeed. But he will need a lot of cash at hand to invest in both!
(Remaining issue: Benefit comes from the fact that success of the projects is uncorrelated (gets samllet with positive correlation). But borrower will like high correlation, it makes his payment point (extreme point) more likely. –> if borrower can choose the benefits of diversification breaks down)
Fixed Investment Model
No collateral
Separation theorem
Theorem: You can think of a household facing a two-staged decision problem.
- I want to retire after x years. How much should I save each year?
- How should I invest my savings over different assets?
Implication –> Portfolio allocation depends only on the risk.return tradeoff, not the time horizon. Makes everything easier (which is why it is assumed in most of the course)
However! For this to hold we must assume Famas Efficient Market Hypothesis –> Return is random walk, and current price is best predictor for future price.
Problem: This goes against mean reversion; downturns tend to be followed by upswing (and vice versa) –> Time horizon affects the optimal asset allocation!
Unclear how big of an issue this is when using the theorem
(PS! Seperation theorem also assume no transaction costs)
Risk in time of death, subdivided
- Idiosyncratic risk: you may die earlier than you neighbor and vice versa
- Tens to cancel out across population
- Easier to insure against - Systematic risk
- Life expectancy of the pop as a whole may rise
Insurance would be great! Public pension is common –> idio risk cancel out (# of people, everybody is forced) –> systematic risk remain a big issue
Annuity (see OneNote) is another option, quite costly and difficult to get :(
Taxation and Savings
Impact?
Taxation affect how much people save because it affects the rate of return on savings. Rp* < Rp
To affect the level of savings a tax has to affect the rate of return –> A net tax affects portfolio composition, but has no impact on the rate of return –> No savings impact
We will look at the effect of gross taxation and/or wealth taxation on savings. Not going to be specific, allt that matters Rp* < Rp.
Income effect
Taxation reduces the PV of your wealth
–> To compensate you consume less and save more.
The Single Index Model
