Assignment 1
Solutions
Summer 2001
Question 1
In 1851 an ancestor of yours made a chair that he sold for
$3.50. In 2001 that piece of furniture is now worth $7,500
to a collector. What would be the holding period return on
this chair if it had been held for the entire time? What would
this be on an effective annual basis?
Holding Period Return = (FV - PV) / PV x 100%
=214,186%
Effective Annual Rate =(1+HPR)^(1/150) - 1
or =(FV/PV) ^(1/150) - 1
=5.25%
Assignment 1
Solutions
Summer 2001
Question 2
Rank the following rates from highest to lowest effective
annual rate; 19% with annual compounding, 18.75% with
semi-annual compounding, 18% APR, with monthly
compounding, and 17.5% with continuous compounding.
Step 1: convert all rates into EAR.
19% with annual compounding = 19% EAR
18.75% with semi-annual compounding
= (1 + 0.1875/2)^2 - 1
= 19.63%
18% APR, with monthly compounding
= (1 + 0.18/12)^12 - 1
= 19.56%
17.5% with continuous compounding
= FVIFA(17.5%,1) - 1
= e^0.175 - 1
= 19.12%
Step 2: rank from highest to lowest.
1. 18.75% with semi-annual compounding
2. 18% APR, with monthly compounding
3. 17.5% with continuous compounding
4. 19% with annual compounding
Assignment 1
Solutions
Summer 2001
Question 3
The grand prize of a lottery promises to pay the winner
$1,000 per week for the rest of their life. How much is this
prize worth if the appropriate effective annual discount rate
is 6% and the winner is expected to live for 30 years after
winning? What if the winner was younger and expected to
live for 45 years?
Step 1: find weekly discount rate.
rw = (1 + 0.06)^(1/52) - 1
rw = 0.001121184
Step 2: find PV for 30 x 52 week annuity
PV = $1,000 x PVIFA(0.001121184, 1560)
PV = $1,000 x (1 - 1/(1 + r)^n)/r
PV = $736,622.96
Step 3: find PV for 45 x 52 week annuity
PV = $1,000 x PVIFA(0.001121184, 2340)
PV = $1,000 x (1 - 1/(1 + r)^n)/r
PV = $827,116.64
Assignment 1
Solutions
Summer 2001
Question 4
To settle a debt of $449.98, a friend of yours has
offered to pay you $39.98 per month for a year. What
rate of interest is implied by these repayment terms on
an APR and effective annual basis?
Since this is a 12-th order polynomial if we try to solve it
algebraically, we will use trial and error (or spreadsheet)
to find the appropriate monthly discount rate.
Find the PV of the annuity payments at rm = 0.5%
PV = $39.98 x (1 - 1/(1 + r)^n)/r = $464.62
This is too high so raise the discount to 1.5%
PV = $39.98 x (1 - 1/(1 + r)^n)/r = $436.08
This is too low, raise the discount rate to 1.0%
PV = $39.98 x (1 - 1/(1 + r)^n)/r = $449.98
This is the correct amount.
If using solver or goal seek,
rm = 0.999929594265684%
Convert to APR = 1% x 12 = 12%
Convert to EAR = 1.01^12 -1 = 12.68%