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semiannual cash inflow is $10012 = $50.

Assume that investors have a required rate of return of 12 percent for this type of bond. Then, the present value

(V) of this bond is:

V = $50(PVIFA6.m) + $l,OOO(PVIF6+,")

= $50(11.4699) + $1,000(0.3118) = $573.50 + $311.80 = $885.30

Note that the required rate of return (12 percent) is higher than the coupon rate of interest (10 percent), and so

the bond value (or the price investors are willing to pay for this particular bond) is less than its $1,OOO face

value.

Review Questions

is a critical consideration in many financial and investment decisions.

1.

and is the reverse of the

2. The process of determining present value is often called

process.

+ i/m)"'"'is a general formula used for

3. F,,= P(l

results in the maximum possible future value at the end of rz periods for a given

4.

rate of interest.

is a series of payments (or receipts) of a fixed amount for a specified number

5.

of periods.

, or effective annual rate, is used to compare investments with different

6. The

on a common basis.

7. The present value of a mixed stream of payments (or receipts) is the of present

values of

8. A(n) is an annuity in which payments go on forever.

9. The is the annual deposit (or payment) of an amount that is necessary to

accumulate a specified future sum.

10. If a loan is to be repaid in equal periodic amounts, it is said to be a(n)

Answers: (1) Time value of money; (2) discounting, compounding; (3) intrayear compounding; (4) Continuous

compounding; ( 5 ) annuity; (6) annual percentage rate (APR), compounding periods; (7) sum, the individual

payments; (8) perpetuity; (9) sinking fund; (10) amortized loan.

165

TIME VALUE OF MONEY

CHAP. 61

Solved Problems

61

. Future Value. Compute the future values of ( a ) an initial $2,000 compounded annually for 10

years at 8 percent; (6) an initial $2,000 compounded annually for 10 years at 10 percent; (c) an

annuity of $2,000 for 10 years at 8 percent; and ( d ) an annuity of $2,000 for 10 years at 10

percent.

SOLUTION

To find the future value of an investment compounded annually, use:

(a)

F,,= P(l + i)" = P *FV1Fj.n

In this case, P = $2,000, i = 8%,n = 10, and FVIF8%,10yr2.1589. Therefore,

=

= $2,000(1 + 0.08)'O = $2,000(2.1589) = $4,317.80

Flo

F,,= P(1+ i)" P.FVIFi,,

=

(6)

Here P = $2,000, i = 10%, n = 10, and FVIFlo,lo= 2.5937. Therefore,

F1o = $2,000(1 + 0.10)'O = $2,000(2.5937) = $5,187.40

For the future value of an annuity, use:

(c)

S, = A * FVIFAj,,

In this case A = $2,000, i = 8%,n 10, and FVIFA8%,loyr= 14.486. Therefore,

=

Slo = $2,000(14.486) = $28,972

S, = A * FVIFA,,,

(4

Here, A = $2,000, i = 10%, and FVIFAlo˜lo 15.937. Therefore,

=

Slo = $2,000(15.937) = $31,874

Intrayear Compounding. Calculate how much you would have in a savings account 5 years

6.2

from now if you invest $1,000 today, given that the interest paid is 8 percent compounded:

( a ) annually; ( b ) semiannually; ( c ) quarterly; and ( d ) continuously.

SOLUTION

A general formula for intrayear compounding is:

F,,= P(l + Um)'"'' = P.FVIFi,,,t,,,,,,,

For this problem P = $1,000 and n = 5 years.

When m 1,i = 8%, and FVIF(8%/l),S.l 1.4693,

(a) = =

y)5'1

F5 = $1,000( 1+ = $1,000(1.4693) = $1,469.30

(6) When rn = 2 and FVIF(8%12),5.2

= FVIF4,10= 1.4802,

F5 = $1,000(1.4802) = $1,480.20

m =4 and FVIF(8%/4),5.4

= FVIF2,20 1.4859,

=

(c)

F5 = $1,000(1.4859) $1,485.90

=

(d) For continuous compounding, use:

F,, = p . e""

Fs = $1,000(2.71828)0˜08'5= $1,000(2.71828)0'4

= $1,000(1.4918) = $1,491.80

[CHAP. 6

TIME VALUE OF MONEY

166

Present Value. Calculate the present value, discounted at 10percent, of receiving: ( a ) $800at the

6.3

end of year 4; ( b )$200 at the end of year 3 and $300 at the end of year 5, ( c ) $500 at the end of

year 4 and $300 at the end of year 6, and ( d ) $500 a year for the next 10 years.

SOLUTION

F4 = $800, and i = 10%.

Here n = 4,

PVIFlo.4 = 0.6830

P = $800(0.6830) = $546.40

P = $200(PVIFio,3)+ $300(PVIFlouS)

= $200(0.7513) + $300(0.6209) = $150.26 + $186.27 = $336.53

P = $500(PVIFlo,4)+ $300(PVIFlo,s)= $500(0.6830) + $300(0.5645)

= $341.50 + $169.35 = $510.85

( d ) For the present value of an annuity, use:

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