Question

A few years out of UCCS you’re earning $125,000 and decide to buy a condominium. The one you like will cost $300,000. The

bank will require a 15% down payment and will lend you the difference in a 30yr traditional mortgage at an interest rate of

4.800%(M).

For three years from the day you bought the condo, Colorado real estate continues to climb. Your condo appreciates 4% every

year for three years. And, interest rates fall to 3.60%(M). You decide to refinance your mortgage. You can refinance into a new

mortgage for (again) 85% of the current market value of the condo.

Now seven more years have passed, a total of ten years since you bought the condo. The real estate market slowed down and

then it crashed. For the next 6 years, your condo appreciated at 1% per year, but then over-building finally caught up with

Colorado and in year 10 the housing market fell 20%. You pay a broker a 5% commission to sell the condo. You also have closing

costs of $2,500.

Q: You made an initial cash outflow (the down payment) and ten years later had cash in the bank. Your initial investment is PV (a negative number), the final cash back is FV (a positive number), and 10 years passed. What was the % return each year? In other words, using TVM, solve for annual I (to three decimal places).

Answer 1

15% downpayment = 300000* 15% = $45000

Therefore a loan of $ 2,55,000 was taken initially.

The value of the house was $3,00,000. It appreciated to $3,00,000 *(1+4%)^3 = $ 3,37,459.

At this time you refinanced the loan at 85% of market value = $3,37,459 * 85% = $2,86,840.

The house then appreciate for 6 years at the rate of 1% = $2,86,840 * (1+1%)^6 = $3,58,220

In the 10th year, it crashes by 20% value = $3,58,330 * (1 - 20%) = 2,86,576

You sold the house and subtracted 5% and $ 2,500 as expenses leaving you with cash of $2,69,747.

However, we also need to reduce the EMI payment from the bank account for the year 10 to calculate final cash in the bank. The EMI payment for the refinanced loan can be calculated using the below formula

EMI Value = (P*R*(1+R/12)^N)/(12*((1+R/12)^N-1)) where P = Loan Principal, R = Annual Interest Rate N= Number of Payments in the Tenure. Substituting the values we get EMI of $1,304.11 and hence an annual payment of $15,649.

This gives the bank balance to be $2,54,097.68

Using the Cummulative Annual Growth Rate (CAGR) formula = (Final Value/Initial Value)^(1/Number of time periods) - 1 we get the Annual return of 18.899%

Note: Alternatively, if the bank reduced the outstanding principal amount from the bank account and we then calculated the values, it would be as before.

Principal Outstanding after 7 years (84 months) of principal payment = $2,44,535. Therefore, amount in bank after principal is deducted by bank = $9,562.97

This gives us a CAGR of -14.348% per annum.