By the definition of regular annuities where the

By the definition of regular annuities, where the first payment
starts one period in the future or at the end of the current year, by reverse logic, this
implies that the PVA is one period from the first payment. What this means is that the
PVA is in period 2, i.e., one period prior to the first payment. This implies we have to
discount this annuity from period 2 to period 0 as follows:
PV = 100 / (1.12) + 150 / (1.12) 2 + 325[1 – (1.12) -6 / .12] / (1.12) 2 = $1274.08
Note that 21 . 1336 $
12 .
) 12 . 1 ( 1
325
6









and we then discount this two periods. Now,
we can combine the two terms, the 150 and the annuity value of 1336.21 over one
common denominator discounting for 2 periods:
 
 
08 . 1274 $
12 1
21 . 1336 150
12 1
100
2


 
.
.
PV
This will be used later in the semester, not now, but you should try to understand this.
2) Financial Calculator: Use the cash flow keys (make sure you hit 2
nd
and C ALL and
have no BEGIN in screen)
Period CF
0
1
2
3-8
This last step means I am putting in a $325 Cash Flow and the Number (N
j
) of these cash
flows are 6. Note, that I could have just hit the CF button 6 times after typing in the 325.
This is what I normally do because I usually forget the sequencing of these things and
have to experiment. After entering the cash flows enter the interest rate: 12, i/yr
Now hit the Net Present Value (NPV) key, which is the 2
nd
and then the
NPV
PRC
button
NPV = $1274.08
Calculating Interest Rates
16

1) Lump Sum PV and FV
From PV and FV we know the basic equations:
 
 
n
1
1
r PV V or F
r
FV
PV
n
n
n  


Now we need to solve for r in one of these, so in FV we divide through by PV:
 
by PV r PV FV
n
n
   1
 
n
n r
PV
FV
  1
Raise both sides to the power of
n
1
 
r
PV
FV
n
n  






1
/ 1
Now get r:
BL r = [ FV
n
/ PV ] 1/n - 1
eg. $100 deposited for 5 years, we end up with $248.9, what is the rate?
r = [ 248.9 / 100] 1/5 – 1 = 1.20 – 1 = .20
Note: This is also the way to get a rate of growth of a PV into a FV.
In the Financial Calculator:
N= 5
PV= -100
FV=248.9
i=??=20%
Finding the Interest Rate in Annuities
 






 


r
r
A PVA
n
1 1
0
Trial and Error
17

We cannot solve the annuity formula for r because there is an r in both the numerator and
denominator, so we have to use trial and error, i.e., by just plugging in different values
and seeing if the PVA equals the PV of the cash flows in the formula.
eg. $500 per year for 5 years which has a 1676  PVA
1) Formula:
From the annuity formula:
 
 
4 . 1895 $
1 .
37907 .
500 ;
1 .
1 . 1 1
500 ? ? 1676
: 10% Try
1 1
5













 







 



r
r
A PVA
n
So 4 . 1895 $  PVA but we were trying to get to 1676. So, we conclude that r is too low,
we did not discount the 500 cash flows enough. Therefore, we need to try a larger
discount rate.
Try 20%:
 
3 . 1495 $ 99 . 2 500
2 .
2 . 1 1
500 ? ? 1676
5
  









So now we discounted the 500 cash flows too much and we conclude that r is too high.
So again we need to try something in between 10% and 20%.
Try 15%:
 
1676 352 . 3 500
15 .
15 . 1 1
500 ? ? 1676
5
  









So this 3
rd
time we luckily narrow in on the answer.
2) Fin. Calculator (Note: either the PV or the PMT must be entered as a negative)
N= 5
PV= -1676
FV= 0
PMT= 500
i=??= you will see that your calculator is “running” but then gives 15
Finding r in uneven cash flows:
Using the original uneven cash flow example above, we know the PV and the cash flows
and we want to find the interest rate.
Year CF PV = 440.19
1 100
18

2 150
3 325
1) Formula