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Question: One year ago an investor purchased a old house for £100,000 with the purpose of refurbishing it and then either letting or selling it. For the refurbishment the investor paid £50,000 one year ago for the materials. Until now (i.e. during the past year) further £40,000 were paid as compensation for the builders in equal quarterly installments. The annual effective rate is 3%.

Assume the investor will let the property to tenants who are going to pay £1,600 p.c.m. at the end of each month. The tenants will enter the property in 6 months and the first payment is due after a month from the time they enter the property. How long does it take before the investor starts making profits?

I've attempted this question once and ended up getting a MATH error on my calculator. I tried equating the value of the Outflows (costs of the house and builders) at t = 0 to the value of the sequence of monthly payments at t=0 and solving for n. Is this the correct way of approaching this question? If so, could someone help me arrive at the answer of 12.31 years? Thanks in advance.

Visual Reference of my Method here on Paint

Nipster
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1 Answers1

1

The formula for the number of months of rent payments is

$$n = -\frac{\ln\left[1-\frac{C}{1.6}r(1+r)^6\right]}{\ln(1+r)} \tag{1}$$

where $C$ is the totol cost at $t=0$, i.e.

$$C=150(1+r)^{12}+10(1+r)^9+10(1+r)^6+10(1+r)^3+10$$

and $r$ is the monthly compounding interest rate, backed out from the annual effective rate given.

$$r=(1+0.03)^{\frac{1}{12}}-1$$

$n$ comes out as 147.7 months, or 12.31 years.

Added responding to the comments:

The value $R$ of the total rents at $t=0$ is

$$R=\frac{1.6}{(1+r)^7}+\frac{1.6}{(1+r)^8}+ \space... \space + \frac{1.6}{(1+r)^{6+n}}, $$

a finite geometric series that can be summed up analytically. Then, setting $R=C$ leads to the formula (1).

Quanto
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