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I would love it if someone could tell me how to solve this. Consider the following cash flow: Year Cash Flow 0 -$400 1 0 2 +$200 3 +$150 4 +$100 5 +$50 Write one equation with i as the onyl unknown for the cash flow. In the equation you are not to use more than two single payment compound interest factors. (you may use as many other fators as you wish). Then solve your equation for i.
This problem is easy to solve when you set up the equation with 4 discounting factors and use a calculator that will iteratively solve for the interest rate. The thing that makes this problem tricky is that you are only allowed to use 2 single payment compound interest factors. Wikipedia has the formula for the present value of a growing (or shrinking annuity). The reason we need 2 factors is because the formula only works when the cash flow grows or shrinks exponentially. In our case, the cash flow shrinks linearly. Therefore, we will need to use the equation once for the cash flows in years 2 and 3, and once for the cash flows in years 4 and 5. The other tricky thing here is that any present value formula solves for the present value one compounding period prior to the first cash flow. Therefore the equation for the cash flows in years 2 and 3 solve for the "present value" from the perspective of year 1. Therefore, we will need to discount (divide) this "present value" by (1+i)^1 to calculate a true present value from the perspective of today. Similarly, the equation for the cash flows in years 4 and 5 solve for the "present value" from the perspective of year 3. Therefore, we will need to discount (divide) this "present value" by (1+i)^3 to calculate a true present value from the perspective of today. The formula for the present value of a growing (or shrinking) annuity is: PV = [A/(i-g)]*{1 - [(1+g)/(1+i)]^n} where A = Annuity payment in the first compounding period i = interest rate g = growth (or shrink) rate n = number of compounding periods For the first equation (years 2 and 3), we know that A = $200, g = -0.25 (since there is a 25% decrease from the first to the second payment), and n = 2. We also need to remember to divide by (1+i) to get the true present value, so I put it in the denominator of the first term: PV = {200/[(i+0.25)(1+i)]}*{1 - [(1-0.25)/(1+i)]^2} For the second equation (years 4 and 5), we know that A = $100, g = -0.50 (since there is a 50% decrease from the first to the second payment), and n = 2. We also need to remember to divide by (1+i)^3 to get the true present value, so I put it in the denominator of the first term: PV = {100/[(i+0.50)(1+i)^3]}*{1 - [(1-0.50)/(1+i)]^2} Therefore the net present value (NPV) of the entire stream is as follows: NPV = -400 + {200/[(i+0.25)(1+i)]}*{1 - [(1-0.25)/(1+i)]^2} + {100/[(i+0.50)(1+i)^3]}*{1 - [(1-0.50)/(1+i)]^2} I am not sure if you are given that the NPV of this stream is equal to 0, but if you are, then you can just use algebraic manipulation to solve for i, since it would be the only unknown. I hope this helps!
Your problem is to find the rate of return or the internal rate of return at which the investment yields an NPV of zero The equation for finding IRR doesn't permit separation of i from other variables thus we resort to finding an approximate value of i with a step wise procedure called linear interpolation To do this, we guess at the rate say 10% for i and we find NPV, if the NPV we found at 10% is greater than 0 we will take a second guess at i higher than 10% to bring down NPV below zero But if at 10% the NPV is negative, we will take a second guess at i lower than 10% say 5% to drive up the NPV above zero Once we have two rates for i which yield an NPV above and below zero, we be in place to use Linear Interpolation to find approximation of i Let's us find NPV for these cash flows at 10% N ---------------- Project Y --- PVIF(10%,n) ---- PV --------------------------------------... 0 --------------- ($400)--- 1 ------------------ $000 ------ 0.90909 --- $0 2 ------------------ $200 ------ 0.82644 --- $165.29 3 ------------------ $150 ------ 0.75131 --- $112.70 4 ------------------ $100 ------ 0.68301 --- $68.30 5 ------------------ $050 ------ 0.62092 --- $31.05 --------------------------------------... ------------------PV of Net Cash Inflows---- $377.34 NPV @ 10% = $337.34 - $400 NPV @ 10% = -22.66 Since at 10%, NPV is negative, we will take a second guess at 5% to bring NPV above zero N ---------------- Project Y --- PVIF(5%,n) ---- PV --------------------------------------... 0 --------------- ($400)--- 1 ------------------ $000 ------ 0.95238 --- $0 2 ------------------ $200 ------ 0.90702 --- $181.41 3 ------------------ $150 ------ 0.86383 --- $129.58 4 ------------------ $100 ------ 0.82270 --- $82.27 5 ------------------ $050 ------ 0.78352 --- $39.18 --------------------------------------... ------------------PV of Net Cash Inflows---- $432.44 NPV @ 5% = $432.44 - $400 NPV @ 5% = 32.44 At 5% , we have a negative NPV, thus the required rate of return lies between 5% and 10% Let us use Linear Interpolation to find an approximation of i i = iL + [npvL/(npvL-npvH)](iH-iL) i = 0.05 + [ 32.44/(32.44-{-22.66}) ] (0.10 - 0.05) i = 0.05 + [ 32.44 / 55.1] (0.05) i = 0.05 + [0.58874773139745916515426497277677] (0.05) i = 0.05 + 0.029437386569872958257713248638838 i = 0.07944 or rate of return i is 7.944% approximately With Forney Excel, we can get a value for i closer to the actual Type in the values in the cells mentioned to the left of the numbers A1 -400 A2 0 A3 200 A4 150 A5 100 A6 50 In cell a7 type in the following =IRR(A1:A6) 7.822%
Future value
I would be interested in finding out more about this too
I'm not completely convinced about this one