Time Value of Money PDF

# 3. TIME VALUE OF MONEY The value of money keeps on changing with time. For this reason, the finance managers should consider the concept of time value of money while making investment decisions, finance decisions etc. ## TIME PREFERENCE FOR MONEY Desire to obtain money today as opposed to waiti...

# 3. TIME VALUE OF MONEY The value of money keeps on changing with time. For this reason, the finance managers should consider the concept of time value of money while making investment decisions, finance decisions etc. ## TIME PREFERENCE FOR MONEY Desire to obtain money today as opposed to waiting to receive the same amount in future. - Reasons are; * Reduction in the value of money due to inflation. * Availability of investment opportunity which may not be available in future. * Availability of discount in the market. * Urgency of current needs, requiring financial support. * Uncertain future - the future cash flows may never be realized. ## IMPORTANCE OF TIME VALUE OF MONEY (TVM) The concept of TVM may be applied in the following areas: 1. Capital budgeting to evaluate capital investments. 2. Inva valuation of assets and CoA 3. In preparation of loan amortization schedules ## TECHNIQUES USED IN TVM * **a) COMPOUNDING techniques:** * Process of determining future value. These techniques refer to the methods used to determine the future value. * The standard compounding PERIOD is usually once a YEAR: **ANNUAL Compounding**. * In this case, the future value is determined as follows: $FV = PV(1 + r)^n$ *FV = future value* *PV = present value* *r = cost of capital* *n = maturity period* *(1 + r/100)^n = (FVIF), Future value interest factor.* * However, in some cases, compounding may be done more than once a year: eg semi-annually, quarterly, monthly, weekly, daily or continuously. * This is multi-period compounding. When the number of times that compounding is done in a year is **DEFINITE**, the future value is determined as follows: $FV = PV (1 + r/100)^n$ *m = number of times compounding is done in a year-semi-annually* * When compounding is done continuously, the future value will be determined as follows: $FV = PV(e)^n$ where; *e = 2.7183* * **Example:** Paul deposited 1M in a commercial bank which promised to pay an annual interest of 12% over a period of 5YRJ. **Required:** * **a)** Assuming compounding is done annually, determine the total future value accumulated interest, and effective interest rate. * **b)** If compounding is done semi-annually, determine the total F.V and accumulated interest. * **c)** If compounding is done continuously, determine the future value, and accumulated interest. * **a)** $FV = PV (1+r/100)^n$ $PV = 1M$ $r = 12%$ $n = 5Yrs.$ *Total FV = 1,000,000 x (1 + 12/100)^5 = 1,762,341.6836h* *Accumulated Interest = 1762,341.683 - 1000,000 = 762,341.6838* * **b)** $FV = PV x (1 + r/100)^n$ $PV = 1M$ $r = 12%$ $m = 2$ $n = 5 x 2$ *Total = 1,790, 847.697h* *FV = Accumulated Interest = 790, 847.6976h* *Effective Interest Rate (r):* $r = (1 + r/100)^m - 1$ x 100% $r = (1 + 0.12/2)^2 - 1$ x 100% $r = 12.36%$ *r = (1 + r/100)^m - 1$ x 100% *1,790, 847 = 1m x (1 + r/100)^5* *r = 12.359%* *r = 12.36%* * **c)** *Continuously compounding* $FV = PV (e)^{rn}$ $FV = PV (e)^n$ $PV = 1M$ $e = 2.7183$ $r = 12$ $n = 5$ $1000,000 x 2.7183^{0.12x5}$ *Total FV = 1,822, 126.109h* *Accumulated Interest = 822,126.109h* *NB Effective Interest Rate; rate of real return on a saving or any interest paying investment when effects of compounding are taken into account.* # COMPOUNDING OF ANNUITIES Annuity: series of cashflows of equal amount receivable or payable each year. There are two types of annuities: 1. Ordinary annuity 2. Annuity due ## COMPOUNDING OF ORDINARY ANNUITY An ordinary annuity is a series of cashflows of (equal) amount receivable or payable at the end of each year. **Eg:** Simon deposited 100,000Gh into a commercial bank at the end of each year over a period of 5YRJ. The bank promised to pay an annual interest of 6%. P.A. **Required:** 1. Total F.V of the deposit. 2. The Interest earned over the 5YR period. **Solution:** **Method 1** | Year | Deposit | FVIF=6% | FV | |---|---|---|---| | 1 | 100,000 | (1.06)4 = 1.2625 | 126,250 | | 2 | 100,000 | (1.06)3 = 1.1910 | 119,100 | | 3 | 100,000 | (1.06)2 = 1.1236 | 112,360 | | 4 | 100,000 | (1.06) = 1.06 | 106,000 | | 5 | 100,000 | (1.06)0 = 1 | 100,000 | | **Total F.V** | | | **563,710 **| **Method 2** $FV = A x ( (1 + r)^n - 1) / r$ $A = annuity$ $((1 + r)^n - 1) / r = (FVIFA rn), Future Value Interest Factor of Annuity.$ $FV = 100,000 x ( (1 + 0.06)^5 - 1) / 0.06$ $FV = 100,000 x 5.6371$ $FV = 563,710h$ **b)** Interest rate (r) $r = (Total FV - Total Deposits ) / Total Deposits$ x 100% $r = (563710 - (100,000 x 5) ) / 500,000$ x 100% $r = 63710 / 500,000$ x 100% $r = 12.74%$ ## COMPOUNDING OF ANNUITY DUE An annuity due is a series of a cashflow of equal amount receivable or payable at the beginning of each year. **Eg:** John deposited 200,0000h in. a commercial bank at the beginning of each year over a period of 5YRJ. The bank promised to pay an annual interest of 8% **Required:** | 1. | Total future value | | 2. | The interest returned over the 5YR period. | **Method 1** | Yr | Deposit | FVIFr=8% | FV | |---|---|---|---| | 1 | 200,000 | (1.08)5 = 1.4693 | 293,860 | | 2 | 200,000 | (1.08)4 = 1.3605 | 272,100 | | 3 | 200,000 | (1.08)3 = 1.2597 | 251,940 | | 4 | 200,000 | (1.08)2 = 1.1664 | 233,280 | | 5 | 200,000 | (1.08) = 1.08 | 216,000 | | **Total Future Value** | | | 1,267,180 | **Method 2** $FV = A x ( (1 + r)^n - 1) / r$ x (1 + r) $FV = 200,000 x ( (1 + 0.08)^5 - 1) / 0.08$ x 1.08 $FV = 1,267,185.66h$ **Interest rate** $r = (Total FV - Total Deposit) / Total Deposit$ x 100%. $r = (1,267,180 - 1000,000) / 1000,000$ x 100% $r = 267180 / 1000,000$ x 100% $r = 26.718%$ $r = 26.72%.$ # b) DISCOUNTING techniques.. Discounting is the process of determining the P.V. Discounting techniques refer to methods used to determine the P.V. Discounting is the reverse of compounding. Thus if $FV = PVC1+r)^n$ => $PV = FV / (1+r)^n$ which also: $PV=FV(1+r)^{(-n)}$ Where $(1+r)^n$ = Present value Interest Factor (PVIFrn) **Example:** What is the P.V of 200K receivable at the end of the 5th year using a discount rate of 10% $FV = 200,000$ $r =10%$ $n =s$ $PV = 200,000 / (1+0.1)^5$ $PV = 124,184.2646$ $PV = 124,184h$ **2.** Determine the P.V of 1M receivable at the end of the 10th year using a discount rate of 14%. $FV = 1,000,000$ $r = 14%$ $n = 10$ $PV = 1,000,000 x (1 + 0.14)^{-10}$ $PV = 269,700h$ **3** Suppose the amount was receivable at the **start** of Yr 10, all other details remaining constant. Determine P.V. $PV = 1,000,000 x (1.14)^9$ $PV = 307,500h$ # DISCOUNTING OF ANNUITIES An annuity is a series of cashflows of equal amount receivable or payable each year. Types of annuities are: 1. Ordinary annuity 2. Annuity due 3. Growing annuity 4. A deferred annuity 5. A perpetual annuity. ## DISCOUNTING ORDINARY ANNUITY An ordinary annuity is a series of cashflows of equal amount receivable or payable at the end of each year. **Example:** Consider the following cashflows receivable at the end of each year over a period of 5YRJ. | Year | Cashflow | PVIF 10% | P.V | |---|---|---|---| | 1 | 10,000 | 0.9091 | 9091 | | 2 | 10,000 | 0.8264 | 8264 | | 3 | 10,000 | 0.7513 | 7513 | | 4 | 10,000 | 0.6830 | 6830 | | 5 | 10,000 | 0.6209 | 06209 | | | | | **Total P.V = 37,907** | The cost of capital in 10%. Determine the total present value. **Method 2:** $Total PV = A x ( 1 - (1 + r)^{-n} ) / r$ Where $(1 - (1 + r)^{-n} ) / r = (PVIFArn), Prevent Value Interest Factor of Annuity.$ $Total PV = 10000 x (1 - (1 + 0.1)^{-5} ) / 0.1$ $Total PV = 37,90862$ ## DISCOUNTING ANNUITY DUE Annuity due is a series of cashflows of equal amount receivable or payable at the beginning of each year. **Example:** Consider the following cashflow receivable at the beginning of each year over a 5YR period. | Yr/1 | Cashflow | DVIFA12% | PV | |---|---|---|---| | 1 | 20,000 | 1 | 20,000 | | 2 | 20,000 | 0.8929 | 17,858 | | 3 | 20,000 | 0.7972 | 15,944 | | 4 | 20,000 | 0.7118 | 14,236 | | 5 | 20,000 | 0.6335 | 12,670 | | | | | **Total P. V = 80,708** | The minimum required rate of return is 12% Determine the total P.v. $Total PV = A x ( 1 - (1 + r)^{-n} ) / (1 + r)$ $Total PV = 20,000 x ( 1 - (1 + 0.12)^{-5} ) / (1 + 0.12)$ $Total PV = 20,000 x 4.03735$ $Total PV = 80,746.99h$ $Total PV = 80,747h$ ## Discounting of AGROWING annuity - Agrowing annuity is an annuity which is expected to GROW or increase at a constant rate each year. The total P.V of an agrowing annuity can be determined as follows; $Total PV = A x ( 1 - (1 + g)^n ) / (r - g)$ *A = Agrowing annuity* * g = growth rate* **Example:** ABC Ltd is considering taking a capital investment which is expected to generate annual cash flows of 1M for 5YRJ. The cash flow shall grow at the rate of 5% P.A. The cost of capital is 10%. Determine the total P.v. of the cashflow. $A = 1,000,000$ *N = 5YRJ* $r = 10%$ *g = 5%* $Total PV = 1,000,000 x ( 1 - ( 1 + 0.05)^5 ) / ( 0.1 - 0.05)$ $Total PV = 4,150,000h$ ## Discounting of ADEFERRED annuity A deferred annuity is a series of cashflows of equal amount receivable or payable in future years. **Example:** MAY 2018 @ 4 (8) | **MAY** | **Yr** | **Cashflow** | **PVIFALo%** | **PV** | |---|---|---|---|---| | | 1-5 | 5000 | PVIFA10%n= 5 = 3.7908 | 6.1446 | | | 6-10 | 9000 | PVIFA10%n= 10 - PVIFA10%. n = 5 = 6.1447 - 3.7908 | 7.6061-3.7908 | | | 11-15 | 4000 | PVIFA10%=10-PVIFA10%.n=10 = 7.6061-6.1446 | 6.1446 | | | 16-20 | 13000 | | | | | | | | | | | | **PV** | | | | | 1-5 | = 5000 x 3.708 | = 18,954 | | | 6-10 | =9000 x 2.3538 | = 21,184 | | | 11-15 | = 4000 x 1.4615 | = 5,846 | | | | | **Total PV = 45,984** | ## Discounting a PERPETUAL annuity - A perpetual annuity is a veries of cashflows of equal amount receivable or payable indefinitely. The Total P.V of a perpetual annuity may be determined using two methods:; * **a)** where **NO GROWTH** or **INCREASE** is expected, the total P.V will be determined av; $Total PV = A / r$ **b)** where a perpetual annuity is expected to **GROW** or **INCREASE** at a constant rate each year. The total PV will be; $Total PV = A / (r - g)$. **Example** Bima LTD pays a dividend of 2ch per share indefinitely. The minimum required rate of return by investors is 10%. Required: * **a)** Total P.V. of dividend.. * **b)** Assuming the dividend should grow at a constant rate of 5%, determine the total P.V of dir. **a)** $Total PV = A / r$ *A = 26h* *r = 10%* $Total PV = 20h$ **b)** $Total PV = A / (r-g)$ *A = 26h* *r=10%* *g=5%* $Total PV = 40h.$ ## c) LOAN AMORTISATION SCHEDULE. - This is a table that shows a breakdown on how a loan is repaid. It's therefore a tabular presentation showing the Principal amount BORROWED, period INSTALLMENTS which include interest and principal, the interest, principal repayment and outstanding balance. - Interest is normally computed on the reducing balance. - Periodic installments may be determined as follows: *Annual installment = Amount borrowed / PVIFArn* *Semi-annual installment = Amount borrowed / PVIFA anxa ...etc* **Example:** DECEMBER 2007, Q3(C) NIR PETER borrowed 1M on 1st Jul 2007, repayable semi-annually over 3YRJ at a rate of 12%. P.A *Semi-annual installment = 1000,000 / (1 + r/2)^n - 1 / (r/2)* *Semi-annual installment = 1000,000 / (1 + 0.12/2)^3*2 - 1 / ( 0.12/2) * *Semi-annual installment = 4.9173* *A = 203,364* | Period | CONSTANT Installment A | (12%)=6% Interest B | Principal Repayment C | Outstanding Balance | |---|---|---|---|---| | 0 | - | - | - | 1000,000 | | 1 | 203,364 | 60,000 | 143,364 | 856,636 | | 2 | 203,364 | | | | | 3 | 203,364 | | | | | 4 | 203,364 | | | | | 5 | 203,364 | | | | | 6 | 203,364 | | | | *B = Ax rate.* *C = A - B.* *D = previous B-C* If interest is paid annually, all other details remain constant. Determine a loan repayment schedule. r = 12% n = 3 *Annual installment = 1000,000 / 1 - (1 + 0.12)^-3 / 0.12* *Annual installment = 416,354* | Yr | Annual Installment A | Interest Payable (pi) B | Principal re-payment C | Outstanding Balance | |---|---|---|---|---| | 0 | - | - | - | 1000,000 | | 1 | 416,354 | 120,000 | 296,354 | 703,646 | | 2 | 416,354 | 84,438 | 331,916 | 371,730 | | 3 | 416,354 | 44,608 | 371,746 | -16 | *Dec 2013 Q3a* *Nov 2011 Q3b* *Dec 2013 Q2 (16)* SHADRACK CHAND borrowed 80,000 monthly over 12 months. rate 1.25%. *Monthly installment = 80,000 x (1 + r/12)^n -1 / (r/12)* *Monthly installment = 8o,000 x (1 + 0.0125)^12 -1 / (0.0125)* *Monthly installment = 7221* | Month | Monthly installment | Interest 1.25% monthly | Principal | Outstanding Balance | |---|---|---|---|---| | 1 | 7221 | 1000 | 6221 | 73,779 | | 2 | 7221 | 922 | 6299 | 67,480 | | 3 | 7201 | 844 | 6377 | 61,103 | | 4 | 7221 | 764 | 6457 | 54,646 | | 5 | 7221 | 683 | 6538 | 48,108 | | 6 | 7221 | 601 | 6600 | 41,488 | | 7 | 7221 | 519 | 6701 | 34,787 | | 8 | 7221 | 435 | 6786 | 28,001 | | 9 | 7221 | 350 | 6871 | 21,130 | | 10 | 7221 | 264 | 6957 | 14,173 | | 11 | 7221 | 177 | 7044 | 7129 | | 12 | 7221 | 19 | 7132 | -3 | *Nov 2011 Q3 (a)* MIKE GUYO borrowed 1M annual rate 14%. 4 yrs repayable. *Annual installment = Total amount borrowed / PVIFArn* *Annual installment = 1000,000 / 1 - (1 + 0.14)^-4 / 0.14* *Annual installment = 343,206* | Year | Annual installment | Int. Payable | Principal repayable | Outstanding Balance | |---|---|---|---|---| | 0 | - | - | - | 1000,000 | | 1 | 343,206 | 140,000 | 203,206 | 796, 794 | | 2 | 343,206 | 111,551 | 231,655 | 565,139 | | 3 | 343,206 | 79,119 | 264,087 | 301,052 | | 4 | 343,206 | 42,147 | 301,059 | -7 | ==End of OCR for page 12==