AAdmin
Apr 03'24

Exercise

You are given the following information about the purchase of a home:

  • The home was purchased on Jan 1 2010 for $300,000
  • The buyer used a 10-year mortgage to purchase the home with a down payment of $100,000
  • The nominal interest rate for the mortgage is 4%, payable at the end of every month
  • The buyer sold the home at the end of 2014 for $440,000
  • If the buyer hadn't purchased the home, he would have rented a three bedroom apartment. The monthly rent for a three bedroom apartment, payable at the beginning of the month, was $1,300 on Jan 1 2010 and $1,800 on Jan 1 2015
  • The rent on the apartment was adjusted at the start of each year and grew at a constant annual rate

Using only the information above (ignore all other expenses), approximate the effective annual rate of return for this real estate investment.

  • 15.26%
  • 18.26%
  • 20.26%
  • 22.26%
  • 24.26%

This exercise is too difficult and created for educational purposes only
AAdmin
Apr 03'24

Solution: D

We first define the following variables:

  • [math]P[/math] is the monthly mortgage payment
  • [math]r[/math] is the monthly rate of return for the investment
  • [math]g[/math] is the annual growth rate for the apartment rent
  • [math]RP[/math] is the remaining principal at the end of 2014

In order to find the internal rate of return, we need to determine the cash inflows and outflows. The cash inflows are the monthly rental payments plus the lump sum of $440,000 minus the remaining principal on the mortgage. The cash outflows are the down payment and monthly mortgage payments.

The present value of the cash inflows equals

[[math]] \frac{440,000 - RP}{(1+r)^{60}} + 1300 * \ddot{a}_{\overline{12}|r} \sum_{v=0}^4 x^v, \, x = \frac{1+g}{(1+r)^{12}} [[/math]]

and the present value of the cash outflows equals

[[math]] 100,000 + P * a_{\overline{60}|r} [[/math]]

The payment [math]P[/math] equals

[[math]] P * a_{\overline{120}|0.04/12} = 200,000 \implies P = 2024.9 [[/math]]

, the remaining principal equals

[[math]] RP = P * a_{\overline{60}|0.04/12} = 109,950 [[/math]]

and [math]1+g [/math] equals

[[math]] (1+g)^{5} = 18/13 \implies 1+g = 1.06725. [[/math]]

Putting everything together, the net present value of the investment equals

[[math]] \frac{330,050}{(1+r)^{60}} + 1300 * \ddot{a}_{\overline{12}|r} * \frac{1-x^5}{1-x} - 100,000 - P * a_{\overline{60}|r}. [[/math]]

We need to estimate [math]r[/math] such that the net present value equals zero. We use trial and error with the five available options:

Effective annual return [math]r[/math] NPV
15.26% 0.01191 38,822.78
18.26% 0.01407 20,568.19
20.26% 0.01549 9,828.85
22.26% 0.01689 124.47
24.26% 0.01827 -8,653.67

We see from the table above that the solution is approximately 22.26%.

The following R code numerically checks that the answer above is correct: 

P <- 2024.9
RP<-109950
g<-0.06725
payments <- rep(P,60)
rent <- rep(1300,60) * (1+g)^{c(rep(0,12),rep(1,12),rep(2,12),rep(3,12),rep(4,12))}
npv<-function(r){
drent<-rent*(1+r)^(-{0:59})
dpayments <-payments * (1+r)^{-(1:60)}
(440000-RP)/((1+r)^60) + sum(drent) - 100000 - sum(dpayments)
}
sapply(c(0.01191,0.01407,0.01549,0.01689,0.01827),npv)
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