The rent for the next five years of an eighty Five-year property contract is set at £4,000 permonth. Thereafter the rent will increase by 20% compound every five years. Whatprice would be paid for the contract in order to achieve a yield of 8% per annumeffective?
The correct answer and explanation is:
Let’s carefully analyze the problem step-by-step:
Problem Summary:
- Property contract length: 85 years
- Initial rent: £4,000 per month for the first 5 years (so £48,000 per year)
- After first 5 years: rent increases by 20% compounded every 5 years
- Required yield: 8% effective annual yield
- Goal: Find the price to pay now for this contract to achieve an 8% yield.
Step 1: Understand the cash flows
- For years 1–5: annual rent = £4,000 × 12 = £48,000
- From year 6–10: rent increases by 20% → £48,000 × 1.2 = £57,600
- From year 11–15: rent increases another 20% on previous increase → £57,600 × 1.2 = £69,120
- And so on, rent increases every 5 years by 20% compound.
Step 2: Define rent at each 5-year period
Let R0=48,000R_0 = 48,000 be the annual rent for the first 5 years.
The rent for each 5-year period nn (where n=0,1,2,…n = 0, 1, 2, …) is: Rn=R0×(1.2)nR_n = R_0 \times (1.2)^n
Step 3: Present Value of each 5-year rent block
We want to calculate the present value (PV) of the rent payments.
- Discount rate i=8%i = 8\% per year (effective).
- Rent is paid annually.
- Rent amount RnR_n is paid for 5 years, starting at year 5n+15n + 1 through 5n+55n + 5.
Step 4: Calculate PV of each 5-year block
The PV of the nthn^{th} block of 5 years of rent, starting at year 5n+15n + 1, is: PVn=Rn×∑t=151(1+i)5n+tPV_n = R_n \times \sum_{t=1}^{5} \frac{1}{(1 + i)^{5n + t}}
This is a 5-year annuity starting at time 5n+15n + 1.
Rewrite sum inside as: ∑t=15(1+i)−(5n+t)=(1+i)−5n×∑t=15(1+i)−t\sum_{t=1}^{5} (1+i)^{-(5n+t)} = (1+i)^{-5n} \times \sum_{t=1}^{5} (1+i)^{-t}
The term ∑t=15(1+i)−t\sum_{t=1}^5 (1+i)^{-t} is a standard 5-year annuity factor: a5‾∣i=1−(1+i)−5ia_{\overline{5}|i} = \frac{1 – (1+i)^{-5}}{i}
Step 5: Total PV of all 17 blocks (since 85 years / 5 = 17 blocks)
PV=∑n=016Rn×(1+i)−5n×a5‾∣iPV = \sum_{n=0}^{16} R_n \times (1+i)^{-5n} \times a_{\overline{5}|i}
Substitute Rn=R0(1.2)nR_n = R_0 (1.2)^n: PV=a5‾∣i×R0×∑n=016(1.2)n(1+i)−5nPV = a_{\overline{5}|i} \times R_0 \times \sum_{n=0}^{16} (1.2)^n (1+i)^{-5n}
Step 6: Calculate annuity factor a5‾∣ia_{\overline{5}|i}
At i=8%=0.08i = 8\% = 0.08: a5‾∣0.08=1−(1.08)−50.08a_{\overline{5}|0.08} = \frac{1 – (1.08)^{-5}}{0.08}
Calculate (1.08)−5(1.08)^{-5}: 1.085=1.4693 ⟹ (1.08)−5=11.4693=0.68061.08^{5} = 1.4693 \implies (1.08)^{-5} = \frac{1}{1.4693} = 0.6806
Therefore: a5‾∣0.08=1−0.68060.08=0.31940.08=3.9925a_{\overline{5}|0.08} = \frac{1 – 0.6806}{0.08} = \frac{0.3194}{0.08} = 3.9925
Step 7: Calculate the sum
Define: S=∑n=016(1.2)n(1.08)−5n=∑n=016(1.21.085)nS = \sum_{n=0}^{16} (1.2)^n (1.08)^{-5n} = \sum_{n=0}^{16} \left( \frac{1.2}{1.08^5} \right)^n
Calculate 1.0851.08^5 again: 1.085=1.46931.08^5 = 1.4693
Therefore: 1.21.4693=0.8166\frac{1.2}{1.4693} = 0.8166
So, S=∑n=016(0.8166)nS = \sum_{n=0}^{16} (0.8166)^n
This is a geometric series with ratio r=0.8166<1r = 0.8166 < 1, so: S=1−r171−rS = \frac{1 – r^{17}}{1 – r}
Calculate r17=0.816617r^{17} = 0.8166^{17}:
Using a calculator, log10(0.816617)=17×log10(0.8166)=17×(−0.0889)=−1.511\log_{10}(0.8166^{17}) = 17 \times \log_{10}(0.8166) = 17 \times (-0.0889) = -1.511
So, r17=10−1.511=0.0308r^{17} = 10^{-1.511} = 0.0308
Hence, S=1−0.03081−0.8166=0.96920.1834=5.285S = \frac{1 – 0.0308}{1 – 0.8166} = \frac{0.9692}{0.1834} = 5.285
Step 8: Calculate total present value
PV=a5‾∣i×R0×S=3.9925×48,000×5.285PV = a_{\overline{5}|i} \times R_0 \times S = 3.9925 \times 48,000 \times 5.285
Calculate stepwise: 3.9925×48,000=191,6403.9925 \times 48,000 = 191,640
Then: 191,640×5.285=1,012,130191,640 \times 5.285 = 1,012,130
Answer:
The price to pay for the contract to achieve an 8% yield is approximately £1,012,130.
Explanation:
This problem involves valuing a long-term lease contract with rent that increases in a stepwise manner every 5 years by 20%. To find the contract’s value at an 8% annual effective yield, we discount all future cash flows back to the present.
Because rent increases every 5 years by 20%, the total rent stream can be split into 5-year blocks where the rent is constant, but each subsequent block grows by 20% over the previous one. For each 5-year block, we calculate the present value of those 5 years’ rents (an annuity), then discount that entire block back to today.
Summing the present values of all 17 blocks (since 85 years / 5 years per block = 17 blocks) gives the total present value. Since the rent increase is geometric and the discount factor is geometric, the sum of the present values reduces to a geometric series that we can calculate explicitly.
This approach yields the price to pay now to achieve an 8% yield on the property lease income stream with the specified rent escalation pattern.