excans:43c3873616

Solution: D

We first define the following variables:

• [math]P[/math] is the monthly mortgage payment
• [math]r[/math] is the monthly internal 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:

IRR	[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)