4. Replacement Analysis Problem

So, now we have to find the equivalent annual cost for the third year of the purchase price 3500000 for year 3, 𝑨 𝒊(𝟏+𝒊)𝒏 𝟎.𝟏𝟓(𝟏+𝟎.𝟏𝟓)𝟑 USCRF = = = = 0.4380 (𝟑𝟓𝟎𝟎𝟏𝟐𝟎𝟎,𝟎.𝟏𝟓,𝟑) (𝟏+𝒊)𝒏−𝟏 (𝟏+𝟎.𝟏𝟓)𝟑−𝟏 EAC = 0.4380 × 35,00,000 = 15,33,000 rupees.

To find the equivalent annual cost (EAC) for a purchase price of 3,500,000 rupees in the third year, we need to use the Uniform Series Capital Recovery Factor (USCRF). For year 3, the USCRF is calculated to be 0.4380. We multiply this factor by the purchase price. Thus, EAC becomes 0.4380 times 3,500,000, which equals 1,533,000 rupees. This indicates the annual cost associated with this investment over the specified year.

Think of it like a loan. If you take a loan of 3,500,000 rupees for a project, the EAC tells you how much you need to pay back each year for the third year, considering interest and other costs. It's like knowing your monthly payments for a car over several years, helping you budget accordingly.

So, like this you are going to calculate for all the years. Now let us find the equivalent annual cost of the operating and the maintenance cost. So, how to find the equivalent annual cost let us go back to the cash flow diagram. So, this 1,13,200 is operating and maintenance cost at the end of year 1. Now you convert it into t = 0, how to convert it into t = 0, find the present worth? So, find the present worth of 1,13,200, so that is a first step. Once you find the present worth of 1,13,200 then you can find it is equivalent annual cost using uniform series capital recovery factor.

To calculate the equivalent annual cost of operating and maintenance costs, you first identify the operating cost, which is 113,200 rupees at the end of year 1. The first step is to convert this future cost into today's terms (i.e., find the present worth). This involves using a present worth factor based on the interest rate and the number of years. Once you find the present worth, you can then apply the USCRF to convert it back into an equivalent annual cost, making it easier to compare with other costs.

Imagine planning a vacation in a year. The cost now is uncertain, but you have an estimate. To understand how much you truly need, you look at today’s value of that future cost. It’s similar to saving money today to ensure you can afford that vacation next year, adjusted for future expenses.

So, we are going to find the present worth of 1,13,200 that is your operating cost. So, you need to find P for the known F, i, n, 𝑷 𝟏 P.W = = = 0.8696 𝟏𝟏𝟑𝟐𝟎𝟎,𝟎.𝟏𝟓,𝟏 (𝟏+𝟎.𝟏𝟓)𝟏 Present worth factor you multiply it by the operating and maintenance cost Present worth value = 0.8696 × 1,13,200 = 98,438.72 rupees.

To determine the present worth of the 113,200 rupees operating cost at the end of year 1, we use the present worth factor, which in this case is 0.8696. This factor accounts for the interest rate of 15% over one year. By multiplying the operating cost by this factor, we derive the present worth, resulting in 98,438.72 rupees. This value now represents the amount needed today to cover that future cost.

Think of planning a retreat. If you need to set aside some money for a trip that’s a year away, the amount you save today should consider possible interest or inflation. If you expect to save 1000 rupees next year, today you would set aside a smaller amount, such as 869.6 rupees, to ensure you can make that trip.

So, like this you are going to calculate the present worth of all the operating and maintenance cost. So, let us workout for one more trial, so that you will understand better.

The process of calculating the present worth for all operating and maintenance costs involves repeating the same method for each year’s cost, using the respective present worth factors. After obtaining the present worth for all years, the next step is to sum these values to find the cumulative operating and maintenance cost. This total gives a clearer picture of future expenses that need to be accounted for in budgeting.

If you run a monthly subscription service, you can estimate how much you will need for the whole year by adding up the costs month by month. If the pricing varies slightly, the total helps you understand how much you really need to prepare financially.

So, now that you have found the present value of your operating and maintenance cost, you have to find it's equivalent value, so go for the uniform series capital recovery factor. So, that means you are going to find A for the given P, i, n, what is a known P? P is nothing but your the present value is 98,438.72, interest rate is 0.15, n is 1.

Once you have calculated the present worth of the operating and maintenance costs (98,438.72 rupees), the next step is to determine the equivalent annual cost using the USCRF. For a known present value, interest rate, and time (one year in this case), the USCRF will allow you to convert this present value into an annual cost, facilitating easier comparisons with other annual expenses.

Think of a gym membership. You may pay upfront for the year, but by calculating how much that is per month, you understand better how to fit it into your budget—just like turning a larger one-time payment into manageable monthly installments.

So, now let us find the equivalent annual cost of the resale value, that is your salvage value. So, let us go back to the cash flow diagram. So, now what you are going to do is, this is the salvage value which is occurring at the end of year 1. So, you can directly convert into a equivalent annual cost using uniform series sinking fund factor or the another method is you find it’s present value.

To determine the equivalent annual cost of the salvage value (or resale value), you can follow one of two methods: either apply the USCRF directly based on the future salvage value or convert the salvage value into its present worth before applying the factor to obtain the equivalent annual cost. This helps in understanding the potential return on investment when considering replacing the asset.

If you buy a car, you can think about its future value when you sell it. Just like estimating whether it’s worth keeping based on how much you can get back when selling it in the future helps in making more informed decisions about whether to upgrade or keep it.

So, you are supposed to add the purchase price and the operating and maintenance cost, your salvage value is in flow cash inflow, so you subtract it. So, what is your equivalent annual cost of the purchase price for the year 1? Equivalent annual cost of the purchase price is 40,25,000 you add it with the equivalent annual cost of the operating and maintenance cost.

To find the total equivalent annual cost, combine the costs of the purchase price and the operating and maintenance costs, while subtracting the salvage value of the asset (considered cash inflow). This total provides a clear perspective on the overall annual cost involved in maintaining the machine, making it easier to assess its economic feasibility.

Imagine owning a rental property. You have monthly mortgage payments, maintenance costs, and want to subtract potential income from tenants. By calculating your actual outflow each month, you can determine if it’s worthwhile to maintain or sell the property.

Now you can see that your cost is high in the initial stage then it reduces, reaches the minimum value then again starts increasing. After year 3, you can see that it is the reaching minimum at the year 3 and after that it is increasing.

The analysis shows that the costs associated with the machine gradually decrease, reaching a minimum point by the third year before increasing again. This pattern indicates that while the machine initially has high costs, it becomes more efficient over time until maintenance and operational costs begin to rise, suggesting that year three is the optimal time for replacement.

Imagine a smartphone. In the first few years, it’s efficient, but as it ages, repair costs, battery replacement, and upgrades can add up, leading to a point where it’s more costly to keep than to buy a new model.

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