ANSWER BOTH PARTS STEP BY STEP F) Suppose that an annuity will make annual payme

Question

ANSWER BOTH PARTS STEP BY STEP

F) Suppose that an annuity will make annual payments of 5400 dollars apiece (starting a year from now), followed by a smaller partial payment a year after the last full payment. If the present value of the annuity is 41000 dollars and the interest rate is 7.6 percent effective, how large is the final partial payment?

G) A perpetuity pays 1600 dollars on January 1 of 1980, 1982, 1984, ..., and pays X dollars on January 1 of 1981, 1983, 1985, ... If the present value on January 1, 1975 is 30000 dollars, and the effective rate of interest is 7 percent, what is X?

Explanation / Answer

F) NPV has to be =0;
Thus NPV = -41,000 + 5400/(1.076)1 + 5400 / (1.076)2 + ......
We are to assume this annuity flow of 5400 until it reaches a year where it crosses an NPV of 0;
Suppose that happens after 'x' years; so in the "x+1"th year, the annyity will be a smaller partial payment as another full payment will make the NPV>0 which is not possible;
Hence by using Excel you will find out that
5400/(1.076)1 + 5400 / (1.076)2 + ...... will be = 41,551.89 after 12 years of pay ins and will be = 39,309.84 after 11 years of payin;

By this we can say that a full annuity payment of 5400 each year will come in for 11 years to make the NPV = -41000 + 39,309.84 = -1690.84$
So the partial payment at the end of the 12th year will be such that its present value will be equal to this deficit of 1690.84$ ;
Let 'P' be the partial payment;

P/ (1.076)12 = 1690.84
P = 1690.84 * (1.076)12 =4072.39~= 4073$

Thus, the partial payment will be 4073$; Note that here you can see that the partial payment is lower than the initial annuity amount of 5400$ hence it is termed as partial payment;

G) Present value of perpetual 1600 dollars inflow after 5,7,9, .. years form now :
1600 / (1.07)5 + 1600 / (1.07)7 + .... = 1600 /(1.07)5 [ 1 + 1/(1.07)2 + 1/(1.07)4 + ... this is an infinite GP with a common ratio of r<1; thus we can apply sum of inifnite GP formula = a/(1-r)
= 1140.78 ( 1/ (1-1/(1.07)2 ) = 1140.78 ( 1/0.12656) = 9013.74~= 9014

So out of the 30,000 investment, 9014 has been recovered through payments on January 1 of 1980, 1982, 1984...
So remaining 30,000 - 9014 = 20986 will come from the cash flows of 'X' dollars on January 1 of 1981, 1983, 1985 and so on till perpetuity;

Equating by the NPV equation we get:
20,986 = X/ (1.07)6 + X/ (1.07)8 + ...
applying the same inifnite GP formula we get:
20,986 = X/ (1.07)6 [ 1 + 1/1.072 + 1/1.074 + ... ]
20,986 = X/ (1.07)6 [ 7.9013]
X = 20,986 * (1.07)6 / (7.9013) = 3985.96 ~= 3986$

Thus, X will be = 3986$

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