- CFA Exams
- CFA Level I Exam
- Topic 1. Quantitative Methods
- Learning Module 2. Time Value of Money in Finance
- Subject 2. Fixed Income Instruments and the Time Value of Money

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**CFA Practice Question**

The present value of a 4-year ordinary annuity of $1000 per year starting in year 1 is the difference of 2 perpetuities.

Perpetuity 2: $1000 per year starting in 5 years' time

Perpetuity 1 : $1000 per year starting in year 1

Perpetuity 2: $1000 per year starting in 5 years' time

Given a 5% discount rate, the present value of this annuity is ______.

A. $3,545.95

B. $5,525.63

C. $16,454.05

**Explanation:**The first payment for Perpetuity 1 is at t = 1.

The first payment for Perpetuity 2 is at t = 5.

The formula for the present value of a perpetuity is PV = A/r, where A = annuity and r = rate. This formula is simple enough to use directly on your calculator.

For Perpetuity 1, PV at t = 0 is = 1000/0.05 = 20,000

For Perpetuity 2, we need to find the present value at t = 0. To do so, we need to first find the present value at t = 5-1 = 4, since an ordinary annuity has its first payment one period away. This also applies to a perpetual annuity. A = 1000, r at period 5 = 0.06. Thus, PV = 1000/0.05 = 20,000.

From the perspective of now (i.e., t = 0), the amount of 20,000 can be considered a future value. Thus, we now need to find the present value of this amount.

Using Texas Instruments BA II Plus:

Don't forget to clear your memories using 2nd QUIT and 2nd CLR WORK

20000 ± FV: FV = -20,000.00

4 N: N = 4.00

5 I/Y: I/Y = 5.00

CPT PV7: PV = 16,454.05

Using Hewlett Packard hp 12 C:

f CLEAR FIN: 0.00000000

f CLEAR REG: 0.00000000

4 n: 4.00000000

5 i: 5.00000000

20000 CHS FV: - 20,000.00000

g END: - 20,000.00000

PV: 16,454.04950

The present value of the 4-year annuity is x - PV(0) = 20,000 - 16,454.05 = 3,545.95.

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**User Contributed Comments**
3

User |
Comment |
---|---|

danlan |
Use CF0=0, C01=1000, F01=4, I=5 and we get NPV=3545.95 |

volkovv |
or: use PMT=-1000; N=4; I/Y=5; CPT PV=3,545.95 |

nrk65 |
4 payments of 1000 -> 4000; so PV has to less than 4000 |