Module-5-Annuties Introduction Annuity: A series of payments made at equal intervals of time. Examples: House rents, mortgage payments, instalment payments on automobiles and interest payments on money invested. . R- The periodic payment of the annuity n- The numer of payments made i- the interest rate per payment interval S- The accumulated value of the annuity; the amount of money you will have at the end of the annuity’s term. A- The present value of the annuity or discounted value, the amount of money which must be set aside today to allow a specified payment for a predetermined period of time. Terms used in Annuity Annuity-certain: An annuity such that payments are certain to be made for a fixed period of time. Example: A 30 year mortgage, because there are precisely 360 monthly payments made during the duration of the loan Contingent annuity: An annuity under which the payments are not certain to be made. A common type of contingent annuity is one in which payments are made only if a person is alive (Life Annuity). Example:A monthly retirement payment paid during the life of a retiree. Term: The fixed period of time for which payments are made. Payment interval or conversion period:The time between payments is called the payment interval. Simple Annuity:When the payment period coinsides with the conversion period, the annuity is said to e simple. Annuity-immediate: An annuity under which payments of 1 are made at the end of each period for n periods. Annuity-Due: The payments are made at the beginning of the period General Annuity:Annuity is not simple is called general annuity. Ordinary Annuity:The payments are made at the end of each payment interval the annuity is called ordinar annuity. Accumulated value The accumulated value of an ordinary annuity is computed by S=RX accumulation factor. Accumulated value of an annuity due is computed by S=Rxaccumulation factorX(1+i) Present Value Present value of an ordinary annuity is computed by A=RX discount factor. Accumulated value of an annuity due is computed by A=Rxdiscount factorX(1+i) Example-1 Find the accumulated value of an annuity of $700 invested at the end of each quarter for 5 years at an annual rate of 8% compounded quarterly. Solution: Accumulation factor for n=20(4X5) and i=0.08/4=0.02 is=24.29737 S=RX accumulation factor=$700X24.29737=17008.159 Example-2 Find the accumulated value of an annuity of $700 invested at the beginning of each quarter for 5 years at an annual rate of 8% compounded quarterly. Solution: Accumulation factor for n=20(4X5) and i=0.08/4=0.02 is=24.29737 S=RX accumulation factorX(1+i)=$700X24.29737X1.02=17348.322 Example-3 Find the present value of an annuity of $300 at the end of each month for 5 years at 9% compounded monthly. Solution: Discount factor for n=60(12X5) and i=0.009/12=0.0075 is 48.17337. A=RX discount factor=$300X48.17337=14452.011 Example-4 Find the present value of an annuity of $300 at the begining of each month for 5 years at 9% compounded monthly. Solution: Discount factor for n=60(12X5) and i=0.009/12=0.0075 is 48.17337. A=RX discount X(1+i) factor=$300X48.17337X1.0075=14560.4011 Sinking Fund Method: In Sinking Fund Method will provide us with an amount of depreciation as well as provide funds for the replacement of this asset when an asset need replacement like the end of life of an asset. Under this method, we charged depreciation on the value of asset but will not be credited to the asset account instead we will credit to sinking fund account. At the end of each accounting year, the total amount of sinking fund credited in a year will be invested in the outside marketable security to provide cash for the replacement of an asset when needed. What is the Sinking Fund Method? The sinking fund method is a technique for depreciating an asset while generating enough money to replace it at the end of its useful life. As depreciation charges are incurred to reflect the asset's falling value, a matching amount of cash is invested. These funds sit in a sinking fund account and generate interest. The sinking fund method is a depreciation technique used to finance the replacement of an asset at the end of its useful life. Companies rarely use the sinking fund method of depreciation because of its complexity.The sinking fund method is mainly used by large-scale industries, such as utility companies, that require expensive, long-term assets to function. Example The accumulation of sinking fund in which $10000 is to accumulated in 1 year if payments are made Quarterly into an account which pays 5% compounded quarterly, for the above case Construct a sinking fund schedule. Solution: 𝑆 10000 𝑅 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 = 4.07563=2453.60 Perio d Initial Interest amoun at 1.25% t Quarterl Increase Final y in fund mount paymen t 1 0 0 2453.60 2453.60 2453.60 2 2453.6 0 30.67 2453.60 2483.67 4953.67 3 4953.6 7 55.50 2453.60 2509.10 7462.77 4 7462.7 7 93.28 2453.60 2546.88 10009.65 Amortization Amortization typically refers to the process of writing down the value of either a loan or an intangible asset. It also refers to the repayment of loan principal over time. Amortization schedules are used by lenders, such as financial institutions, to present a loan repayment schedule based on a specific maturity date In business, amortization refers to spreading payments over multiple periods. The term is used for two separate processes: amortization of loans and amortization of assets Amortization is a method of spreading the cost of an intangible asset over a specific period of time, which is usually the course of its useful life. Intangible assets are nonphysical assets that are nonetheless essential to a company, such as patents, trademarks, and copyrights. The goal in amortizing an asset is to match the expense of acquiring it with the revenue it generates. Example Construct an amortization schedule for a one and half year loan of $5000 at 5% interest which is to repaid in quarterly installments over one and half years. Solution: 𝐴 5000 𝑅 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 = 5.74601=870.1690 Paymen Paymen Interest t t Number amount Reducti Principa on to l principa balance l $5000 1 870.169 0 62.5 807.669 4192.33 1 2 870.169 0 52.40 817.769 3374.56 2 3 870.169 0 42.18 827.989 2546.57 3 4 870.169 0 31.83 838.339 1708.23 4 Difference between depreciation and Amortization • The key difference between amortization and depreciation is that amortization is used for intangible assets, while depreciation is used for tangible assets. • Another major difference is that amortization is almost always implemented using the straight-line method, whereas depreciation can be implemented using either the straight-line or accelerated method. • Finally, because they are intangible, amortized assets do not have a salvage value, which is the estimated resale value of an asset at the end of its useful life. Depreciated assets, by contrast, often have a salvage value. An asset's salvage value must be subtracted from its cost to determine the amount in which it can be depreciated. Capital Budgeting Capital budgeting is the process a business undertakes to evaluate potential major projects or investments. Construction of a new plant or a big investment in an outside venture are examples of projects that would require capital budgeting before they are approved or rejected. As part of capital budgeting, a company might assess a prospective project's lifetime cash inflows and outflows to determine whether the potential returns that would be generated meet a sufficient target benchmark. The process is also known as investment appraisal. Capital budgeting is a process of evaluating investments and huge expenses in order to obtain the best returns on investment. Capital budgeting is a predominant function of management. Right decisions taken can lead the business to great heights. However, a single wrong decision can inch the business closer to shut down due to the number of funds involved and the tenure of these projects. FEATURES OF CAPITAL BUDGETING • It involves high risk • Large profits are estimated • Long time period between the initial investments and estimated returns Example Which would you prefer to receive(i) $12000 at the end of 3 years plus $25000 at the end of 5 years, or (ii) $2250 at the end of each quarter for the next 5 years.Assume that money is worth 10% annually and is compounded quarterly. Solution: The present value of $ 12000 , 3 years from now is 𝑆 12000 12000 𝐴 = (1:𝑖)𝑛=(1.025)12 = 1.34489=8922.66(n=3X4=12; i=0.10/4=0.025) The present value of $25000, 5 years from now is 𝑆 25000 25000 𝐴 = (1:𝑖)𝑛=(1.025)20 = 1.63862=15256.74(n=3X4=12; i=0.10/4=0.025) The total present value under option(i) is $24179.4004 Under option (ii) the present value of $2250 per quarter for 5 years is A=Rxdiscount factor=$2250X15.58916=$35075.61 Option (ii) is preferale, since the present value of the money is larger. Thank You