5. A property developer is constructing a block of offices. It is anticipated that the offices will take six months to build. The developer incurs costs of KSh.40 million at the beginning of the project followed by KSh.3 million at the end of each month for the following six months during the building period. It is expected that rental income from the offices will be KSh.1 million per month, which will be received at the start of each month beginning with the seventh month. Maintenance and man- agement costs paid by the developer are expected to be KSh.2 million per annum payable monthly in arrears with the first payment at the end of the seventh month. The block of offices is expected to be sold 25 years after the start of the project for KSh,60 million. Calculate the discounted payback period using an effective rate of interest of 10% per annum. [10 marks]

To calculate the discounted payback period, we need to find the time it takes for the net present value (NPV) of the cash flows to become positive. The NPV is calculated by discounting the cash flows using the given effective rate of interest and summing them up.<br /><br />Let's calculate the NPV for each cash flow:<br /><br />1. Initial cost: -$40 million<br />2. Monthly cost for the next six months: -$3 million (for 6 months)<br />3. Monthly rental income starting from the seventh month: +$1 million (for 25 years)<br />4. Annual maintenance and management cost: -$2 million (for 25 years)<br /><br />Now, let's calculate the NPV for each cash flow:<br /><br />1. NPV of initial cost: -$40 million / (1 + 0.10)^0 = -$40 million<br />2. NPV of monthly cost for the next six months: -$3 million * [(1 - (1 + 0.10)^-6) / 0.10] = -$15.16 million<br />3. NPV of monthly rental income starting from the seventh month: $1 million * [(1 - (1 + 0.10)^-300) / 0.10] = $23.83 million<br />4. NPV of annual maintenance and management cost: -$2 million * [(1 - (1 + 0.10)^-300) / 0.10] = -$5.56 million<br /><br />Now, let's sum up the NPVs:<br /><br />NPV = -$40 million - $15.16 million + $23.83 million - $5.56 million = -$37.89 million<br /><br />Since the NPV is negative, we need to find the time it takes for the NPV to become positive. We can do this by adding the cash flows until the NPV becomes positive.<br /><br />1. After the first month: -$40 million - $3 million = -$43 million<br />2. After the second month: -$40 million - $6 million = -$46 million<br />3. After the third month: -$40 million - $9 million = -$49 million<br />4. After the fourth month: -$40 million - $12 million = -$52 million<br />5. After the fifth month: -$40 million - $15 million = -$55 million<br />6. After the sixth month: -$40 million - $18 million = -$58 million<br />7. After the seventh month: -$40 million - $21 million + $1 million = -$60 million<br />8. After the eighth month: -$40 million - $24 million + $1 million = -$63 million<br />9. After the ninth month: -$40 million - $27 million + $1 million = -$66 million<br />10. After the tenth month: -$40 million - $30 million + $1 million = -$69 million<br />11. After the eleventh month: -$40 million - $33 million + $1 million = -$72 million<br />12. After the twelfth month: -$40 million - $36 million + $1 million = -$75 million<br />13. After the thirteenth month: -$40 million - $39 million + $1 million = -$78 million<br />14. After the fourteenth month: -$40 million - $42 million + $1 million = -$81 million<br />15. After the fifteenth month: -$40 million - $45 million + $1 million = -$84 million<br />16. After the sixteenth month: -$40 million - $48 million + $1 million = -$87 million<br />17. After the seventeenth month: -$40 million - $51 million + $1 million = -$90 million<br />18. After the eighteenth month: -$40 million - $54 million + $1 million = -$93 million<br />19. After the nineteenth month: -$40 million - $57 million + $1 million = -$96 million<br />20. After the twentieth month: -$40 million - $60 million + $1 million = -$99 million<br />21. After the twenty-first month: -$40 million - $63 million + $1 million = -$102 million<br />22. After the twenty-second month: -$40 million - $66 million + $1 million = -$105 million<br />23. After the twenty-third month: -$40 million - $69 million + $1 million = -$108 million<br />24. After the twenty-fourth month: -$40 million - $72 million + $1 million = -$111 million<br />25. After the twenty-fifth month: -$40 million - $75 million + $1 million = -$114 million<br />26. After the twenty-sixth month: -$40 million - $78 million + $1 million = -$117 million<br />27. After the twenty-seventh month: -$40 million - $81 million + $1 million = -$120 million<br />28. After the twenty-eighth month: -$40 million - $84 million + $1 million = -$123