You invest $100 (W0) in a one year certificate of deposit that pays W0x(1+i0) at

Question

You invest $100 (W0) in a one year certificate of deposit that pays W0x(1+i0) at the end of the year where i0 is the current interest rate.

The current level of the interest rate is equal to 5% per year. During the first year, the interest rate rises from 5 to 10% per year (stays constant afterwards)

What is the realised return if the investor's time horizon is:
1) 1 year?
2) 5 years?
3) 10 years?

Future Value coefficient

Years 1 2 3 4 5 6 7 8 9 10
5% 1.05 1.10 1.16 1.21 1.28 1.34 1.40 1.48 1.55 1.62
10% 1.10 1.21 1.33 1.46 1.61 1.77 1.94 2.14 2.36 2.59

Explanation / Answer

Here, annual realised return = (Vt/V0)^1/t - 1
V0 face value of the bond, take the face value of the bond is $100.
Time period is 10 years.
For the first year the interest rate is 5%, after that it is 10%.
For the first year annual payment is = 0.05*100
= $5
Then after, it is 10%. Coupon payment is $10.
For finding annual realized return, here we have to calculate the Vt.
Vt is must be account for reinvestment of annual coupon $5 and $10.
These reinvestments are as follows.
year1
Year2
Year3
Year4
Year5
…
Year9
Reinvest
Year10
1st coupon
$5
*(1.05)^10
$8.144
5th year coupon
$10
*(1.1)^5
$16.1051
10th year coupon + Principal
$10
*(1.1)
$11
100
$111
$ 135.24
Therefore Vt is - $135.24
Annual realized return = (135.24/100)^(1/10) - 1
=(1.3524)^(1/10) - 1
=1.0306 - 1
=0.0306 or 3.06%
Therefore Annual realized return is 3.06%

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