Median home price is used as a measure of cost of homes by almost everyone trying to quantify housing affordability. Sometimes this price is compared to a median income, sometimes to the past median price or a median price in another country. In all of cases it's the median price that almost always is used and compared. The argument is that median price is a good metric because it makes sure than half of homes are priced less than median. It's clear that median is much better than some other values like mean price but is median as good measure as it appears? Some of the questions that rarely gets asked (and answered) in home affordability studies are: how affordable are nonmedian homes? How affordable homes are for people who are not median income earners? Is price distribution fixed over time and in various places around the world so it can be used for comparison?
Half of population makes less than median income and half of homes are priced below median but these studies don't show how much below. They don't give clear picture how affordable homes are for people who make 2/3 or half of median income. They don't show how many homes are priced 2/3 or half of median. These studies also don't show what kind of homes are priced at median, 2/3 of median or half of it; are those houses or units, how far from city centers and employment hubs, how large they are (3bdrs or studios, ...). It is somehow assumed that distribution of income roughly follows distribution of house prices making ratio same for all income earners and that income distribution hasn't changed over time making affordability for all income earners changing same as median . It is also assumed that cost of living is proportional to income so that someone making 2/3 of median can use the same percentage of income for mortgage repayments and deposit saving. These studies assume that affordability is equal issue for all income earners and all types of buyers (first buyers, upgraders, downgraders ...). It is also assumed that house price distribution is the same in cities, states and countries across the globe. These studies also ignore the fact that most of people who make above median are middle aged people who already own at least a portion of their home purchased some time ago when prices were different. These people may comprise large part of buyers but income multiple is not relevant as a measure of affordability since large number of these buyers have large equity to start from. All of these assumptions seem to question adequacy of median price in affordability studies.
To try to understand this a bit more, first, we'll take a closer look into house price distribution. Home value data is not so easily available so we'll use freely available sales data for Sydney over the last year to estimate house price distribution. Based on numbers of sales per region (higher turnover in cheaper areas) and number of sales per housing type (more unit sales) we think that actual price distribution is noticeably worse in the sense that lower priced home sales are high disproportionately to their share of total housing. Also median home price is likely to be higher than median sale price due to higher market activity in lower price range. Almost half of all sales in Sydney in past year were unit sales while they represent just over a quarter of housing stock. This makes sale price distribution not perfect representation of actual home price distribution but we have to start somewhere. Real distribution is likely to be much worse in sense that there will be less lower priced homes than the sale statistics shows.
Figure 1  Sydney Home Sale Price Distribution 2016 (all homes) 
Figure 1 shows sales price distribution in Sydney. Median sale price for Inner Sydney is over a million ($1.08m) while mean is almost $1.5m. Urban area has median around $855k and mean over $1.15m while Sydney Metro has median of $820k and mean at almost $1.1m. Mean is always significantly higher than median which indicates that distribution is significantly skewed to the right meaning that there are fewer homes much lower than median and more homes priced much over median.
Sydney Inner
There are almost no homes (less than 2%) in Inner Sydney are priced under half of already high median ($550k ), those few are all small units (mostly studios). Less than 10% of all homes are priced under 2/3 of median ($700k) mostly 1 bdr units and studios. There are no houses of any size in Inner Sydney priced under 2/3 of median home price ($700k) and only few under 3/4 of median ($810k). Less than 8% of homes under median ($1060k) are houses. Median house price in Inner Sydney is just under $2m ($1.96m). Median home sale price ($1.06m) can buy an average 3bdr unit in an ordinary suburb or small older 2bdr terrace in some of the least desirable inner suburbs.
Sydney Urban
Situation is only marginally better in Sydney Urban area. Less than 2.5% of all homes is priced under half median ($425k) mostly studios and 1bdr units. There are around only 10% of all homes priced under 2/3 of median ($570k)  mostly 1bdr units. Only 2.5% of all homes are houses priced below 2/3 of median ($570k)  all houses under 2/3 of median are located more than 30km from city centre. Only handful of 'for demolition' houses are priced below half median ($425k). Just over 10% of all homes priced under median ($855k) are houses (mostly in areas 30km from CBD). Median house price in Urban Sydney is $1.25m. Median home price ($855k) can buy 2 bdr unit in suburbs ~ 5km from the city, 3 bdr unit in suburbs ~10km from city or a cheap house in less desirable suburbs more than 30km from the city.
Sydney Metro
In Metro Sydney, including the cheapest outer suburbs (over 50km from CBD), only around 3% of homes is priced under half of median ($410k) and only around 13% of hokes are priced under 2/3 of median ($550k). Affordability improvements only come in size and quality of housing in areas far from the city: compared to Urban and Inner areas where almost no houses were priced under 2/3 of median, large portion of homes between half median and 2/3 of median in outer areas are houses. These houses increase overall distribution of house prices but still manage to make make only 5% of all homes priced under 2/3 of median ($550k) houses  mostly in areas 50km from the city. Despite that, only 20% of homes below median ($820k) are houses. Median house price in all of Sydney is around $1.15m. Median price ($820k) can buy a older 2bdr unit in suburbs ~5km from CBD, new 2bdr or older 3bdr unit in suburbs 10km from CBD, an ordinary house in suburbs 30km from CBD or new house in outer suburbs (over 50km from CBD).
Figure 2  Income and Home Sale Price Distribution 
Figure 2 shows home sale price distribution in Sydney and income distribution in Australia. Shape of income distribution in Australia is likely to be a good representation of income distribution in Sydney with the difference that median in Sydney is higher (because of higher wages for average earners). There is also to expect a higher percentage of households well below median because low wage earners are not paid more in Sydney (minimum wage is the same in whole of Australia). Income distribution is also skewed distribution but to a less degree. Sydney median income is estimated to be around $84k gross (this figure includes fictitious income like imputed rent and nondisposable income like superannuation). Simple ratio of median home sale price to median gross income equal to around 10.
The chart shows that the most homes (mode price) are priced close to median (3/4 of median), while most of income earners (mode income) earn between half and 2/3 of the median income. One third (32%) of all households make below 2/3 of median ($56k gross) income. At the same time only around 13% of all homes are priced below 2/3 of median home price (mostly smaller units). Some may argue that very low income earners are not part of housing market so comparison is not valid. Even if we assume that none under half median income would never take part in housing market (this is a poor assumption because many young families fall under this category due to low wages at the start of a career and inability of both partners to work full time while having small children) we there would be 20% of people who make between half and 2/3 of median income competing for only around 13% of all homes in Sydney metropolitan area that are priced below 2/3 of median. There are also large percentage of investors who the most active in this price range pushing prices up and making it even more competitive and harder for young families to buy. It is also important to notice that most of those cheaper homes are not suitable for majority of first home buyers due to small size and poor location. Age group that is likely to buy below median homes (young families under 35) have median income lower than overall median and around 20% lower than 3555 age group.
Because there are very few homes priced under 2/3 of median that are adequate for young families, large number of these lower paid first home buyers are forced to buy homes that are more expensive than 10 times their income. In many instances this forces young families to pay 1520 times of their income just to buy a cheapest satisfactory home e.g. 3bdr house or unit still far far from city but with option of tolerable commute time.
As mentioned before, two other points make this situation likely worse: real median home price is higher than median sale home price while real median income is lower than gross median income used in calculations. Also, lower income earners can afford to use much lower percentage of income for a mortgage repayments compared to median earners and don't have much equity to use as deposit. This makes homes prohibitively expensive and totally unaffordable for almost anyone making under 2/3 of median income. To illustrate this consider the case where a young family with two children that makes 2/3 of median gross income ($56k) would after saving 20% of income for almost a decade need over two thirds of their gross income just to repay loan on the cheapest 3bdr home 50km from city (priced $550k) leaving them with less than $400 per week for all other costs.
Figure 3 Sydney Home Sale Price Distribution 2011 and 2016 
Figure 3 shown home sales price distribution for Sydney Metro area in 2011 and 2016. Median price increased by over 50% making median homes less affordable. What is more important is that price distribution got more skewed over the same period. To illustrate this we created Figure 4.
Figure 4. Low priced home distribution Sydney 2011 and 2016 
In 2011 almost 20% of homes were priced below 2/3 of already much lower median price ($580k). In 2016 percentage of homes under 2/3 of higher median fell to 13% of greatly higher median ($820k). Share of homes priced under one half of median halved from 6% to 3% over the same period. While median price increased 50% in just 5 years, prices of lower priced homes increased much more than prices of median priced homes. The lowest priced homes (priced around $200k in 2011) went up by 75% to $350k in 2016. This shows that affordability for below median earners deteriorated much more than the headline median price to median income figure would suggest. Using simple median price to median income is, in this sense, deceiving because it hides true extent of fall in affordability for large number of below average income earning first home buyers. Situation is much worse and it deteriorated much more than what median multiple numbers would indicate. Home affordability in Sydney situation reached level of crisis where large majority of young people without a home cannot even think of buying a home.
Median multiple (ratio of median income to median home price) is clearly not very good measure of affordability, it's not even good measure to estimate changes in affordability over time but is it good to compare affordability around the world? To check this we'll do house price distribution comparison between some other large world cities. Sales data is freely available for San Francisco, New York and London, but not for other Australian cities. These large and major cities with unaffordable housing are commonly used to compare affordability around the world and to detect housing bubbles.
To make comparison better these cities will also be divided in three areas: Inner, Urban and Metro. In some cases these areas will not directly correspond to Sydney areas but that will be noted. These areas are defined as described in notes 2 to 4. Table 1  Population density people per km2 
Table 1 shows density in different areas of these cities. It's obvious that price comparison is going to be favorable to Sydney and some degree SF because of their much lower density (higher availability of residential land). Inner Sydney area has almost the same density as all of NYC metro area that includes large areas in 14 counties across two states. Next few figures show home sale price distribution in these cities.
Figure 7  London Home Sale Price Distribution 2016

Median sale price in inner London is around £530k with almost 20% priced less than half median and 7.5% under half of median (mostly small units). In Urban area (Grater London) median price is £435k with over 20% of homes under 2/3 of median and over 10% under half of median. London Metro median drops further under £400k with around 20% priced under 2/3 of median and 7% under 1/2 of median.
These least affordable cities in USA and UK do not only have lower median home price to median income ratio than Sydney but also much more favorable home price distribution where much larger percentage of homes is priced below 1/2 and 2/3 of median: Sydney Metro has only 3% under half median while SF Metro has 10%, Sydney Metro has only 13% under 2/3 of median while London Metro has over 21%. Situation is even more extreme for areas closer to city center: Sydney Urban area has only 2.5% under half and 10% under 2/3 while NY Urban area (NYC) despite being much smaller and having lower median has almost 10 times larger proportion of homes (20%) under half and 25% under 2/3 of median.
Figure 8  Distribution of under median priced homes in selected cities and areas 
Sydney performs extremely poor in lower priced home affordability in comparison with other expensive cities especially in areas closer to the city center. This means that home affordability for below median income earning first home buyers in Sydney is much much worse than what headline median 'price to median income' suggests. Affordability in Sydney for almost anyone making under median and has no equity or help is literally nonexistent. It would be interesting to see actual statistical distribution of home prices to real income ratio. It wouldn't be very surprising to find out that the ratio is well over 10 making all those mortgages highly risky and subprime. Even family with median income buying sub median home (e.g. 3/4 of median) would not be more than few paychecks far from default in case even third of income gets suddenly lost.
This analysis shows that median multiple is very poor tool not only to estimate home affordability (especially for first home buyers) or compare change in affordability over time, but also for comparing housing affordability between various cities around the world. Even more importantly this analysis shows that home affordability is nonexistent for all first home buyers who make under median income and many who make even more than median.
 Note 1  Sydney Subdivisions: Inner Sydney includes suburbs within ten kilometers from CBD including central, eastern, southern and inner western suburbs (post codes 20002050 and 22032204) and Lower North Shore (post codes 20602069, and 20882090). Inner Sydney as defined here has area of 190km2 and population of around 690k. Urban Sydney area includes most of Sydney: Inner Sydney, rest of Inner West and North Shore, Western Sydney, Norther Beaches, Northwest areas including Hills District, South Western suburbs, St George and Sutherland Shire. Area of Urban Sydney is around 2800km2 with population of almost 4 million. Metro Sydney area includes Urban Sydney area as well as Outer Western Suburbs, parts of Hawkesbury (areas around Richmond and Windsor) and Macarthur region (areas around Campbelltown) excluding Wollondilly. Area of Metro Sydney is over 4000 km2 and population of 4.5 million. Almost all suburbs in outer Sydney Metro area are more than 50km from CBD with one way commuting times well in excess of one hour.N
 Note 2  San Francisco subdivisions: Inner SF is defined here as SF city and county proper, Urban SF is defined as area of following counties: San Francisco, urban parts of Alameda, San Mateo, Santa Clara and small urban part of Contra Costa. Metro SF is defined as urban SF area plus city of Vallejo and some parts of Marin County (San Rafael and near surroundings). Metro SF as used here is smaller than comparable Sydney area and much smaller than commonly used San Francisco Bay Area (excludes nonurban parts of Alameda, San Mateo, Santa Clara, most of Solano and Marin counties and excludes Napa and Sonoma counties completely.
 Note 3  New York subdivisions: Inner NY is defined here as Manhattan, Urban NY area is defined as NY City proper (5 boroughs  Manhattan, Bronx, Queens, Brooklyn, and Staten Island), Metro NY includes Urban area plus Nassau, Westchester, Rockland, Bergen, Hudson, Union, Essex, Passaic, and Middlesex counties in states of NY and NJ. It has to be noted that both Urban and Metro NY areas used here are much more relatively smaller than comparable areas in Sydney. Urban NY city area as defined here (NYC) has density of over 10k per km2 and makes only one third of population of typically used wide NYC metro area. Compared to Urban Sydney with density of 1500 per km2 and 90% of population of total metro area. NY metro area used here (with population of 16 million) is much smaller than commonly used New York–Newark–Bridgeport, NY, NJ, CT, PA area with population of 24 million.
 Note 4  London subdivisions:: Inner London is defined here as Inner London (as defined by London Government Act 1963) plus City of London. Urban London is defined here as Greater London (density 5.3k per km2) while Metro London is defined as London metropolitan area or sometimes called London commuter belt (including parts of Hertfordshire, Essex, Kent, Surrey, Berkshire and Buckinghamshire Counties) with population of over 14 million and density over 1500 per km2.
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